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,(g) — l is conditionally positive definite, that is, 2 ——V X 0. llv? It is easy to verify that c = coe~ " , where c0 is an absolute constant. Let Fx and F2 correspond to the elements £t and £2 °f E. We compute It is easy to see that (Fl9 F2) = ^ ( ^'^) 9 and thus is the isometric mapping ofd^ into L2(E, JU) given in §7.1. It can be verified that this mapping is an isomorphism. An elementary calculation shows that for a given T(g) the representations in SS and L2(E, (i) are equivalent. It is well known that the space L2(E, /x), where M is the standard Gaussian measure, can be represented in the form 00 ( r x > n , ]I) (v is the measure on Nn that is 1 at each point). We consider the isomorphism of Hilbert spaces Ll(TXn, W) -* I j ( r ^ ) ® / 2 (iv«, H^) induced by (4). It is easy to verify that the image of the subspace H^np C Ll(TXn, W) on which C/jJ acts is Ll(Tx) 0 ^ p and that the operators U^(\jj) in H^n>p go over to operators of the form (3). 4. The decomposition of the space of the representation £/£ of Diff X into a sum of subspaces that are primary with respect to the subgroup Diff Xk. Let (5)
&1./ > 0
i, j
under the condition that 2 5» = 0. It then follows that exp (\i (p? ^ ) is positive definite for /u > 0. Since the positive definite functions form a weakly closed set, the limit exp (a *\«
1 n ) = limexp (u ^ ^ p " )
is positive definite. DEFINITION. The unitary representation of G defined by the positive definite function \px(g), X > 0, is called canonical. A cyclic vector £x in the space of a canonical representation for which (T(g)Z\, £^) = \px(g) is called canonical Let us construct a canonical representation. We consider the space Y — K\G, which is a Lobachevskii plane. Let y0 be the point of Y that corresponds to the coset of the identity element. We define the kernel 2 ), where X > 0, on the Lobachevskii plane by the formula where yx = yogi, y2 = ^0^2- By Theorem 1.1 this kernel is positive definite. We consider the space of all finite continuous functions on Y. We denote by Lx the completion of this space in the norm:
II/IP =
20
A. M. Vershik, I. M. Gel'fand, andM. I. Graev
where dy is an invariant measure on Y. A canonical representation of G is defined by operators in Lx of the form ( 7 W ) ( y ) = fiyg). (That the operators T(g) are unitary and form a representation of G is obvious.) THEOREM 1.2. / / X > 1, then a canonical representation in Lx splits into a direct integral over the representations of the principal continuous series of G. If 0 < X < 1, then where HK is the space of the representation 7\ of the supplementary series, and Lx splits into representations of the principal continuous series only. PROOF. It suffices to verify that \jjx(g) can be expanded in zonal spherical functions of the corresponding irreducible representations. We may limit our attention to the matrices g =(c°s,
rsm t
' ) . For these
Vsinhfcoshr/
matrices we have i|?Hg)= COsh 2*4-1 Y*' F u r t n e r m o r e > w e know that the zonal spherical functions of the representations of the principal continuous series have the following form: cpi-npteHP-i-ip (cosh 20, 2
where
1-ip
is the Legendre function.
Let X > 1; then (-^Tf) is square integrable on [1, °°) and can therefore be expanded in an integral of functions P-i-iP (x) (the Fock-Mehler 2
transform). Thus, we have o
The coefficients ax(p) in this expression can be calculated by the inversion formula for the Fock-Mehler transform; we obtain 1
"K (P) = - s j r
(
) ( f27X]
!
) Ptanh
Now let 0 < X < 1. It is known that the zonal spherical function = ^-x/2( c °sh 2t) of the representation Tx of the supplementary series has the following asymptotic form:
Representations of the group SL(2,R)
21
It follows that the function 2 W2
"^l1"!")
p_w{x)
is square integrable on [1, ©°) and can therefore be expanded in an integral of functions P - i - i P (x). 2
In what follows we say that the canonical representation Lx for 0 < X < 1 is congruent to the representation Hx of the supplementary series modulo representations of the principal series. The explicit separation of the component of the supplementary series in Lx will be carried out a little later (see Theorem 1.3). In conclusion we give another two expressions for the kernel ^N>i, y2) = tHgigi1), where yx = yogu y2 = yog2. From the definition of \p(g) it follows easily that ^CVij yi) ~ cosh~xp(yly
y2),
where p(yi, y2) is the invariant metric on the Lobachevskii plane. We suppose further that the Lobachevskii plane Y is realized as the interior of the unit disk \z\ < 1 in the complex plane, where G acts by fractional linear transformations: z-+az realization ^
x
_ . It is easy to verify that in this
has the following form: (1-1*1 I2) (1-1*2 I2) "W2
We observe that the invariant measure on the unit disk is ( 1 - \z\2)~2dz dz. Thus, in the realization on the unit disk the norm in the space of the canonical representation has the following form: (5)
i-i*i i2) ( i - i _ |i_Zlz2|2
2b. A SECOND METHOD OF SPECIFYING A CANONICAL REPRESENTATION. Suppose that the Lobachevskii plane is realized as the interior of the unit disk \z\ < 1. Then the norm in the space L\ of a canonical representation is given by (5). If we now go over from the functions f(z) to the functions (1 — |z|2)"*"x~2/(z), then we obtain a new and very convenient realization of a canonical representation. In this realization the space Lx of a canonical representation is the completion of the space of finite (that is, vanishing close to the boundary) continuous functions in the unit disk | z | < 1 with respect to the norm
22
A. M. Vershik, I. M. Gel'fand, and M. I. Graev
(6)
||/1| 2 =
\i-ziz2\~Kf{zi)f(z2)dzidzidz2dz
j
The representation operators act according to the formula
We note that for finite continuous functions /(z) in the disk | z | < 1 the norm ||/|| 2 can be written in the following convenient form: oo V u/
j/J l m, n=0
11/11
where Cmn 00 = — In particular, if/ depends only on the modulus r, then the norm ||/|| 2 takes the following simple form:
To derive (8) it is sufficient to represent the kernel | 1 — zxz2 \~x as the product (1 - z 1 z 2 )" x/2 (l - z 1 z 2 )" x/2 , and then to expand each factor in a binomial series. This space is very interesting. We shall see now that it contains a large store of generalized functions. We consider the space K of test functions 0(z) that are continuous and infinitely differentiable in the closed disk |z| < 1. Let /(0) be a generalized function, that is, a linear functional on K. From / we construct a new functional in the space Ko of functions /(z) that are infinitely differentiable and finite in the disk |z| < 1 (that is, vanish near the boundary): (l,f)=
2 m,n—0
cmn.(K)l(zmzn)
j
f(z)zm~zndzdz.
|z'l
If this series converges absolutely and | (/, / ) | < c \\f\\, then the functional (/, f) can be extended to a continuous linear functional in Lx and therefore specifies an element / in Lx. We claim that the delta function £x = 5(z) concentrated at the point z = 0 belongs to LK. In fact, since £ x (l) = 1 and £K(zmzn) = 0 for m 4- n > 0, we have (9)
(6x, /) = (6, f) = j f(z)dz dz.
Representations of the group SL(2,R)
23
Consequently, by the definition of the norm in Lx, I £\, / ) I ^ 11/11 and hence 5(z) G Lx. It can be shown that £x is a canonical vector in Lx, that is, This is easily derived from (9) if we replace / by T(g)fn, where fn is a sequence of test functions converging to £ x , and then proceed to the limit. 1 It is not difficult to verify that Lx contains not only 5(z), but also all its derivatives, 6(m> n)(z)= g(2n+i> ^ __
°m " i
dzm dzn
. In particular, all derivatives
lie in the subspace of Lx consisting of functions2* that
dr2nX
depend only on the modulus r. We now look at the generalized functions / = a(z)5(l - | z | 2 ) , where a(z) is a continuous function on the circle | z | = 1 (that is, 1
We can obtain this result formally by substituting in (9)
' The functions 6 ^ \z) form an orthogonal basis in Lx. Thus, each element / of this space can be written in the form 7__
v ^
dm+nS(z) dzm dzn
u
m, n—0
with
!IZH2= S
m, n=0
(m!rl!)2%nW|Vn|2.
In particular, in the subspace of functions depending only on the modulus r there is an orthogonal basis consisting of the functionsfi<2n+i>( r ) _ -_.
written in the form / = >j bn
d
2n+1
6 (r)
2n+\
' , with
2n+i
• Thus, any element of this subspace can be
24
A. M. Vershik, I. M. Gel'fand, andM. I. Graev
LEMMA 1.3. The functions
I = a ( z ) 5 ( l - \z\2) belong to the space Lx
for 0 < X < 1. PROOF. Since 2JI
) = 1 j a (eif ) «*<*-»>' dt =
am_n,
we have
\(l,f)\< m,n=:0
Hence, by the Cauchy inequality |(J,/)|<( 2
m, n=0
CmnM | CCm-n|2)1/2 || /||.
Thus, it remains to prove the convergence of the series (10)
2
C m n (X)|a m . n |
m, n=0
2
= S
|a,|2( S
ft=-oo
cmn(X)).
m-n=fe
To do so we use the following estimate for the coefficients
If follows from this estimate that for 0 < X < 1 the series 2
c
nn (X)
n n-0
converges. On the other hand, it is not hard to see that < 2 Cnn(h) and, therefore, n=0
2
2
m—n=k
c mn (X)<6' 1, where Q does not depend
m—n=h oo
on /:. Since 2 I a^ |2 converges, the convergence of the series (10) follows.^ fc0
^ We note that the function 6(z) can also be obtained by a limit passage. Let us consider the sequence of functions fn G L\ of the form cn
for
0
for
— <|z|
where c n > 0 is defined by the condition II /„ II = 1. It is easy to verify that this is a fundamental sequence in L\ and that its limit is equal to 6(z). Similarly, we can obtain 6(1 - |z|2) as the limit of a fundamental sequence of functions of the form cn, for
where cn is determined from the relation \ fn (z) dz dz = n.
Representations of the group SL(2,R)
25
We state without proof two simple propositions on the functions , = 5(z) and / = / ( z ) 5 ( l - | z | 2 ) . LEMMA 1.4. Let lx = fx(z) 5 (1 - | z | 2 ) , l2 = / 2 ( z ) 5 ( l - | z | 2 ) , then (11)
l)(Ji, « x = 2 - -
2n2n j j | sin — 2 — 0
(12)
0
2) (6(z), 6 ( l - | z | 2 ) ) 3
From (11), it follows, in particular, that
LEMMA
e function
1 . 5 . 77ze representation
operator
T(g),
where
g = (?_), 'P a '
/ ( z ) 5 ( l - Iz I 2 ) i/ifo / ( ^ £ ± J ) | p 2 + a |*-26(1 -
fate
\z | 2 ).
The next result follows immediately from Lemmas 1.4 and 1.5. THEOREM 1.3. For 0 < X < 1 the subspace Hx C Lx generated by the functions /(z)5(l - |z| 2 )/s invariant and the representation of G acting in it is the representation Tx of the supplementary series. In what follows we find it useful to know the projection of the canonical vector £x onto H\. This projection is obviously invariant under the maximal compact subgroup and is consequently proportional to 5(1 — |z| 2 ). Let where c(X) = ||5(1 — Izl 2 )!!" 1 . Then, according to Lemma 1.4, 2) we have (£\> V\) = KC(\); consequently the projection of the canonical vector %x onto the subspace Hx is equal to 7rc(X)rjx, where c2(X) = || 5 (1 - |z| 2 )||" 2 = 2X7T"3/2 r( 1 - X / 2 ) / r ( | ( l - X ) ) . The next result follows easily from standard estimates for F(X). LEMMA 1.6. ( J x , T?X) = 1 + 0(X 2 ) as X^
0.
COROLLARY. As X -• 0, the distance from the canonical vector £x in Lx to Hx is 0(X2). 3. Theorems on tensor products of representations of G. THEOREM
1.4.
Let Tx and Tx be two representations of the supplementary series. If Xt + X2 < 1, then Tu ® Tk2 = TXi+X2 0 T, where T splits into representations of the principal continuous and the discrete series only. If Xt + X2 ^ 1, then TXi (g) 7\ 2 splits into representations of the principal continuous and the discrete series only. The tensor product of two irreducible unitary representations of which at least one belongs to the principal continuous or the discrete series splits into representations of the principal continuous and the discrete series only. For the proof see [12].
26
A. M. Vershik, I. M. Gel'fand, andM. I. Graev
We now let Gn = G x . . . x G. Since G is of type 1, it is standard n
knowledge that any irreducible unitary representation T of Gn can be obtained as follows. We are given irreducible unitary representations r ( 1 ) , . . ., r (w) of G, acting in Hilbert spaces Hu . . ., Hn, respectively. A representation T of Gn acts in the tensor product Hi ® . . . ® Hn according to the following formula: (14)
T(gu . . ., gn){lt ® . . . ® En) = (Tto(gl)ll)<8> ' ' ' ®(T
Here two representations T' and T" of Gn are equivalent if and only if all the corresponding representations T'^ and T"^ of G are equivalent, i = l , . . . , n. We say that a representation T of Gn of the form (14) is purely of the supplementary series if all the T^ are representations of G of the supplementary series. In this case, if T^ = TXi, i = 1, . . . , « , then the corresponding representations of the group Gw are denoted by 7\w>. . . , \ n , and the representation space by // X i > .. . , \ n . The next theorem follows from the one stated above. THEOREM 1.5. Let T^,...,^ and Tx"v...xn be two representations ofGn purely of the supplementary series: and let \\ + X{' < 1, / = 1, . . ., n. Then T\'v...xn ® ^ j , • • • Xn ~ ^M+^i'• • • '^n+rn ® ^ ' where in the decomposition of T into irreducible representations there are no representations purely of the supplementary series. In what follows we find it useful to specify explicitly an embedding of H\l + \2{\ + X2 < 1) in the tensor product//"^ 0 Hkr Let G be defined in the first form. Then H\i + \a is the completion of the space of finite continuous real functions fix) with the norm -f-oo -J-oo
j j — oo — oo
Now //"M ® H%2 is the completion of the space of finite continuous functions Fixi, x2) of two variables with the norm x2-x'2
\-** F(xu X2)F(x[, x'2) dxi dx2 dx\ dx2.
In Hu (g) HX2 there are many generalized functions. The precise meaning of this statement is the following. Let liF) be a linear functional on the space of infinitely differentiable
functions P(xu x2) such that \F(xu x2)\ < C(\ + x\Y^12
0 + xlr*J2,
By means of / we construct the following linear functional T on the space of finite infinitely differentiable functions F(xi, x2): (/, F) = /(F), where
+
OO
=J
+CX,
— oo — o o
\Xi-x[\-^\x2-x'2\-^F{x[,
x'2)dx[dx2.
Representations of the group SL(2, R)
27
If the functional (7, F) is defined and continuous in the norm ofHki
then the functional (7, F) is continuous in the norm of Hu ® HX2. This lemma is proved by standard calculations involving the Fourier transform, and we omit the proof. Next we can establish the following lemma. LEMMA 1.8. Let 71 and T2 be defined by the generalized functions h = /i(*i)5(*i ~ x2) and l2 = f2(x1)5(x1 - x2). Then + OO+OO
(h, h)= j J \Xl-X2\~Xi~k2
fl(xl)f2(x2)dxldx2.
— (XI —OO
THEOREM 1.6. / / Xj > 0, X2 > 0, Xx + X2 < 1, then the mapping defines an isometric embedding of HXi+x2 in Hkl
defines an isometric embedding Hu+...+xk-+- HM 0 . . .
i
then the mapping i, 3
is the composition of the mappings and H%. -v (g> H^... §2. Construction of the multiplicative integral of representations of G = PSL(2, R).
Let G = PSL(2, R) and let X be a compact topological space with a given measure m. We define a group operation on the set of functions
28
A. M. Vershik, I. M. Gel'fand, andM. I. Graev
g: X -• G as pointwise multiplication: (g\g2)(x) = gi(x)g2(x). We define Gx as the group of all continuous functions g: X -• G with the topology of uniform convergence. We give here a construction of an irreducible unitary representation of the group Gx, which we call the multiplicative integral of representations of G. Following the definition of the integral as closely as possible, we replace Gx by the group of step functions and define an integral on it as a tensor product of representations. As we decrease the length of the intervals of subdivision and simultaneously allow the parameter on which the representations in the tensor product depend to approach a certain limit, we obtain a representation of Gx', which we call the integral of representations. It is remarkable that the integral of representations is an irreducible representation. We now proceed to precise definitions. 1. Definition of the group G°. For every Borel subset Xf C X we denote by Gx' the group of functions g: X -» G that are constant on X' and equal to 1 on the complement of X'. It is obvious that there exists a natural isomorphism GX' = G. A partition v: X = U Xj of X into finitely many disjoint Borel subsets is called admissible. On the set of admissible partitions we define an ordering, setting vx < v2 if v2 is a refinement of vx. It is obvious that the set of admissible partitions is directed (that is, for any p1 and v2 there exists a v such that Vi C v and v2 < v). h For any admissible partition v: X = [} Xt WQ denote by Gv the group of functions g: X -> G that are constant on each of the subsets Xt. It is obvious that Gv = GXi
X . . . X GXj}.
Observe that for vx < v2 there is a natural embedding: GVi -+ Gv^. We define the group of step functions G° as the inductive limit of the Gv\ G° = l i m Gv.
In this section we construct a representation of G°. We make the transition from this representation to a representation of Gx in §3. 2. Construction of a representation of G°. Let m be a positive finite measure on X, defined on all Borel subsets of X. We always assume that m is countably additive. Let us consider the Hilbert spaces in which the representations 7\ of the supplementary series act. In § 1 we denoted these spaces with the action of G defined on them by HK, 0 < X < 1. Next, we denote by i/ 0 the one-dimensional space in which the identity representation of G acts. Let v: X = \j Xt be an arbitrary admissible partition such that
Representations of the group SL(2, R)
29
X,- = m{Xt) < 1. We set In SBv we define a representation of the group Gv = GXi x . . . X GXfe, supposing that GXi = G (i = \, . . ., k) acts in HXi. The representation of Gv in G^V is irreducible (see §1.3); we have agreed to call such representations purely of the supplementary series. Note that since GVi C GVt for vx < v2, a representation of each of the groups Gv', v < J>, is also defined in SBv • LEMMA 2 A. If v i < v2, then S£V2 splits into the direct sum of subspaces invariant under GVi: SBX2 = SBVi 0 SB' where SB' does not contain invariant subspaces in which a representation of GVi purely of the supplementary series acts. k
PROOF. Let v2 > vu that is, vx\ X = U Xh v2\ X = U Xih
where
Xi = U Xtj. We set \ t = m(Xt), \tj = m (Xtj)\ thus, SBVl = ® HK., SBV = 0 H^... We also set SB\ = 0 ^ . . ; then SBV = 0 d^i . It is evident i, j
lJ
i
2
i
that Gxt acts diagonally in c^v2 = 0 ^ ^ i ; (that is, acts simultaneously on each factor HXij). Thus, the representation of Gxt — G in <^J is a tensor product of representations 7\ l7 of the supplementary series. h = l!i *>u, w h e r e H\i i s From this it follows that SB\ = H%i 0 H\, the space in which the representation TXi of the supplementary series acts, and H*x splits only into representations of the principal, the continuous, and the supplementary series (see §1.3). Forming the tensor product of the spaces SB\2 and bearing in mind 0 Hj,. = SBVl^nd GXi x . . . X ^xh = GVi, we obtain: SBV2 = SBVi® SB', where %SB' does not contain representations purely of the supplementary series. THEOREM 2.1. There exist morphisms of Hilbert spaces defined for each pair vx < v2 of admissible partitions of X satisfying the following conditions: 1) h2vv commutes with the action of GVi in S£Vi and SBV2; 2) h]v2 ° U%vx = h,Vl for any vx
30
A. M. Vershik, I. M. Gel'fand, andM. I. Graev
Let v2 > vu that i s ^ : X = (J Xt, v2: X = U. xih
where Xt = [) Xu.
set \f = m{Xi), \if = m(Xz/); thus, 38Vi = O # v <^va = 0 i=l
Hk..
i, j
For each / = 1, 2, . . ., k we define a mapping i/\ { ->- (g> H%..r i,i ii
^i = 2 Xiy, compatible with the action of Gxt — G, just as this was done 3
in §1.3. These mappings induce the mapping 7*V2Vi
:
$£vi
~*~ 3&V2 ,
which is compatible with the action of Gv in these spaces. From the definition of the mappings H^—*-
lj
that the mappings / ^ so defined satisfy the compatibility requirement 2) of the Theorem. REMARK. Another method of specifying the compatibility of the system of morphisms / ^ will be given in §3. DEFINITION. We assign to all possible pairs p1 < v2 of admissible partitions of X the morphisms /^ ^ : 3£Vi ->- SSV2, which commute with the action of GVi in S£Vi and 3£V2 and satisfy the compatibility condition KV2 ° iv2vx = U^v, f° r v\
Representations of the group SL(2,R)
31
jv%Vx into jliVi and consequently induce an isomorphism of the spaces St° and SB'0 compatible with the action of G°. THEOREM 2.2. The representation U of G° in SB is irreducible. u PROOF. Let v\ X = U Xt be an admissible partition such that X,- = m(Xi) < 1. We restrict the representation of G° in SB to Gv. k
An irreducible representation of Gv acts i n c ^ v = <8> H%. . By Lemma 2.1 i=l
l
SBV occurs with multiplicity 1 in each SBV> for v > v. Since d^v is irreducible, it follows that it occurs with multiplicity1 in the whole space SB. We now suppose that SB splits into the direct sum of invariant subspaces, $£ = SB' © Si". Then Stv is contained in one of the summands, for example, inS6'. Now let v > v. Since S£v> => S£v and an irreducible representation of Gv> acts ino^V, we have SBV> cz SB'. Consequently, SB' contains all the subspaces SBV', y' >.v, and therefore coincides This completes the proof. THEOREM 2.3. Let mx and m2 be two positive measures on X, and U^ representations of G° defined on these measures. If mx =fc m2, then the representations U^ and £/(2) are inequivalent. PROOF. We denote by SB0* and St(2) the representation spaces of ifi* and U^2\ Since mt ¥= m2, there exists an admissible partition v: X = U Xt such that X^1} = m^Xf) < 1, X[-2) = m2{X{) < 1 for / = 1, . . ., k and miiXi) ^ m2(X{) at least for one /. We claim that the representations of Gv C G° in SBO) and ^ ( 2 ) are inequivalent. It then follows that the representations of the whole group G° in these spaces a fortiori are inequivalent. Let us first consider the spaces SB^ = I Hiso andSB™ = ®H%<», in which the irreducible representations of Gv act. Since X} =£ Xf for at least one /, the representations of Gv in SB™ and SB™ are inequivalent. Furthermore, (^'v2'} = SB™ 0 c^i, for every v > ^, and Mv does not contain representations purely of the supplementary series of Gv, therefore does not contain representations equivalent to SB™ (see Lemma 2.1). Consequently the whole space SB™ does not contain representations equivalent to d^^. But then c^"(2> also does not contain representations of Gv equivalent to SB™. Since obviously SB™ cz SBa\ the representations of Gv in SBO) and S£{2) are inequivalent.
32
A. M. Vershik, I. M. Gel'fund, and M. I. Graev
§3. Another construction of the multiplicative integral of representations of G = PSL(2, R).
The concept of the multiplicative integral of representations of G = PSL(2, R) introduced in §2 can also be obtained starting out from the canonical representations of G. Here we explain this second method. It is surprising that although the representations in the product are significantly more "massive", their product turns out to be the same as before. 1. Construction of the representation. As before, let X be a compact topological space on which a positive finite measure m is given, defined on all Borel subsets and countably additive. We consider the canonical representations of G in the Hilbert spaces Lx introduced in §1.2. Next we denote by Lo the one-dimensional space in which the identity representation To of G acts. We recall that in each space Lx we have fixed a cyclic vector £x which we have called canonical. For this vector where \jj(g) is the function defined in §1.2. h
With each admissible partition v\ X = U Xh we associate a Hilbert space where A, = m(Xi). We define in Xv a unitary representation of Gv = GXi
For each admissible partition v\ X = U I j we specify a vector ?v 6 Xv\ ?v = hi ® - • • ® &tft» where £Xl- is the canonical vector in LXi. It is obvious that £„ is a cyclic vector in Xv, LEMMA 3.1. For any pair of partitions vx < v2 the mapping £vi I - > £v2 can be extended to a morphism /v2vi • %vi -+
XV2,
which commutes with the action of GVi. PROOF. It is sufficient to verify that for any gVi G GVi, where the parentheses denote the scalar product in the
Representations of the group SL(2, R)
33
corresponding space. h
According to hypothesis we have v^ X= \J Xt,
v2: X= U Xtj, where
i=l
i, j
Xt= U Xu. Let gVl G G^, that is, gVi = gx . . . gn, where gt E G x . ~ G. Then (i)
j
ft
fe
(T (gVi) HVI, ^ ^ = _ n (T (gt) hr ih)Lt=.n tf> (gt),
where X,- = m(Zz). Similarly, let gVj E G^, that is, gV2 = ]f ^ . , where i,3
gij E G^. s G. Then (2)
( r (gv2) ^v2, ?v 2 )^ V2 = JI. ^^" (gu),
where X// = m(Xij). If now g,,2 = gVi, this means that g^ = gj for every i = \, . . ., k. In addition, since Xt = 2 ^o"' f° r a n y z = 1, • • •, ^ we have j
Ijip^j (g/<7-) = ip^K^)- Consequently the expressions (1) and (2) are the 3
same and the lemma is proved. It is obvious that the morphisms jVj Vi satisfy the compatibility condition jv^2 ojv^ = j ^ V i for vx
34
A. M. Vershik, I. M. Gel'fand, andM. I. Graev
where g = # ( • ) E
G°.
LEMMA 3 . 2 .
where \p(g) is the function on G introduced in §1.2. PROOF. For any g = g(») G G° we can find an admissible partition ft
v. X= U Xt of X such that g E Gv, that is, g = gx . . . g^, where i=l
Si
G
GXi = G. But then, setting X/ = m(Xt), we have (Z1 (S) iv, ?v)^v
n *** (^)=exp ( 2 i = exp ( f In (if (g (x)) dm (x) 3. Equivalence of the two constructions of the representations of G°. We claim that the unitary representation of G° in X constructed here is equivalent to the representation in Si constructed in §2. LEMMA 3.3. There is an embedding morphism Si -> X that is compatible with the action of G°. PROOF. In §1.2 we have shown that for 0 < X < 1 the canonical representation Lx of G is congruent to the representation Hx of the supplementary series modulo representations of the principal continuous series; in other words, Lk == Hk ® Lk, where L'x can be expanded in an integral over representations of the principal continuous series. k
Hence it follows that the representation space Xv =
^
l
Gv = GXi X . . . X GXk, where v: X = [} Xh X,- = m{Xt) < 1, splits into a direct sum of invariant subspaces: k
where Siv = (8) HK., and X'v for any v < v does not contain represeni=i
l
tations purely of the supplementary series of Gv\ This also shows that under the morphism ]v
Representations of the group SL(2, R)
35
In §2 we have given another method of specifying such a compatible system of morphisms. However, the method given there has the advantage that it does not make use of the concept of a vacuum vector and does not depend on the choice of a maximal compact subgroup. THEOREM 3.1. The morphism SS -> X defined above is an isomorphism. Thus, the representations of G° in S£ and X are equivalent. PROOF. Since the vacuum vector l0 6 X is cyclic in X, it is sufficient for us to verify that £ belongs tod^7. To do this we find the projection of £0 onto each 3£v. k
Let v. X ~ (J 1^ be an arbitrary admissible partition such that i=i
k
X; = m(Xi) < 1. Then we represent £0 as an element of Xv =
*=*
h
the form %0 = (g) | ^ , where £Xl. is the canonical vector in L\v According to § 1.2b the projection of i*x ^ L\ onto Hx C LK is equal to cxrix, where r?x is the fixed unit vector of HK explicitly constructed there; here 1 - cx = O(\2) as X -• 0. k
k
Hence it follows that the projection of i 0 = 0 t%. onto S£v= (g) H^. »=i
l
i=i
l
is equal to k
Tiv = ( U %) T]v, ft
where r^ = 0 %. is a vector of unit norm. From the estimate c x = 1 + O(K2) it follows that for an indefinite refinement of v, as max X,- tends to zero, the norm of r\v tends to 1, hence that y\v itself tends to £ 0 , and the theorem is proved. THEOREM 3.2. The vacuum vector in X is uniquely determined to within a factor. PROOF. Let £o and £o be two vacuum vectors in X, and let r\v and 77" ft
k
be their projections onto S8V = ® -^"x^ where v: X = U X/, Xz- = mCX,) < 1. i=i
1= 1
Since Jo and go a r e invariant under Kv = Kxx X . . . X ^Xfe c ^^5 their projections 17[, and 97[I onto ^Kv have the same property. But in 3£v there is, to within a factor, only one vector that is invariant under Kv\ consequently, r)'v and 17[I are proportional. On the other hand, since $£° = lira 3&\ is everywhere dense in X (by Theorem 3.1), the vectors r\v and 77^ converge, respectively, to £0 a n d Jo when v is refined indefinitely provided that max Xz- -* 0. Consequently, since r\v and r\l are proportional, so are the limit vectors Jo and £{J. This completes the proof. REMARK. We emphasize that the vacuum vector £0 is contained in each subspace Xv whereas it is not contained in any of the subspaces $£?.
36
A. M. Vershik, I. M. GeVfand and M. I. Graev
4. A representation of Gx. So far we have constructed a representation Ug of the group G° of step functions X -* G. We now show how a representation of the group Gx of continuous functions on X ->• G can be defined in terms of this representation. Namely, we claim that the representation U-g of G° (second construction) can be extended to a representation of a complete metric group containing both G° and Gx as everywhere dense subgroups. This then defines an irreducible unitary representation of Gx. For simplicity we assume further that the support of m is the whole space X. We first construct a certain metric on G°. Let p(yi, y2) be the invariant metric on the Lobachevskii plane Y = K\G. We define on G a metric d{g\, £2) invariant under right translations and such that d(gu £2) > Phfogu yog2)
for any gt, g2 €E G, where j>0 is the point on the Lobachevskii plane Y that corresponds to the unit coset. (Such a metric exists; for example, we may set d(gl9 g2) = p(y081, yogi) + piy\g\, yigi)> w n e r e ^1 ^ JV) We now introduce a metric 5 on the group G° of step functions, setting
Completing G° in this metric we obtain a complete metric group Gx, consisting of all m-measurable functions #(•) for which f d(g(x), e) dm (x)< 00. Observe that the completion of G° in the metric 5 contains, in particular, the group Gx of continuous functions; Gx is everywhere dense in this completion. We claim that the representation 1/% of G° constructed above can be extended to a representation of Gx. For this purpose we consider the functional ^ on G° introduced earlier:
where 5o 6 <^ is a vacuum vector. It was shown above that (3)
T(i) = e x p ( jln^(x))cfoii(«)) •
LEMMA 3.4. The functional ^(g) can be extended from G° to a continuous functional on the whole group Gx. PROOF. It is sufficient to verify that ty(g) in (3) is defined and continuous on the whole group Gx. We use the following expression for i//(g) introduced in §1.2a: \jj(g) = cosh'1 p(yog, g).
Representations of the group SL(2, R)
37
From this expression it follows that | In \jj(g) | < piyog, ^o) ^ dig, e)> therefore the integral In \p(g(x))dm(x) converges absolutely. The continuity of ^(g) follows immediately from the following estimate: | j lny(gl(x))dm(x) - J lnq(g2(x))dm(x) | < 6(^(0, &(•))• We shall prove this inequality. We set 77 Qt) = piyogjix), y0) and use the bound I Incosh ln ilj(gl(x))dm(x)
•Jl.In
=
We have
T2
- -JJ 'i n
In
dm(x)
§ digx(x), g2(x))dm(x) = fifeO), &(•)). THEOREM 3.3. TTie representation^^ °f G° can be extended by continuity to a unitary representation of Gx. This follows immediately from the preceding lemma and the following proposition. LEMMA 3.5. Let G be a topological group satisfying the first axiom of countability, G° C G a subgroup everywhere dense in G, and T a continuous unitary representation of G° in a Hilbert space H with a cyclic vector J. Further, let ®(g) = (T(g)£, £), g G G°. If $(•) can be extended to a continuous function on G, then the representation T of G° can be extended by continuity to a unitary representation of G. PROOF OF THE LEMMA. We first show that for any r\u r\2 £ H the function ^ ^ ( g ) = (T(g)rii, r)2) can be extended by continuity from G° to G. Let HQ be the space consisting of finite linear combinations of vectors T(g)%, g £ G°. Since £ is a cyclic vector, Ho is everywhere dense in H. It is evident that if r?1? 7?2 G Ho, then $ ^ ^ ( 0 can be extended by continuity from G° to G. For if r?j = 2a,T(g{)S, T?2 = EbjT(gj% then
We now let r}1, r\2 be arbitrary; without loss of generality we may suppose that H77JI = ||r?2|| = 1. For any r?i, T?2 £ H, \\rj\\\ = ||i?2|| = 1, we have
I < i w (g) - ^ t , ; (g) l < I N i - TI;H +11 ri2 - -niiiHence the family ^^^(g) is equicontinuous in 971? T?2, and since it can be extended to all g E G for an everywhere dense set of vectors 1715 r?2, this proves that ^ ^ ( O c a n be extended to G for all 7?l5 r?2. We claim that the operators T(g), g E G, so obtained are unitary, that is,
38
A. M. Vershik, I. M. Gel'fand, andM. I. Graev
(T(g)£, T(g)£) = (£, £) for any { G E Let {gn} be a sequence of elements of G° that converges to g. For any e >> 0 there exists an TV such that for ra, n > N
(4)
I W s J E , r(fir n )|) — (g, 0 | < e
(since (7Xsm)f, rfew)£) = CTfe,^ )£, £), hence converges to (£, £)). On the other hand, we can find m and w, greater than N, such that (5) | (T(gm)h T(gn)l) - (T(g)l, T(gn)l) | + T(gn)l) - (T(g)t, T(g)l) | < e. Comparing (4) and (5), we see that \(T(g)l,
T(g)t)-(Z,
I) | < 2 e ,
which proves that Tig) is unitary. From the fact that the Tig) are unitary and weakly convergent it follows that \\T(gn)$ - Tig)i\\ -> 0, as gn -+ g, gn G G°. From this it follows automatically that Tigt)Tig2)£ = Tigxg2)^ for all gl9 g2 G G. Thus, we have constructed an irreducible unitary representation of the complete metric group Gx in X. Restricting it to the everywhere dense subgroup Gx of Gx, we obtain the required irreducible unitary representation of Gx mX. Since Gx is dense in Gx, the vacuum vector £0 is also cyclic relative to x G , and for every gG Gx we have (6)
(U~t0, go) = exp ( j In ^{g{x))dm{x)) .
We see that the metric 5 does not figure in (6). Hence our representation of Gx does not depend on the choice of the metric 5 in the construction. 5. The representation U^ commutes with the transformations of X that preserve the measure m. We consider continuous transformations a: x H-* XQ of X. They induce automorphisms of Gx: I = S(') »-* ? a = £ a ('), x
where g%r)
If t/^ is the representation of G constructed in this section and o is an arbitrary continuous transformation of X, then we can define a new representation U~ of Gx by setting U~ = U%a. THEOREM 3.4. The representation U~ is equivalent to the representation of Gx defined in terms of the measure ma on X, where ma(Xr) = m(X'a) for any measurable subset X' C X. In particular, if o preserves m, then U~ and Ug are equivalent. PROOF. Let U~ be the representation of Gx defined in terms of the measure ma on X. We compare the spherical functions (U^o, £0) and ^o)» where £0 is the vacuum vector. On the one hand,
Representations of the group SL(2, R)
39
(U^to, £0) = exp ( Jin cosh"1}fo(x))dm°(x))
.
On the other hand, (U-to, to) = (UTU, g0) = exp ( j In cosh"1 i / / ^ " 1 ) ) ^ * ) ) = = exp ( f In cosh"1 \p(g(x))dma(x) Thus, «7~£o, £o) = ( ^ ? o , So) for any g E G x , hence Uj and Of are il equivalent. 6. Invariant definition of a canonical representation. In § 1 a canonical representation of G was defined constructively. We wish to demonstrate that its connection with the representation of Gx in X we have constructed is not accidental. There is a natural embedding of G in Gx. When we restrict our representation of Gx in X to G, we obtain a certain representation of G. We look at the vacuum vector Jo in X, that is, the vector £0, ||£0|| = 1, that is invariant under Kx. We consider the minimal G-invariant subspace that contains £0 and denote it by Lx. It follows from the construction performed in §3.1 that Lx is a canonical representation of Gx with X = m(X). We draw attention to the following interesting fact. If in X we consider the restriction of the representation of Gx to G, naturally embedded in G x , then for X = m(X) < 1 there is precisely one representation of the supplementary series Hx, with X = m(X), that occurs as a discrete component in the decomposition. The orthogonal complement to this space splits into representations only of the principal continuous and the discrete series. For m(X) > 1 this representation of the supplementary series is absent. This follows from the construction of the representation of Gx carried out in §2. §4. A representation of Gx associated with the Lobachevskii plane. Here we give an explicit form of the multiplicative integral of representations of G = PSL(2, R). 1. Construction of a representation of Gx. Let I be a compact topological space, m a positive finite measure on X defined on all Borel subsets and countably additive. For simplicity we assume that the support of m is the whole space X. Let 7 be a Lobachevskii plane on which the group of motions G acts transitively. We consider the set Yx of all continuous mappings y = y(*): X -+ Y. We introduce the linear space $£°, whose elements are formal finite linear combinations of such mappings: 2 U o yt,
Xt 6 C ,
yte
Yx,
In other words, 36° is a free linear space over C with Yx as a set of generators.
40
A. M. Vershik, I. M. Gel'fand, andM. I. Graev
We introduce a scalar product in $£°. Let p O i , y2) be the invariant metric on the Lobachevskii plane. For any pair of mappings in Yx, Vi = J>i(0 and y2 = y2(') we set
($1, y2) = exp/l In cosh^pOiOO,
y2(x))dm(x)\
and then extend this scalar product by linearity to the whole space St°. The Hermitian form so defined on S6° is positive definite (for a proof see the end of §4.2 below). Let S£ be the completion of S£Q in the norm
n\\2 - «, *). We define a unitary representation £/~ of Gx inSS. For this purpose we observe first that an action of Gx on the set Yx of continuous mappings X -> Y is naturally defined. Namely, an element g = #(•) G Gx takes y = y(-) into yg = ^i(-), where ^ ( x ) = y(x)g(x). We assign to each g E G x the following operator £/~ in c^.°: THEOREM 4.1. T/ze operators Ug are unitary on 36° and form a representation of Gx. PROOF. The fact that the operators U^ form a representation is obvious. That they are unitary follows immediately from the invariance of p(yi, y2) on Y. Since the operators U~ are unitary on SS°, they can be extended to unitary operators in the whole spaced^. So we have constructed a unitary representation of Gx m&8. 2. Realization in the unit disk. We provide explicit expressions for the scalar product in $6° and for the operator U^ when Y is realized as the interior of the unit disk \z\ < 1.
fa p.\
Let G be given as the group of matrices g= I s - I, let Y be the interior of the unit disk \z\ < 1, and let G act in the unit disk in the following manner: z -> zg'1 = ^ z + j . Then c^° is the space of finite formal linear combinations where z(«) are continuous mappings of X into the unit disk \z | < 1. For a pair of mappings zx{>) and z 2 («) the scalar product in (0 0 has the following form: (1)
( Z l (.), «,(-)) = ex
The representation operator f/_, g=r=/^il^LU ^, takes z(-) into z ( . ) r = , , ( . ) , where ^ ^ ^ ^ (2) oWi We indicate another convenient realization of the representation (1). (It can be obtained from the first by the transformation z(-) -> X(z(«)) ° z(-),
Representations of the group SL(2,R)
where X(z(-)) = exp j In (1 -
41
\z(x)\2yldm(x).)
In this realization, as before, the elements of 3£° are formal finite linear combinations of continuous transformations of X into the unit disc \z\ < 1:
but the scalar product has the simpler form: (3)
( Z l ( . ) , z 2 ( . ) ) = ex
The representation operator Ug is given by the formula:
*V(-) = exp( j In |^)z(x) + ^ ) ' | - 1 ^ ( where zt(x) is defined by (2). In conclusion we verify that the Hermitian form introduced in 36° is positive definite. It is simplest to confirm this for the Hermitian form given by (3). We introduce the following notation: f zi(x) zi(x) fi(x, n) = < l
for
n>0,
l i zff (x) ( ) for
Fi(xi,
.. ., xh; nu . . .,nh)=
h
[\ ft (xs, ns), where i = 1, 2.
It is not hard to check that the scalar product (3) can be represented in the following form: oo
(4) ( Z l (.),
2s(.))=S
2
X I ^1(^1, . • .,xk; nu . . ., nh)F2(xi,
ITKUiX . . ., xh; nu . . .,rcft)dm(xi)
.. .
dm(xk).
To obtain this expression from (3) we have to expand first the function In 11 - z1(x)z2(x)\~1 in a series: In I 1 — zi (x) z2 (x)l"1 = j 2 J ^ J ) /1 (^, /*) /2 (^, n) dm (x). Then we expand exp 1/ in a power series, where ( J ' n)h(x, n)dm(x), and obtain the required expression (4). It is evident that each term in (4) gives a positive definite Hermitian form on 36°; consequently, the Hermitian form given by (4) for any pair of mappings ZiO) and z2(«) is positive definite. 3. Equivalence of the representation constructed here with the preceding ones. THEOREM 4.2. The representation U~ of Gx in 3£ is equivalent
42
A. M. Vershik, I. M. Gel'fand, andM. I. Graev
to the representation constructed in §3. Hence it follows, in particular, that U-g is irreducible. Let >^o be the point of the Lobachevskii plane Y = K\G (where K is a fixed maximal compact subgroup) corresponding to the unit coset. We denote by y0 = yo(*) the mapping that takes X into y0. The vector y0 belongs toS£, and it is clear that for every g G Kx. Thus y0 is a vacuum vector in SB. LEMMA 4.1. The vector y0 is cyclic in SBPROOF. It is sufficient to verify that as g ranges over Gx, U^Vo ranges over the whole of Yx. It is known that the natural fibration G -> Y = K\G is trivial, hence there exists a continuous cross section s: Y -> G. Now s induces the mapping Yx -• Gx under which y = >>(•) G F x goes into if = g(.) G Gx, where g(x) = s[y(x)]. It is also clear that >>(•) = J^o^C*)- This completes the proof. Let us find the spherical function (U~y0, y0), where y0 is a vacuum vector. Since U~yQ = yog'1, we obtain by the formula for the scalar product in SB In cosh"1 p(yog""1 (*)»
fa)
l
= exp J In cosh~ p(y0g(x), y0) dm (x) = exp \ In ty(g(x)) dm (z). We proceed now to the proof of Theorem 4.2. In §3 the representation £/- of Gx was defined in the Hilbert space X with the cyclic vacuum vector £0. It was also established that (U~%0, £0) = exp f In \p(g(x))dm(x). So we see that (U^o, %Q)X = (Ugy0, ^ 0 ) c ^ ' S m c e t h e vectors g0 and y0 are cyclic in their respective spaces, it follows that the mapping |0«—*• 1}Q can be extended to an isomorphism X-+ SB that commutes with the action of Gx in X a n d ^ . This proves Theorem 4.2. §5. A representation of Gx associated with a maximal compact group K C G
1. Construction of a representation of G. We take G to be the group of
/a
p\
matrices Ig- — I, | a |2 — | p |2 = 1. As before, let X be a compact topological space with positive finite measure m. For simplicity we assume that the support of m is the whole space X and that m(X) = 1. Henceforth we write dx instead of dm(x). Although the method of construction that we use here for the representation of Gx is cumbersome, it has the advantage that all the formulae can be written out explicitly and completely and are to some extent
Representations of the group SL(2, R)
43
analogous to the expression of representations of the rotation group by means of spherical functions. We suggest that on first reading the reader should omit the simple but tedious proof in the second half of §5.1 of the fact that the formulae gives a unitary representation. Formulae for representations of the Lie algebra of Gx are given in two forms at the end of this section. We introduce the Hilbert space S£ whose elements are all the sequences where f0
*• = (/<>, fu • • ., h, . . . ) . E C, and fa for k > 0 are functions X x . . . x X
X Z X . . . x Z->- C, satisfying the following conditions:1 ft
1) fa(*i, • • • > xk\ n\> • • • > nk) is symmetric with respect to permutations of the pairs (*/, «/), (XJ, rij)\ i, . . ., xk; nu . . ., nk) | n . = 0 = ft-ife,
. . . , # * , • • ., ^ ; ^ i , . . ., w f , . . ., nh) (i =
(1)
3)
2
2
fcT | nx ... nh | J l ^ ^ i ' • • • ' • r ^
n
u
••
ft=l ni, ..., nft (n i ^ 0)
REMARK. Nothing would be changed in the definition of S£ if we were to assume that all the integral indices nt are non-zero. Then, of course, condition 2) is unnecessary, and the norm, as before, is given by (1). We construct a unitary representation of Gx inSS. First we introduce on G functions Pmn(g) and pn(g). We define Pmn(g) for n > 0 as the coefficient of zm in the power series expansion of/ ?z _ \n, where ry
ft
For n < 0 we define Pmn(g) by:
x,-, «,• indicate that the corresponding variables are omitted.
^
PZ +
a
'
44
A. M. Vershik, I. M. GeVfand, andM. I. Graev
From this definition it follows that 1) P-m-n(g) = Pmn(g); 2) Pmn(g) = 0 if mn < 0; 3) Pmo(g) = 1 for m = 0 and Pm0(g) = 0 for m =£ 0; 4) Pon(g) = ( x ) n for n > 0 and POn(g) = (-J-) W for « < 0. Next we set pwfe) = (~|-) n for n > 0, p nfe) = ( - | " ) ' n | for « < 0, = ILet ? = ( a - \ be an arbitrary element of Gx. We associate with it VP(-) «(•) / the operator f/~ in ^f that is given by the formula U7{h) = {
S
{ n *-,«,<*<*«»* 2 2 i l
Z
(*i ¥= 0)
s x
J I I Pij(S(fj))fh+a{xif
. . . , x h , t u . . .,ta; m
u
. . . , / T i k , Zlf . . . , Z 8 ) ^ i . . .
and ¥(g)-exp j l n ^ 2 ^ ^ ) ) ^ .
(3)
Here \p(g) denotes lal"1. THEOREM 5.1. The operators £/~ are unitary and form a representation of Gx, that is, {UtFu U7F2) = (Fu F2) for any g G Gx and Fx, F2e\S6 and TJ~U~F= U~~F x
for any*gl9g 2 G G and F ^S£. We verify these relations for the vectors F of a certain space 3£° everywhere dense in SB, which we now introduce. We denote by M the set of sequences of the form F = (1, fu fzf . . .,
/A,
• • 0,
where fk{xu . . ., xh\ nu . . ., nk) = w ^ , w4) . . . w(a;A, rcA), and w(x, «) is a function continuous in x for any fixed n such that M(*, 0) = 1 and
Represen tations of the group SL (2, R)
(4)
45
2 fir J i»(*.») !•<**< oo.
We denote by 3t° the space of all finite linear combinations of elements of M. We must verify that Si0 a SB. Now it is evident that the vectors F G M satisfy conditions 1) and 2) in the definition of SB. Furthermore, if F = (1, u(x, n)9 . . ., u(xi9 n), . . ., u(xk, nk), . . . )
(5)
is a vector of M, then its norm \\F\\ can be represented in the form: (6)
| | * T = ex
Consequently, by (4), F also satisfies condition 3), and hence F 6 SB. We observe that if F G M, that is, if it has the form (4), then the expression for UgF reduces to the following simple form: (7)
UgF = X(g, M ) ( 1 , y(a:, » ) , . . ., v{xu n ^ , . . . , v ( x h , n h ) , - - .)»
where X(?f ")
(8)
(9)
v(x,n)=2Pmn(g(x))u(x,m). m
LEMMA 5.1. The space SB0 is everywhere dense in SB. PROOF. We assume that all the indices «,- are non-zero (see the Remark on p. 115). Suppose that SB0 is not dense in SB, hence that there exists a non-zero vector F = {/&}, orthogonal to SB0. We consider in SB the vectors of t h e f o r m F% = {khfk}, w h e r e f0 = 1, f k ( x u . . ., x k 9 ; n l 9 . . ., nk) = = u(xl9 nx) . . . u(xk, nk) for k > 0, u(x, n) is a continuous function, and X is an arbitrary number. Since Fx £ SB0, we have (F x , F) = 0, that is, CO
2 ~w ( 2
I wi • • •n* I"1 x
X \ /k (^i, • • •, %h; ^ i , . . . , ^fe) fu {xi, . . . , Xh; 724, . . . , rih) dxx .. . dxk 1 = 0
f o r a n y X. H e n c e i t f o l l o w s t h a t f o r e v e r y k = 0 , 1 , . . . w e h a v e t h e relation | ^i . . . nh I"1 x ni,...,nft (n-^0) 4,
. . ., nh)fh(xu
...,xk;
nt, .. .,^)rfx 1 . .. da;fc = 0
46
A. M- Vershik, I. M. Gel'fand, and M. I. Graev
or (H)
2
\ni--nh\'1
X
X \ u {xi, Hi) ... u (xu, nh) fh (xu . . ., xh; nil . . ., nh) dxx . . . dxh = 0 for any continuous function u(x, n) such that I n I ] I u\xi
2J
n
) I "^
Since f°k(xu . . ., x^; nXi . . ., % ) is symmetric under permutations of the pairs (x/, nt) and (x;-, «7-), it follows from (11) that / £ = 0 (fc = 0, 1, . . .)• Thus, F = 0, in contradiction to the hypothesis. To prove Theorem 5.1 we need certain relations for the functions Pmn{g)
and piig): (
a)
~ V n Pmn(g)^- {7.!}m Pnm(g)
for m ^ 0, ^ ^ 0;
b) for any compact subset V C G there exist constants C > 0 and r, l l | |
0 < r < 1, s u c h t h a t l ^ f e ) ! < S
mm> (g2) - P
d)
Ml
-2In
e) I "i ri6mm»
2-
1 ' Pmn (g)
for m ¥= 0, m' ¥= 0,
- ( - * ) - | . » | - ' M g ) for m ^ 0, i n ' = 0, — ( —I) m '|m'|- 1 p m ,(g-)for m = 0, m' ^ 0,
Pm-n (*) =
— 4 In i|?(g)
for m = m' = 0
1
(5 w m ' is the Kroneker delta). ) LEMMA 5.2. The function \(g, u) defined by (8) satisfies the following functional relation: (12)
\(gl9 v)\(gu u) = X&gi, u)>
where V{X,
Tl) = S ^ m n f e ^ ) ) ^ , m).
' We can derive a) from the relation Pmn(g)
=
I
(-5^—1L j z~m~1 dz, n > 0; the bound
b) for Pmnig) follows from the fact that the radius of convergence of the series ^ j Pmn (§) zm is greater Trt
than 1; the relation c) follows immediately from the definition of the functions Pmn', d) follows easily from the definition of Pmn and p\ and a).
Representations of the group SL(2, R)
47
PROOF. It follows from the definition of X that
) exp ( 2 2 "TTT^ J Pi (Si (*)) pmi We sum over /, apply d), 1 and obtain )u(x,
LEMMA 5.3. £7- ^
m)dx
for any F G M and any gu g2 e Gx.
= U^^F
PROOF. Let ' F = (1, w(a:, 72), . . ,, Mfo, «i), -.., u(xk, nk), . . .)• Then 67r/
= X(g2,
M )(l,
I;(O:, »),
f
. . , vfo,
ni)
. . . v(^,
nk),
. . .),
where M)(1, W{X, n), . . ., w(xu
nt), ., .,w(xh,
nk), . . .)»
where M; (x, 72) = 2 ^rn'n (#1 (^)) y tei ^ ' ) =
P
2
m'
m'n (gi ($)) Pmm* (#2 (x)) U (x, m).
m', m
It follows from Lemma 5.2 that \(gi, v)\(g2, u) = ^digi, u)- O n t n e other hand, by c) for Pmn(g), we have w(x, n) = 2 Pmn((jgig2)(x))u(x> m )Consequently, UgU^F = U^F. COROLLARY.1 The operators Ug form a representation of Gx in &£?. LEMMA. 5.4.(0-^, U?F2) = (Fl9 F2) for any Fu F2 G M and g G Gx. PROOF. Let F i = ( 1 , Mite, » ) , . . . , _ . ^ 2 = ( 1 , ^ t e r W)» • • •» ^ 2 t e l , « l ) , . . . , M 2 t e f t » ^fe)» • • •)• Then
/V
1
f
\
On the other hand, using the expression (7) for UgF we have (U~Fi, U~F2) = exp J4 j In a|) (gr (x)) dx + —
——
Ml
I
fry
7\ y7^,
i v » / P
J7J S Q
T)l(cj('r\\il
J n
X
I
\TJ
/
^j
P
J ™ (S ( V rn'n (g (x)) Ut (x, Hi) U2 (x\ '
Reversal of the order of the summations is permissible in view of b).
48
A. M. Vershik, I. M. Gel'fand, and M. I. Graev
In the last expression we sum over n under the exponential sign, then use e) and Uf(x, 0) = 1, and obtain (UgFl9 UgF2) = (Fl9 F2) after some elementary simplifications. COROLLARY. The C7~ are unitary operators in SB0. 2. Irreducibility of the representation Ug. The representation operators Ug assume a specially simple form when restricted to the subgroup of matrices
0
c-iV(-
2/-
Namely, (13)
U~ {/„} = {fk}, ft
where / i ^ , . . . , a : A ; 7it, . . . , 7ife) = exp ( i " 2
rcs
. . . , £ * ; /ij, . . . , n f t ) .
8=1
From this expression it is clear that the family of commuting operators Ug has a simple spectrum in SB. The vacuum vector in SB is lo = (*! /i(*» n)i • •
where fk(xl9 . . ., xk; nl9 . . ., nk) =* 0 // K l + . . . + \nk\ > 0. THEOREM 5.2. 77ze representation U^ of Gx in S£ is irreducible.
PROOF. Let A be a bounded operator ind^ that commutes with all the operators Up in particular, with the U% of the form (13). Since the family of operators Up has a simple spectrum, an operator A that commutes with them has the form (14)
Mh} = W*}>
where fljt(^i» • • • > * * ; wi> •• •» wi) a r e measurable functions (satisfying the same relations as the fk). Let £0 be the vacuum vector in SB defined above, It follows from (14) that A£o = 0o£oWe apply to ^ 0 , the operator Ug, g G G, where / cosh r sinh r\ \sinhr cosh r/
,
o
and r does not depend on x. We obtain k
where fk(xl9 . . ., xk\ nx, . ..., %) = cosher ff tanh'^'r. i
From U-gh£Q = A Ufa it follows that ao{/ft} = {ahfk}, hence ^0A = "kfkConsequently, since the fk do not vanish, a^ = a0 for every /:, that is, A is a multiple of the unit operator. 3. Equivalence of the representation Ug to representations constructed earlier. THEOREM 5.3. The representation Ug of Gx in SB is equivalent to
Representations of the group SL(2, R)
49
the representations constructed in the preceding sections. PROOF. It follows easily from the definition of Ug and the scalar product in SB that the spherical function corresponding to the vacuum vector £0 has the form
=exp(2 So we see that this spherical function coincides (with suitable agreement of measures on X) with the spherical functions of the representations of Gx constructed in the preceding sections. The theorem follows immediately from this fact and the irreducibility of Uj. 4. Infinitesimal formulae for the representation. We give formulae for the representation operators of the Lie algebra of Gx in SB (that is, of the algebra (&x of continuous mappings X -> @ of X to the Lie algebra <& of G with the natural commutation relations). We take the following matrices as generators of ©:
it
\
r #
'
are
where
.. ., xh; nt, . .., nk);
(A~f)h (xi, . . . , xu; nu ..., nu) = h
j , . . . , z h i i; nu . . ., nh, ~/)k (a?i, . . ., xh; nu . . ., wft) = ft
= * 2 ^sT(^s)(/fe(x1, . . .,a:ft, «!, . . ., n« +
+ fh(xu
...,xh,nu..
., ns — 1, . . .,
It is convenient to go from Af and ^4^ to
50
A. M. Vershik, I. M. Gel'fand, andM. I. Graev
The operators A~ and A~ act in the following manner: (4ffh (*i, • • •, Xh\ nu . . . , nh) =
— I
. . . , s f t , *; / i t , . . . , nkf
1 ) dt,
f /)ft (^i, • • •, xh; nu . . . , nk) = - k
= ^
^ s T ( ^ s ) / f e ( ^ ! , . . ,,xk;
nu
. . . , ns
— 1, ..
.,nh)
Since ^1~, ^4~, and A~ form a representation of the algebra @z, the same commutation relations hold for them as for the corresponding elements of the Lie algebra @z, namely, U+ A\] = -iA%r
U r , A%] = iAr^
U+ ^ - ] = - 2 ^ ^ 2 .
5. Another method of realizing the operators A~, A*~ and A~. The method of realization proposed here seems very interesting to us. We specify the elements of SS not as sequences of functions of xl, . . ., x^, Hi, . . ., nk, but as sequences of functions of the x-parameters alone. Let us consider, for example, the function f3(xl, x2, x3; 2, 1, - 4). We assign to it the function /3,4(*i, xl9 x2; y\, yi9 y\, yt). More generally, If a function fk(xl9 . . ., x^; nl9 . . ., n^) is given, we first discard the zeros among the numbers nl9 . . ., n^. We then pick out the positive numbers among the nt and denote their sum by m and the sum of the absolute values of the negative nt by n. xx is then repeated \nx\ times, x2 is repeated \n2\ times, etc. Because of the symmetry of / in the pairs (Xj, rii) we can write down first all the arguments X( with positive «,-, then all those with negative «,; we denote the resulting function by . . ., ^ i , . . ., Xk, • •
InlI
I nk
f
Now we give a precise definition that does not depend on these arguments. We introduce the space S£°, whose elements are infinite sequences / = {/mn},m,n=o, i, ..., where U 6 C, fmn: X x . . . X X -> C for m + n > 0, m-[-w
satisfying the following conditions: 1) the functions / m n (^i, . , ., xm ; yl9 . . ., yn) are continuous;
Representations of the group SL(2, R)
51
2) the functions / mw (x l5 . . ., xm ; yl9 . . ., yn) are symmetric in the first m arguments and in the last n arguments:
3) l|/||2= 2 21
{
| » |
7711
i, .. .dxhdyi . .. where the inner sum is taken over all partitions m = m1 + . . . + mk, n = n
t
+ . . . + n
k
, a n d \ m \ = m
. . ., m
l 9
k
, \ n \= n
l f
. . ., n
k
. ( F o r
m = Owe take |m| = 1.) An isometric correspondence between 38° and the previous space is given by {fmn) *-+ {fh}, where fh (xu
...,xp,yl,...,yq;mi,...,mp,ni,...,
nq) =
= fmn ( * i , ...,xl,...,xp,...,xp;
y { , ...,
mi
mp
y
u
...,
yq, . . . , y q )
|m|
..+^,
TI = |
|n^|
n^ \ -f . .. + | nq\).
Now ^4^, A^f, and >1~ act in 36° according to the formulae: , Xm\
yu
S 5=1
i, n (^i, • • •, xm, xs; yu . . ., yn) 4-
— J T(0/m+i, n(^i, • • - , x
., xm;
yu
. . ., yn)
m
, t; y
=
m 2J
T
( ; r * ) / m - i , n ( ^ i , • •., ^ s , • . . , x
—j
m
;
y
u
.. . , y n ) —
u
..
.,yn)dt,
52
A. M. Vershik, I. M. Gel'fand, andM. I. Graev
§6. Another method of constructing a representation of Gx
1. The general construction, a) Construction of the representation space. Let G be an arbitrary topological group and H a linear topological space in which a representation T(g) of G is defined. Further, Let X be a compact topological space with a positive finite measure m. As before, we assume that m is a countably additive, non-negative Borel measure whose support is X. We suppose that in H there is a linear functional / (/ =£ 0) invariant under T(g). Then we construct a representation £/~ of the group Gx of continuous mappings X -> G from the representation T(g) of G and the functional /. We denote by Hx the set of all continuous mappings / = /(*) : X -* H such that /(/(») does not depend on x and l{f{x)) - 0. We introduce a new linear space 36° whose elements are formal finite sums of elements of Hx: Here we set XJ + \2f = (kt + X 2 )/ if \ t + X2 =£ 0, and / + ( - / ) = 0. We emphasize that if fx (x) and f2 (x) are not proportional, then f — f\ + Ii a n d f\ + A are regarded as distinct elements. In <2%>° operations of addition and multiplication by a factor X E C are defined in the natural way. Namely, the product o f / = / ( • ) by X G C is defined as X o /(JC) = \f(x) if X =£ 0, and 0 o /(*) = 0. As a result, SB becomes a linear space. b) Action of the operators in 38°. We define a representation £/~ of Gx inc^ 0 by (2)
(£/;/)(*) = %T(g[x))f(x),
Where X(f, /) is a function of £ and / such that X(g, cf) = \(g, f) for any c =£ 0, and we extend L^ by additivity to all elements (1). LEMMA 6.1. The operators £/~ form a representation of Gx if and only if the function \(g, f) satisfies the following additional condition for any Su 82 e Gx and f G Hx: X(gu A)X(f2, / ) = X g ^ , / ) where / / f/ze weaker relation
the f/~ /orm <3 projective representation. The proof is obvious. c) Construction of a unitary representation. Let Ho C i/ be the set of elements £ such that /(£) = 0. By the invariance of /, Ho is an invariant subspace of H. Suppose that an invariant positive definite scalar product (£i> £2) is defined in H.
Representations of the group SL{2, R)
53
We construct a scalar product in the space S£Q from the scalar product (£1, £2) in H. To do this we first fix a vector £0 e H such that /(£0) = 1For any pair of elements /i = / i O ) and / 2 = / 2 ( # ) of i / x we define our scalar product as follows:
(3)
<£, f2> = z(7i)7(^j exP ( — L = , \ (/;
where /?(*) = ft(x) - /(/p£o are elements of Ho for any x G Z. (We recall that l(f(x)) is independent of x. Instead of l(f(x)) we write /(/), where We extend this scalar product to the whole space 36° by linearity. LEMMA 6.2. The Hermitian form on S£° defined by (3) is positive definite. PROOF. We choose arbitrary elements/j, . . .,fn of Hx and prove that the matrix (ft, fj) is positive definite. We have
(4)
^JJL =2 J ^ ( _ _ 4 = f (/;(*), Hfi)l(fj)
Since a,-/ =
nZo
V
Hfi)l(fj)
J
f-(x))dm(x))n. '
X
~ ~ yJlix), fj(x))dm(x) is positive definite, by Schur'
lemma each term of (4) is positive definite, and the lemma is proved. Thus, U^ acts in a pre-Hilbert space. Let us see how the multiplier Mg, / ) can be chosen so that the representation is unitary. For this purpose we first construct from £0 a function /3(g) with values in H: Q(g) = T(g)%0 - £0- Since / is invariant we have /(/3(g)) = l(T(g)%0) = 0, that is, P(g) G Ho for any g G G. It is not hard to see that is a cocycle with values in Ho, in other words, it satisfies the relation 0fei) + T(gl)(3(g2) = P(glg2) for any gl9 g2 G G. We observe that j3(g) depends on the way we have fixed the vector LEMMA 6.3. / / we set
(5) M?,fi = /'(x) = fix) - l(f)Zo, \c(g)\ = 1, then the operators Uj defined by (2) are unitary and form a projective representation. Specifically, U z> UZ = c « , & ) £ f c j r , , where (6)
X exp (i j Im (71 (g (i)) p ( ft (*)), p( gl (x))) dm (a:)) . The proof comes from a direct verification.
54
A. M. Vershik, I. M. Gel'fand, and M. I. Graev
REMARK. This condition on X(g, f) is also necessary. We now state our final result. A linear topological space H is given and also a representation T(g) of G in H. A linear functional / is given in H that is invariant under the action of G, that is, KT(g)%) = /(£) for any g E G and £ E H. We define a scalar product (£1? £2) m the subspace Ho of elements £ such that /(£) = 0. A representation of Gx is constructed as follows. We consider continuous mappings / = /(.)'• X -> H such that l(f(x)) = const =£ 0. Weintroduce the space S£° whose elements are the formal sums fx + . . . + fn with the relations \J + X2f = (X1 + X 2 ) / if Xi + X2 =£ 0, / + ( - / ) = 0. We construct the scalar product: f (/; (x), f'2(x))dm(x)\ J / an( s a where fl(x) = ff(x) — /(/j-)£0> i £o * fixed vector in H such that ^(£o) = 1- This scalar product is then extended to the whole spaced 0 . We denote by S£ the completion of $£° in this scalar product. The operators Uj are defined by the formula (Ugf)(x) = X(g, f)T(g(x))f(x), where ^ 5 l(fi)l(f2)
c ( | ) e x p ( ^
kg) = ng)%0 - g0, \c(g) i = i, and are extended by additivity to sums of the form (1) and then to the completion. These operators are unitary and form a projective representation of Gx, namely, £/~ Ug = c(gig2)U^, where c(gl5 g2) is defined by (6). 2. Construction of a representation of G x , where G = PSL(2, R). We now apply the general construction described above to the case of the group G = PSL(2, R), given in the second form. We define H as the space of all continuous functions on the circle IJI — 1 in which the representation acts according to the following formula: (7) ! p
and the invariant linear functional / is 0
In the subspace Ho of functions /(?) for which l(f) = 0 we specify a scalar product as follows:
or, in integral form, (9)
2n2n
It is clear from (8) that this scalar product is positive definite, and from
Representations of the group SL(2, R)
55
(9) that it is invariant under the operators T(g) of the form (7). (We recall that We now construct ^°.We fix in H the function £0 = / 0 (f) = 1. Then + al" 2 " 1, where g=(?
P(g) = T(g)f0 - / 0 ; hence, P(g, f) =
-) .
Note that j3(g, f) takes only real values. We examine the set Hx of continuous functions f(x9 f) satisfying the following condition: 2n
-^ [ /(a;, elt)dt = l
for all
^6^-
The elements of 3£° are all possible finite formal linear combinations of such functions: ^]hofi(^, £)> ^* e C. A scalar product is defined for any pair of functions fx = fx(x, f) and f2 = fi(x, f) in Hx by the formula
where f. = f. — \9 a n d | s then extended by linearity to the whole space The representation operator U^ is defined by the formula
feHx,
where
*, /) = exp(— c f In
sin
X
/ ' — f — 1, and is then extended by linearity first to the whole space 3£° 9 and then to its completion S£m We note that in the case considered here the scalar product of the vectors T(g1(x))@(g2(x)) and (l(gi(x)) is real. Therefore, by Lemma 6.3 (see the expression for c(gl9 g2)), the operators Ug form a representation of Gx. We make here an essential remark. We can take for H the subspace of functions on the unit circle | f | = 1 that are boundary values of analytic functions analytic (or anti-analytic) in the interior of the unit disc. Then we obtain other representations of Gx, which are projective. 3. Another construction of a representation of G x , where G = PSL(2, R).
It is sometimes convenient to define representations of G not on functions on the circle but on functions on the line. Let G be given in the first form. We consider the space H of all real continuous functions that satisfy the following condition: f{t) = O(t~2) as t -> ± <»; and we define a represen-
56
A. M. Vershik, I. M. Gel'fand, andM. I. Graev
tation of G in H by the following formula:
We further define in H a G-invariant linear functional /(/): *(/)= ]
f(t)dt.
Let HQ C // be the subspace of functions on which /(/) = 0. In Ho there is an invariant positive definite scalar product +00+00
(/i, /1) = — J J In I *!-< — 00 —00 Jt
1
We now fix the function fo(t) = , 1 + ^2 in H, for which l(f0) = 1. We also set 0fe, f) = (7W 0 )(r) - / 0 ( f ) . We proceed to the construction of 3£°- We consider the set Hx of continuous functions f(x, t), x G X, t E R, satisfying the following conditions: 1)/(X, 0 = 0 ( r 2 ) as t ~> ± oo; 2) J /(x, r)t/r = 1 for any x G X. The elements of SS° are all possible formal linear combinations of such functions: 2^i o /i(s» *)» ^ 6 C. A scalar product is defined for any pair fi(x, t), f2(x, t) by the following formula:
(—j In where / - ( ^ 0 = / f ( ^ , 0 The representation operator acts as follows:
where
f In I ^ — REMARK. If we use instead of // only the subspace H+ (or H~) of functions which are boundary values of analytic functions in the upper (or lower) half-plane, then we obtain other (projective) representations of Gx. 4. Equivalence of the representation Ug to the representation constructed
in §5. THEOREM 6.2 The representation Ug of Gx, G = PSL(2, R) constructed here is equivalent to the representation constructed in §5. Hence
Representations of the group SZ(2, R)
57
it follows that Ug is irreducible. PROOF. For the proof it is sufficient to construct an isometric mapping of SS° into the representation space of § 5 that commutes with the action of Gx in these spaces. Let us examine the construction of the representation in §6.2. In it 3£° consists of formal linear combinations of functions f(x, f), |?| = 1, such that -L f f(x, eil)dt = 1 for any x G X. We expand f(x, eif) in a Fourier b series in t: f(x, eu)= 1 + ^an(x)eint. We associate with f(x, t)£3t° n=£0
an element of the space SSy constructed in §5: F = (1, u(x, n), . . .,
. . ., u(xu nx\ . . ., u(xk, nk), . . .), where u(x, n) = (- \fan{x). We extend this to a linear mapping of the whole space S£° onto $£\. From the definition of the norm in these spaces it follows easily that the mapping so constructed is an isometry. Furthermore, it can be shown that it commutes with the action of Gx in these spaces. §7. Construction with a Gaussian measure
1. To explain the construction of this section we find it convenient to make some modifications in the general constructions in §6.1. Let G be a topological group, E a real Hilbert space, and suppose that an orthogonal representation T(g) of G in E and a cocycle with values in E are given, that is, a function ]3: G -* E satisfying the relation From the part (T, 0) we construct a new unitary representation of G. In what follows we change the notation for the group and write T in place of G, because in the examples F can be both G and Gx. First we construct a new (complex) space M° whose elements are formal finite linear combinations of elements £,- G E: (l)
Xi ° *i + . . • + xw o £„,
x,
ec.
In contrast to the preceding section, X o £ and X£ are now regarded as distinct. Thus, the original space E has a natural embedding in 36° as a subset (not as a subspace!). We now define a scalar product in 36°. Namely, for elements £1? £2 e H we define a scalar product by the formula <£x, £2^ = e x P (£i> £2)* where the round parentheses denote the scalar product in E, and then we extend this scalar product by linearity to all formal linear combinations like (1). It is easy to verify that the Hermitian form so introduced is positive definite (see the proof of Lemma 6.2). Let SS be the completion of S^ in the scalar product just introduced. We define a representation Uy of F in 36°. The action of operators Uy,
58
A. M. Vershik, I. M. Gel'fand, andM. I. Graev
7 E r, on elements £ E E is given as follows: *7v-iE = exp ( — i || P(V) | | a - ( r v 5 , P(v)) and then we extend these operators by linearity to the whole space d&°It can easily be established that the U7 are unitary operators, hence can be extended to the whole space $£, and that these operators form a representation of F. Later we shall see how this construction is related to that of Araki and Streater. 2. Let us consider two examples. a) Let T = PSL(2, R). Let T(g) be the representation of PSL(2, R) constructed in §6.3 (or §6.2), and j3(g) the cocycle defined there. We denote by E the real subspace of Ho also given there. From it we construct SS and the representation Ug. THEOREM 7.1. The representation so constructed coincides with the canonical representation introduced in §1. b) Let T = (PSL(2, R)) x . We consider the space of all mappings f\ X -• E that are measurable on X and satisfy the condition f (x)\\* dm (x)
distribution is a finitely additive, normalized, non-negative measure defined on the algebra of cylinder sets in K, that is, sets of the level of Borel functions of finitely many linear functionals.) It is known that v can be extended to a countably additive measure in an arbitrary nuclear extension K of K. We call this the standard Gaussian measure, and we quote two properties of this measure that we shall need. 1) The standard Gaussian measure in K is equivalent (that is, mutually absolutely continuous) to its translations by elements of K.
Representations of the group SZ(2, R)
59
2) Every orthogonal transformation of K can be uniquely extended to a linear and measurable transformation in K that is defined almost everywhere and preserves the Gaussian measure. We now consider the space E, introduce the pair (T, j5) (See §7.1), and choose the standard Gaussian measure /z in some nuclear extension E of E. We examine the space L2(E, n) of all square integrable complex-valued functional on E (more precisely, of classes of functional that coincide almost everywhere). From the pair (T, 0), we construct in L2(E, JU) a representation Ug:
4
-"4"l\&8)\I2-
, o / \\
= a 2 F (21 (g) cp + P (g)), 2 where F G L (^, p),
L*(E, n) = e x p f f s ^ ®-^FH®
•••®H> n
where H is the complexification of E' and H 0 . . . 0 H is the subspace of n
generalized Hermite polynomials of degree n. Hence the preceding investigations show that our representation of Gx is realized in S£ = exp H by means of the cocycle j3 (See §7.1). This realization coincides with the general scheme of Streater and Araki, which they have examined, however,
60
A. M. Vershik, I. M. Gel'fand, andM. L Graev
only for certain soluble groups. In the terms used here the problem of the construction of a representation reduces to that of the discovery of the cocycle j3 and to the proof of the irreducibility of the representation of Gx. We have done this in the present paper for G — PSL(2, R). References
[1] I. M. Gel'fand and M. I. Graev, Representations of the quaternion groups over locally compact fields and function fields, Funktsional. Anal, i Prilozhen. 2 (1968) no. 1, 20-35. MR 38 #4611. [2] I. M. Gel'fand and I. Ya. Vilenkin, Nekotorye primeneniya garmonicheskogo analiza. Osnashchennye gilbertory prostranstva, Gos. Izdat. Fiz.-Mat. lit. Moscow 1961. MR 26 #4173. Translation: Generalized Functions, Vol. 4: Applications of harmonic analysis, Academic Press, New York and London 1964. MR 30 #4152. [3] I. M. Gel'fand, M. I. Graev, and I. Ya. Vilenkin, IntegraVnaya geometriya i svyazannye s nei voprosy teorii predstavlenii Gos. Izdat. Fiz.-Mat. lit. Moscow 1962. MR 28 # 3324. Translation: Generalized Functions, Volume 5: Integral geometry and representation theory, Academic Press, New York and London 1966. MR 34 # 7726. [4] H. Araki, Factorizable representations of current algebra, Publ. Res. Inst. Math. Sci. 5 (1969/70), 361-422. MR41 #7931. [5] I. Dixmier, Les C*-algebres et leurs representations, second ed., Gauthier-Villars, Paris 1969. MR 30 # 1404, 39 # 7442. [6] A. Guichardet, Symmetric Hilbert spaces and related topics, Lecture Notes in Mathematics, Springer-Verlag, Berlin-Heidelberg-New York 1972. [7] D. Mathon, Infinitely divisible projective representations of the lie Algebras, Proc. Cambridge Philos. Soc. 72 (1972) 357-368. [8] K. R. Parthasarathy, Infinitely divisible representations and positive functions on a compact group, Comm. Math. Phys. 16,148-156 (1970). [9] K. R. Parthasarathy and K. Schmidt, Infinitely divisible projective representations, cocycles, and Levy-Khinchine-Araki formula on locally compact groups, Research Report 17, Manchester-Sheffield School of Probability and Statistics, 1970. [10] K. R. Parthasarathy and K. Schmidt, Factorizable representations of current groups and the Araki-Woods embedding theorem, Acta Math. 128, 53-71 (1972). [11] K. R. Parthasarathy and K. Schmidt, Positive definite kernels, continuous tensor products, and central limit theorems of probability theory, Lecture Notes in Mathematics, Springer-Verlag, Berlin-Heidelberg-New York 1972. [12] L. Pukanszky, On the Kroneker products of irreducible representations of the 2 x 2 real unimodular group, Part I, Trans. Amer. Math. Soc. 100 (1961) 116—152. [13] R. F. Streater, Current commutation relations, continuous tensor products, and infinitely divisible group representations, Rend. Sci. 1st. Fis. E. Fermi, 11 (1969), 247-263. [14] R. F. Streater, Continuous tensor products and current commutation relations, Nuovo CimentoA53 (1968), 487. [15] R. F. Streater, Infinitely divisible representations of lie algebras, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 19,1971, 67-80. Received by the Editors, Translated by W. J. Holman 15 June 1973
To the memory of Sergei Vasil'evich Fomin
REPRESENTATIONS OF THE GROUP OF DIFFEOMORPHISMS A. M. Vershik, I. M. Gel'fand and M. I. Graev This article contains a survey of results on representations of the diffeomorphism group of a noncompact manifold X associated with the space Y% of configurations (that is, of locally finite subsets) in X. These representations are constructed from a quasi-invariant measure n on Y%. In particular, necessary and sufficient conditions are established for the representations to be irreducible. In the case of the Poisson measure ju a description is given of the corresponding representation ring.
Contents Introduction §0. Basic definitions and some preliminary information § 1. The ring of representations of Diff X associated with the space of finite configurations §2. Quasi-invariant measures in the space of infinite configurations §3. Representations of Diff X defined by quasi-invariant measures in the space of infinite configurations (elementary representations) §4. Representations of Diff X generated by the Poisson measure . §5. The ring of elementary representations generated by the Poisson measure §6. Representations of Diff X associated with infinitely divisible measures §7. Representations of the cross product S - C°°0Y)-Diff X . . . Appendix 1. On the methods of defining measures on the configuration space Fx Appendix 2. S^-cocyles and Fermi representations Appendix 3. Representations of Diff X associated with measures in the tangent bundle of the space of infinite configurations . . . . References 61
62 63 66 72
82 89 94 95 100 102 104 108 109
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A. M. Gershik, L M. Gel'fand andM. I. Graev
Introduction This article is a survey of results on the representations of the group Diff X of finite diffeomorphisms of a smooth non-compact manifold X. As for many infinite groups, it is rather difficult to see what the complete stock of irreducible unitary representations of this group might be. Therefore, it is of some interest to single out certain natural classes of representations. We consider the space Tx of infinite configurations (that is, locally finite subsets) in X on which the group Diff X acts in a natural way. If n is a quasi-invariant measure in Tx and p is a representation of the symmetric group Sn(n = 1, 2, . . .), then we construct a unitary representation of Diff X from JU and p, which we call elementary. There is, therefore, a close connection between the theory of elementary representations of Diff X and the theories of quasi-invariant measures on Tx and representations of the symmetric groups. We note that quasi-invariant measures on Tx are studied in statistical physics (Gibbs measures and the simplest of them - the Poisson measure) (see, for example, [12]); and in the theory of point processes (see, for example, [17] and elsewhere). The space of infinite configurations Tx is, in its own right, a very important example of an infinitedimensional manifold, and its study is one of the interesting problems of topology, analysis, and statistical physics. Representations of Diff X that are of finite functional dimension, that is, representations associated with the space of finite configurations, were considered in [8] and [9]. In §1 we incidentally prove by a new method that the representations of a wide class are irreducible. However, we are basically interested in representations of infinite functional dimension associated with Fx; they can be regarded as limits of "partially finite" representations. In this paper necessary and sufficient conditions are obtained for elementary representations to be irreducible. In the case when JLX is the Poisson measure it is proved that the set of elementary representations is multiplicatively closed, that is, the tensor product of two elementary representations splits into the sum of elementary representations, and the structure of the corresponding representation ring is described. An important property of Diff X, which distinguishes it from locally compact groups and which will become apparent in the situations we discuss, is that to a single orbit of Diff X in Tx there is no corresponding representation; however, one can construct a representation from a measure on Tx that is ergodic with respect to the action of Diff X. More interesting and more widely studied is the class of representations associated with the Poisson measure on Fx (see §4). The representation of Diff X in the space L^(TX), where ji is the Poisson measure, arose (as an N/V limit) in [15]; however, the role of the Poisson measure was not noted here. It is
Representations of the group of diffeomorphisms
63
remarkable that this same representation can be realized in a Fock space as EXPp T, where T is a representation of Diff X in L^ (X) and j3 is a certain cocycle (see §4). This circumstance links the theory that we discuss here with [1] and [2]; Representations of Diff X associated with the Poisson measure on Fx are studied by another method in [22]. As far as we know, up to now, no measures, and in particular no Gibbs measures apart from the Poisson measures, have been discussed in connection with representations of Diff X. Representations of the cross product of the additive group C°°(X) and the group Diff X are investigated in a number of very interesting physics papers (see [15] for a list of references; see also [19] and [20]). All the representations of Diff X discussed in this article extend to representations of the cross product C°°(X)*Diff X\ the mathematical part of the results of [15] is contained in this paper. §0. Basic definitions and some preliminary information
1. The group Diff X. Everywhere, X denotes a connected manifold of class C°°. Diff X denotes the group of all diffeomorphisms \p: X -> X that are the identity outside a compact set (depending on i//). The group Diff X is assumed to be furnished with the natural topology: a sequence 4/n is regarded as tending to \p if i// and every \jjn is the identity outside a certain compact set K and if \pn, together with all its derivatives, tends to \p uniformly on K. If Y C X is an arbitrary open subset, then Diff Y denotes the subgroup of diffeomorphisms \p € Diff X that are the identity on X \ Y. 2. The groups S°° and S^. We denote by S°° the group of all permutations of the sequence of natural numbers, by S^ C S°° the subgroup of all finite permutations, and by Sn the group of all permutations of the numbers 1, . . . , n (n = 1, 2, . . .). We regard the Sn as subgroups of S^; thus, Sx C . . . C Sn C . . . and S^ = lim Sn. In what follows, So is understood to mean the trivial group. 3. The configuration spaces Fx and Bx. Any locally finite subset of X is called a configuration1 in X, that is a subset ) C I such that y n K is finite for any compact set K C X. By this definition, any configuration is either a finite or a countable subset of X\ if X is compact, then all configurations in X are finite. Let us denote by Fx the space of all infinite and by B^ the space of all finite configurations in X. The group of diffeomorphisms Diff X acts naturally on Fx and Bx. The space of finite configurations B^ decomposes Sometimes a configuration is defined differently, allowing points x e X to be included in 7 with repetitions; with such a definition a configuration is not a subset of X.
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A. M. Vershik, I. M. Gel'fand andM.L Graev
into a countable union of subsets that are transitive under Diff X: B x = U B*fP\ where B^ 0 is the collection of all «-point subsets in X. We note that B ^ consists of a single element — the empty set 0. For any subset Y C X with compact closure the space Tx splits into the product Tx = BY X TX\Y- Consequently, since By = U B(y \ we have Tx = LJ B(y *X r x x y , and all the subsets in this decomposition are invariant n>0
under Diff Y. 4. The space X°° and the topology in Tx. Let us consider the infinite oo
product X°° = II X/? X{ = X, furnished with the weak topology. The group S°° acts naturally on X°°. We define the subset r C I°° as the set of all sequences (xl9 . . . , xn, . . .) E I ° ° such that: 1) xt ¥= Xj when / ^= / and 2) thejequence x l 5 . . . , xni . . . has no accumulation points in X. The space X°° is invariant under the action of Diff X and S°°, and the S°°-orbit of any point of X°° is closed. There is a natural bijection X^/S00 -• F ^ . We introduce the corresponding quotient topology in Tx; this topology is Hausdorff and metrizable. Similarly, the bijections Xn/Sn -> B^ } , where Xn = {(*!, . . ., xn) 6 Z n ; ^ =^=^ when i ^=/}, and 5W acts on Xn as the permutation group of the coordinates, give the topology on B(Fw)0i = 1, 2, . . .) and hence on BY = LJ ( ) It is easy to see that F^ is, as a topological space, the projective limit of the spaces B^. Namely, F^ = lim (B^-, irKK>)9 where K runs through the open submanifolds in X with compact closures, and irKK>: BK -> BK{K' C K) is the restriction of the configuration 7 £ B^ to K\ that is, ITKK> y = y n Kf. 5. Quasi-invariant and ergodic measures. Let G be a group acting on a space y. A measure JU given on some G-invariant a-algebra in Y is said to be quasi-invariant under G if the inverse image of any measurable set of positive measure, under any transformation g: Y -» Y with g £ G, has positive measure. If /z is quasi-invariant, then the measures n and gii (where gfx is defined as the image of fi, that is, gfi(Q = Kg"1 Q) are equivalent; the density of gn with respect to /1 at a point y € 7 is denoted by
) . The class of a-finite measures equivalent to /z is called the type of A quasi-invariant measure jit in Y is said to be ergodic with respect to the action of G if every measurable set A C Y such that ix(gA A A) = 0 for
Representations of the group of diffeomorphisms
65
any g G G is either a null set or a set of full measure. We discuss measures on Fx, and other spaces connected with Fx, that are quasi-invariant under the action of Diff X, and we construct from these measures unitary representations of Diff X. 6. Measures in the configuration space Fx. We define,1 as usual, the a-algebra $ (F^) of Borel sets on Fx. Henceforth, when we talk of measures on F^,we mean2 complete, non-negative, Borel, normalized, countably-additive measures ju. Since the structure of a complete metric space can be introduced in Fx, for any Borel measure /z the space (Fx, 11) is (after taking the completion of the a-algebra $ (Tx) with respect to M) a Lebesgue space [11], and the technique of conditional decomposition (conditional measures, and so on) can be applied. The same applies to other spaces and fibre bundles over Fx that occur in this paper. Many measures on Fx arising for various reasons in statistical physics and probability theory turn out to be quasi-invariant and ergodic under Diff X. The following example is classical. POISSON MEASURE. Given any positive3 smooth measure m on a manifold X, we consider the union Ax - Bx U F ^ of all configurations on X. We define the measure of each subset { ) G Ax; | 7 O U | = n } by
where X > 0 is fixed. By Kolmogorov's theorem there exists a unique measure on $[(FX) defined by these conditions. It is called the Poisson measure with parameter X (associated with the measure m on X). Let us note the following important properties of the Poisson measure 11, which follow immediately from its definition. 1) When m (X) < °°, the measure fx is concentrated on the set Bx of finite configurations, and when m (X) = °°, it is concentrated on Fx. 2) Suppose that the manifold X = Xx U . . . U Xn is split arbitrarily into finitely many disjoint measurable subsets, that Ax = Ax X . . . X Ax is the corresponding decomposition of Ax into a direct product, and that jz,- is the projection of the Poisson measure JJL onto Ax. (i = 1, . . . , n). Then (x = nxX . . . X fin. This property of the Poisson measure is called infinite decomposability. 3) The Poisson measure is quasi-invariant under Diff X and invariant under the subgroup Diff(X, m) C Diff X of diffeomorphisms preserving m. Here, 1
Note that VL(rX) is a-generated by sets of the form CUn= {yerx;\y n U\= n), where [/runs over the compact sets in X (n = 0, 1,. . .)• In statistical physics a measure JU on Tx is usually called a state (see, for example, [12]) and in probability theory and the theory of mass observation it is usually called a point random process (see, for example, [17]). By a positive smooth measure we mean a measure with positive density at all points x GX.
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A. M. Vershik, I. M. Gel'fand and M. I. Graev
dutyiy) d\x.{y)
^ >
_ yy dm (ijr1*) 1 1 dm{x) x£y
(the product makes sense, because by the finiteness of \p9 almost all the factors are equal to 1). 4) If m(X) = oo5 then the Poisson measure JU is ergodic with respect to Diff X. Furthermore (see §4), if dim X > 1, then the Poisson measure is ergodic with respect to Diff (X, m). Any measure in Bx that is quasi-invariant under Diff X is equivalent to a sum of smooth positive measures on B*£\ In particular, any two quasiinvariant measures on B ^ are equivalent. Let us note that for any F C J , where Y is compact, the projection of any quasi-invariant measure in Tx onto By = LJ B& is non-zero for all n. § 1. The ring of representations of Diff X associated with the space of finite configurations
We discuss here the simplest class of representations of Diff X. These representations have finite functional dimension; from the point of view of orbit theory they have been discussed in detail by Kirillov [9]. 1. The representations Vp. We associate with each pair (n, p), where p is a unitary representation of the symmetric group Sn in a space W(n = 0, 1, . . . ) , a unitary representation Vp of Diff X. The construction of Vp is similar to Weyl's construction of the irreducible finite-dimensional representations of the general linear group. Given a positive smooth measure m on X, we define mn in Xn to be the product measure: mn = m X . . .X m. We consider the space L2 (Xn, W) of functions F on Xn with values in the representation space n
W of p such that 2
=J
i, . - . ,
xn)\\lvdm(xl)...dm(xn)
A unitary representation Un of Diff X is given on L2m (Xn, W) by the formula (1)
{Un 0 0 F)(xit...,Xn)=
where J^ix) = 2
n
dm(\p lJC) dm(x) dm(x)
[\ 4 / 2 (Xk)
. Let us denote by Hn
the subspace of functions
>p
F G L mn(X , W) such that F(xa(l), . . . ,x a(w) ) = p" 1 (a)F(x l5 . . . ,xn) for any a G 5W. It is obvious that Hn p is invariant under Diff X. We define the representation Vp of Diff X as the restriction of Un from / . * ( * » , W)to// f l > p .
Representations of the group of diffeomorphisms
67
In the particular case when p is the unit representation of Sn, then Vp acts by (1) on the space of scalar functions F(xly . . . , xn) that are symmetric in all the arguments. It is obvious that if m is replaced on X by any other smooth positive measure, each Vp is replaced by an equivalent representation. Le^t us construct another realization of Vp, which will be useful later on. Let Xn C Xn be the submanifold of points (xi9 . . . , *W)JE Xn with pairwise distinct coordinates. We consider the fibration p of Xn by the orbits of Sn, p: Xn -> B(^\ Note that p o \p = \p o p for any \p G Diff A\ Suppose that we are given any measurable cross section s\ B ^ -> Xn. Obviously, for any \jj G Diff X and 7 G B*^ the elements ^(^" 1 7) and i//~1(57) lie in the same fibre of p, and we define a function a on Diff I X B ^ with values in Sn by the formula sW'iy)
= [^(sy)]
o(\p, 7 ) , w h e r e 1 (x l 9 . . . , x n)o = (x O{1)9.
. . ,^ff(n)).
Let /i = pmrt be the projection onto B^* of the measure mn = m X . . . X m on Xn. We denote by ££(B^°, V) the space of functions F on B^z) with values in W such that \\F ||2 = We define the representation F^ of Diff X in LjCB^0, PV) by (2)
{
It is not difficult to check that this representation is equivalent to the one constructed earlier. To see this it is sufficient to consider the map s*: HntO-» Ll(B^\ W) induced by the cross section 5, ((s*F)(y) - F(sy)). It is easy to verify that s* is an isomorphism and that the operators Vp(\p) in Hn p go over under 5* to operators of the form (2). In the particular case when p is the unit representation of Sn, then Vp acts on L ^ ( B ^ ) according to the formula
2. Properties of the representations Vp. From the definition of Vp we obtain immediately the following result. PROPOSITION 1. For any representations pj and p 2 of Sn (n = 0, 1, . . .) there is an equivalence Vp**p* = F Pi © Vp*. DEFINITION (see [18]). The exterior product pxo p 2 of representations Pi of Sn and p 2 of Sn^ is the representation of Sn^n^ induced by the
a is a 1-cocycle of Diff X with values in the set of measurable maps B ^ l ) -> Sn (see Appendix 2).
68
A. M. Vershik, I. M. Gel'fand andM. I. Graev
representation px X p 2 of Sn^ X Sn^\ p{ o p 2 = Ind 5 Wl x ^2 (pt x p 2 ) . We are
assuming that Sn and £„ are embedded in Sn + n as the subgroups of permutations of 1, . . . , nx and of nx + 1, . . . , nn + n2, respectively. Note (see [18]) that exterior multiplication is commutative and associative. The following fact parallels standard results about representations of the classical groups in the Weyl realization. PROPOSITION 2. For any nlt n2 = 0, 1 , 2 , . . . and any representations Pi and p 2 of Sn and Sn , respectively, there is an equivalence ypx»fh ^
ypi 0
yp2t
COROLLARY. The set of representations Vp is closed under the operation of tensor multiplication. THEOREM 1. \) If p is an irreducible representation of Sn, then the representation Vp of Diff X is irreducible. 2) Two representations VPi and VPi, where px and p 2 are irreducible representations of Sn and Sn , respectively, are equivalent if and only if nx = n2 and px ~ p 2 . PROOF. We consider P^", where pn is the regular representation of Sn (n = 0, 1 , 2 , . . .). It is easy to see that VPn is equivalent to the representation in <§) L2m (X) given by (vPnwF)(Xi, ...,xn) = ft J ^ / 2 ^ ) , P ( r ^ ! , . . . , r 1 ^ ) . Results of Kirillov ([9], Theorem 4) imply that the VPn are pairwise disjoint and that the number of interlacings of VPn is «!, that is, equal to the number of interlacings of pn. Hence and from Proposition 1 the assertion of the theorem follows immediately. When dim X > 1, a stronger assertion is true, which we prove independently of the results in [9]. Namely, let m be an arbitrary smooth positive measure on X such that m(X) = «>. We denote by Diff(Z, m) the subgroup of diffeomorphisms \p G Diff X that leave m invariant. THEOREM 2. If dim X > 1, then the assertion of Theorem 1 is true for the restrictions of the Vp to Diff(X, m). The proof will depend on the following two assertions. LEMMA 1. For any natural number n and any set of distinct points xl9 . . . , xn in X there exist neighbourhoods Ox, . . . , On, corresponding to xi9 . . . , xn> with the following properties'. 1) the closure Oj of Ot is C°°-diffeomorphic to a disc, Ot dOj = 0 when 0 ^ 0 2) for any permutation (kx, . . . , kn) of 1, . . . , n there is a diffeomorphism \p G Diff(X, m) such that \p(Ot) = Ok.(i = 1, . . . , « ) .
PROOF. It is sufficient to consider the case when X is an open ball and m is the Lebesgue measure in X. In this case it is easy to check that for any xt and Xj, i =£ /, there is a diffeomorphism \pfj G Diff(T, m) with the following properties:
Representations of the group of diffeomorphisms
69
1) for any sufficiently small 8 > 0 we have ^ijD%i = Dzx., i>ipx- = D%p where Dex is a disc of radius e with centre at x G X\ 2) the diffeomorphism i//zy is the identity in neighbourhoods of xk for which k =fc /, /. Hence the assertion of the lemma follows immediately. LEMMA 2. For any^open connected submanifold Y C X with compact closure, the subspace L^ (T) C L^ (Y) of functions f on Y such that \ f(y)dm(y) = 0 is irreducible under the operators of the representation of V Diff(7, m): (U(^)f(y) = ft^y). PRCKDF. First we claim that for any non-trivial invariant subspace X C L^iY) and any neighbourhood 0 C Y, where 0 is C°°-diffeomorphic to a disc, there is a vector / G X , / =£ 0, such that supp / C O . For let us take an arbitrary vector / ( 1 ) G «£ , / (1 > =£ 0. Since Z ^ 1 ^ const on Y, there is a y0 G 7 such that / ( 1 ) ^ const in any neighbourhood 0' of >v Consequently, there exists a diffeomorphism \jj E Diff(y, m) such that supp i// C o ' and / (1) (i//y) ^ / ( 1 ) (y). We put f&y = fa)(\py)-f(1)(y). 2 (2) (2) Then /< > G X, / ¥= 0 and supp / C 0 ' . If the neighbourhood 0' is sufficiently small, then, by Lemma 1, there is a diffeomorphism i//i G Diff(y, m) with ^ 0 ' C 0 that carries / ( 2 ) into a vector / with supp / C 0. ^ Let us suppose that L^(Y) -%i® ^2»where «^x and %2 are non-zero invariant subspaces. We fix neighbourhoods 0 and 0 ' in 7 such that 0 and 0 ' are C°°-diffeomorphic to discs, 0 Pi 0 ' = 0, and m(0) = m(0'). From what has been proved, there are f( E X (, ft ^= 0, such that supp/;- C 0 (i = 1, 2). It is obvious that we can find a neighbourhood 0X C 0, where Ox is C°°-diffeomorphic to a disc, and a diffeomorphism \J/ G Diff(0, m) such
i
that \ fi(4iy)f2(y)dm(y)
=£ 0; without loss of generality we may assume
that \p = 1. For any € > 0 we can write_0 = O± U 0 8 U(0 \ ( 0 ! U 0 8 )), where O 8 is C°°-diffeomorphic to a disc, Ox O 0 e = 0, and m(0 \ (0! U 0 8 )) < 8 . It is not difficult to prove that there is a diffeomorphism \//e G Diff(T, m) that is the identity on 0 t and such that m(\pe Oe \ O') < e (see, for example, [3], Lemma 1.1). Since 0 O 0 ' = 0, we have
0\(0iU0 8 )
70
A. M. Vershik, I. M. Gel'fand andM. I. Graev
Consequently, because Xi and X* are orthogonal, \fi{y)hW)dm{y)+ j
fd^)h^)dm(y) +
Oe\$Q1Ot
01
Since the second and third terms in this equation can be made arbitrarily small, we have
\ fi(y)f2(y)dm(y)
- 0, which is a contradiction.
PROOF OF THEOREM 2. Let us realize the representation Vp = Vn>p of Diff X as acting on the subspace Hn p C L2m (Xn, W), where W is the space of the representation p of Sn ( for the definition of Hn , see §1.1). In this realization the operators of the representation of Diff(Z, m) have the following form: (y n ' P (^)F)(xXi . . ., xn) = Fiq^Xi, . . ., if-^n), if 6 Diff (X, JII). Let Ol, . . . , Ow be arbitrary disjoint neighbourhoods in X satisfying conditions 1 and 2 of Lemma 1. We denote by H~'p the sub" i , . . . , Ufi
space of functions of Hn p that are concentrated on U (0frl X . . . X 6kn ) C Xn where (A:l5 . . . , kn) runs over all permuta(klt...,kn)
tions of (1, . . . , « ) ; obviously there is a natural isomorphism
We consider the subspace Hno\l..,on^Zl(Ox)
0 . . . ® Li(On)
(g) PF,
2
where L^ (O,-) C L m (Ot) is the orthogonal complement to the subspace of contsants. From the definition it follows that H^fP Q is invariant under under the subgroup Go ^0 of diffeomorphisms \p E Diff(Z, m) such that MOi U . . . U ^ ) ' = O 1 U . . . U O r t . W e denote by V£p Q the P restriction of the representation Vn>p of GQ Q 0 to HQ> Note that the subgroup GJ? of diffeomorphisms n C Gn n ult...
,un
ulf...
,un
that are the identity on (9^ . . . , Ow acts trivially on Hg>p i* • • • '
by Lemma 1 the factor group Gn ult...
n
IG%
,un
n
Q
and that
/I
is isomorphic to the
ul,. .. , un
cross product of Diff(0!, m)X . . . X Diff(Ow, m) with *Sw. The assertion how follows easily from this and from Lemma 2. tP
The representations V^ Q ofGQ i » • o , where p runs over the inequivalent irreducible representations of Sn, are irreducible and pairwise i > • • • >
fi
• •
J
pj
Representations of the group of diffeomorphisms
71
inequivalent. We now claim that the representation V^>p Q ofGQ occurs in Q n>p V with multiplicity 1 and not at all in representations Vn>p , where p + p, nor in representations Vn>p>, where ri <^n. For let H' be the orthogonal complement to H^>p Q ini/ w p -We split H' into the sum of subspaces that are primary with respect to Diff(Olf m)X . . . X Diff(Ow, m). It is not difficult to see that in each of these subspaces at least one of the subgroups Diff(6^, m) (/= 1, . . . , « ) acts trivially. But the representation of each subgroup Diff(0,-, m) in HQP O is a multiple of a non-trivial irreducible representation. Consequently, the representations V£>p Q are not contained in H', nor for the same reason in Hn> p , n < n. From the properties of V^>p Q we have just established it follows immediately that the representations Vp of Diff(X m) are pairwise inequivalent. We claim that they are irreducible. Let X C Hn p be a subspace invariant under Diff(X, m), X =£ 0. Then for any collection Ou . . . , 0W of disjoint neighbourhoods satisfying conditions 1 and 2 of Lemma 1 either H"tP C ^ , or ^
H
ult...
,un
o[P...,o n X = 0. It is not difficult to see that the spaces Hg'p 0 generate Hn p, therefore, H£p Q C X for some collection Ou . . . , On. But then, by Lemma 1, X contains the whole of H^jp and hence coincides with Hn . The theorem is now proved. 0 REMARK. Let us denote by 21 the group of all (classes of coinciding mod 0) invertible measurable transformations of X that preserve the measure m (the dimension of X is arbitrary); we furnish W with the weak topology. The representation Vp of Diff(X m) C 21 extends naturally to a representation of 21 and the resulting representation Vp of 21 is continuous in the weak topology. It is easy to show that in the weak topology Diff(X, m), for dim X > 1, is everywhere dense in 21. This makes it possible to prove Theorem 2 anew, reducing its proof to those of the analogous assertions for 21, which are easily verified. On the other hand, this path enables us to establish Theorem 1 for any weakly dense subgroup of 21, that is, to prove the following proposition. THEOREM 3. The assertions of Theorem 1 are true for the restrictions of the representations Vp of 21 to any subgroup G C 21 that is weakly dense in 21. 3. The representation ring 'M . We consider the free module !J9 over Z on the set of all pairwise inequivalent irreducible representations Vp of Diff X as basis. By the propositions in §1.2, the tensor product j/Pi 0 yPi o f irreducible representations VPi and Vp* decomposes into a sum of irreducible representations Vp and therefore is an element of Jl'.
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In this way a ring structure is defined in 91, where multiplication is the tensor product. Let us introduce another ring R(S) associated with the representations of the symmetric groups Sn (see [18]). We denote by R(Sn) the free module over Z on the set of pairwise inequivalent irreducible representations of Sn (n = 0, 1, 2, . . . ) as basis (where R(S0) - Z). We consider the Z-module R(S) = © R(Sn) and give a ring structure to R(S) by defining multiplication n—0
as the exterior product. From the propositions in §1.2 we obtain immediately the following result. THEOREM 4. The ring ffi generated by the representations Vp of Diff X is isomorphic to R(S). For the map p -> Vp, where p runs over the representations of Sn (n = 0, 1, . . . ) extends to a ring isomorphism R(S) -*~ ffi. REMARK. There exists a natural ring isomorphism 0: R(S) -+• Zlfli, a 2 , . . . ] ,
where an is the n-th elementary symmetric function in an infinite number of unknowns, n - 1, 2, . . . ; for the definition of 6 see, for example, [18]. By the theorem we have proved, there is a ring isomorphism X -> Z[al9 a2, . • • ], where to each representation V9 there corresponds the symmetric function 0(p). These symmetric functions in an infinite number of unknowns have the usual properties of characters: each representation Vp is uniquely determined by its symmetric function, on adding two representations their corresponding symmetric functions are added, and on taking the tensor product they are multiplied. §2. Quasi-invariant measures in the space of infinite configurations
Before turning to the discussion of representations of Diff X associated with the space of infinite configurations Tx, we ought first of all to study in detail measures in Vx, and in fibrations over it, that are quasi-invariant under Diff X. We have already recalled that there are many such measures with various properties (see §0.6); these measures arise (in another connection) in statistical physics, probability theory, and elsewhere. The ergodic theory for infinite dimensional groups differs in many ways from the theory for locally compact groups (see, for example, [6]). In particular, the action of Diff X in Yx is such that in Tx there is no quasi-invariant measure that is concentrated on a single orbit.1 In addition, care is needed because an infinite-dimensional group can act transitively, but not ergodically, on an infinite-dimensional space [13]. This explains the I
For locally compact groups such a measure exists and is equivalent to the transform of the Haar measure on the group.
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somewhat lengthy proof of the lemma in §2.1, which at first glance would appear obvious. 1. Lemma on quasi-invariant measures on B ^ X Yx y.* LEMMA 1. Let Y C X be a connected open submanifold with compact closure, let fin be a measure on B^^X Fx_Y that is quasi-invariant under the subgroup Diff Y, and let iin and ju^' be the projections of (xn onto B ^ and ^x_r respectively. Then fxn is equivalent to n'n X JUJJ (n = 0, 1, 2, . . . ). REMARK. If ixn is the restriction to B(yw) X Tx_Y of a fixed quasi-invariant measure JJL on Tx, then the measures ju^' on F y _ r are, generally speaking, not equivalent. It is easy to show that the equivalence of the measures fx'n' on rx_Y {n = 0, 1 , 2 , . . . ) corresponds precisely to the equivalence of the measures [i and ju' X //' on Tx, where n' and /z" are the projections of ix onto BY and I ^ y 1 First we prove the following geometrically obvious proposition.
PROPOSITION \. In Diff Y there is a countable set of one-parameter subgroups Gl such that the group G C Diff Y generated by them acts transitively in B ^ (n = 1, 2, . . . ) . PROOF. We suppose first that dim Y = 1. We specify in Y a countable basis of neighbourhoods Ur, Ur C Y, that are diffeomorphic to R1. We fix for each r a diffeomorphism $/. R1 ->• Ur. Under $r the group of translations on R1 goes over into a one-parameter group of diffeomorphisms x -+ ft(x) on Ur (-°° < t < °°), which acts transitively on Ur. The map <pr can always be chosen so that the diffeomorphisms x -* ft(x) on Ur extend trivially to a diffeomorphism on the whole of Y. It is not difficult to check that the sequence of groups {Gt} constructed in this way satisfies the required condition. Now let dim Y = p, where p > 1. We specify in Rp a countable set of one-parameter subgroups Hx C Diff Rp such that the group generated by them acts transitively in Rp; the construction of such a family presents no difficulty. Now we take a countable basis of neighbourhoods Ur in Y, diffeomorphic to R p , and fix diffeomorphisms \pr\ Rp ->- Ur. Let us denote by Glr the image of Hj under <pr. The elements of Glr can be extended trivially to diffeomorphisms over the whole of Y, and so Glr can be regarded as a one-parameter subgroup of Diff Y. For a fixed r, the subgroups Glr generate a group which acts transitively in Ur and leaves the points of Y \ Ur fixed. Hence it is obvious that the group G C Diff Y generated by all the Glr acts ^-transitively in Y (n = 1, 2, . . . ), and Proposition 1 is proved. The following proposition is concerned with the theory of measurable currents of a quasi-invariant measure. When ju is the Poisson measure, the equivalence n ~ ju' X n" is a direct consequence of the property of being infinitely decomposable. The assertion of the lemma in this case is trivial.
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PROPOSITION 2. Suppose that R1 acts measurably1 on the Lebesgue space (X, n) with quasi-invariant measure JU and that f is a measurable partitioning of (X, /x) that is fixed mod 0 under R1. Then for almost all C G f the conditional measures iic on C are quasi-invariant under R1. PROOF. For any t G R and C G f we put q(t, C) = i A
where the inf is taken over all subsets A C C, with MCC4) = 1- We also use the notation qo(t, C) = q(T, C)q(-t, C). Obviously, the condition yC ^ Tl[ic is equivalent to qo(t, C) = 1. Since the action of R1 on (X, JU) is measurable and f is a measurable partitioning, q{t, Q, and hence also qo(t, C), are measurable as functions on R1 X JT (Xt = X/Z). Since JJL is quasi-invariant, for any fixed ( G R 1 we have ixc ~ Tx\f for almost all C E f with respect to the measure JU? on X^ (fx^ is the projection of pi); hence qo(t, Q = 1 almost everywhere with respect to /i? on X^. Hence, by Fubini's theorem for (R1 X X^, m X )uf), where m is the Lebesgue measure on R1, for almost all C G f with respect to pif we have: m { t e R1; (jfo(r, C) =£ 1 } = 0. On the other hand, the set of t G R1 for which JUC ~ ^rMC (for a fixed C) forms a group. Thus, for almost all C G f the set { t E.Rl\qQ{t, C)= \) is a subgroup of R1 of full Lebesgue measure. But every subgroup of a locally compact group with full Haar measure coincides with the whole group [4]. Consequently, for almost all C G f { t G R 1 ; qo(t, Q = 1 } = R 1 , that is, for almost all C G f the measure JJLC is quasi-invariant under the action of R1, and Proposition 2 is proved. PROOF OF LEMMA 1. Let {GJ be a countable set of one-parameter subgroups of Diff Y such that the group G generated by them acts transitively in B ( ^ ; such a set exists by Proposition 1. Since Gt = R1, it follows from Proposition 2 that for almost all, (in the sense of Mn')> configurations 7 G Tx Y the conditional measure JU^ on B^ is quasi-invariant under each Gl (/ = 1, 2, . . . ), hence also under the whole group G generated by them. On the other hand, the measures on B(rw) that are quasi-invariant under G are all equivalent to each other, and consequently to fi'n. For on B^\ as on every smooth manifold, there is, up to equivalence, a unique measure that is quasi-invariant under a group of diffeomorphisms acting transitively, namely, the smooth measure with everywhere positive density. That is the map R1 X X -* X ((g, x) -* gx) is measurable as a map between spaces with measures m X M and /u respectively, where m is the Lebesgue measure in R 1 .
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Thus, for almost all 7 G TX_Y ( m * ne s e n s e °f Mp the conditional measure y?n on B ^ is equivalent to ixn, and the lemma is proved. 2. Measurable indexings in Tx. We say that / is an indexing in Tx if for each configuration 7 6 ^ there is a bijective map i(y, -): y ^ N, N= {1, 2, . . .}. We denote by Fx 1 the subset of elements (7, x) G F^ X X such that x G 7, and we associate with each indexing / a bijective map Fx 1 ->• F^ X iV, defined by (7, x) -* (7, /(% x)). If this map is measurable in both directions (with respect to Borel a-algebras on F^ l and F^ X AO, then the indexing / is called measurable.1 Let / be a measurable indexing. We introduce a sequence of measurable maps ak\ Tx ->• X (k = 1, 2, . . . ) defined by the conditions: ak(y) G 7, /(7, ^ ( 7 ) ) = k (that is, 0^(7) is the k-th element of the configuration 7). We associate with / a cross section s: Fx -> X°, defined by 5(7) = (a 1(7), . . . , an(y), . . . ). It is not difficult to verify that the set 5 Fjf is measurable and that the bijective map F^ -> sFx is measurable in both directions. ^ For any ^ G Diff X and 7 G F ^ , the elements s^'1 y) e X°° and ijj'1 (sy) G X°° belong to the same ^-orbit in ^°°. We define a map a: Diff XX Tx -> 5°° by ^ i / / " ^ ) = [i//"1 (57)] a(0, 7); the notation here means (xl9 . . . , xn, . . . ) o = (x a(1) , . . . , x a(w j, . . . ). Let us now introduce the idea of an admissible indexing. We are given an increasing sequence X{ C . . . C Xk C . . . of connected open subsets with compact closures such that X = U A^. DEFINITION. We say that a measurable indexing / is admissible (with respect to the given sequence Xx C . . . C Xn C . . . ) if the map a: Diff XX Fx -> *S°° defined by it satisfies the following condition: if supp \// C Xk and | 7 O Xk \ = n, then o(\p, 7) G ^ (k = 1, 2, . . . ; * = 0, 1, . . . ) . In particular, o(\p, 7) G 5^ for any \// G Diff X and 7 G F x . It is not difficult to construct examples of admissible indexings. For example, the following indexing, which was proved to be measurable in [17], is admissible. Let a continuous metric be given on X. With each positive integer k we associate a covering (Xkl)l= ^ 2j of X by disjoint measurable subsets with diameters not exceeding \/k, satisfying the following two conditions. 1) the partitioning X = Xkl U . . . U XkiU . . . is a refinement of X = XlU(X2\X1)U .. .U(Xn\Xn^)U .. . ; 2) Each set Xn is covered by finitely many of the sets Xkl. 1
If a measure JI is given in Tx, then indexings need be given only on subsets of full measure in r x , and we make no distinction between indexings that coincide mod 0.
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It is obvious that such a covering exists. We number its elements so that if Xki C Xn and Xkj C X \ Xn, then i < j (n = 1, 2, . . . ). For any x E X and ^ E i V w e put Z^OO = /, if x E X^. The correspondence x -* (/i(x), . . . , fkix), . . . ) is a morphism from X to the set of all sequences of positive integers. We define an ordering in X by putting x ' < x " if ( / A x 1 ) , .. . f k ( x ' ) 9 .. . ) < ( f l ( x " ) , . . . , f k ( x n ) , . . . ) i n t h e lexicographic ordering. For any 7 E Tx and x E 7, the_set { x' E 7 ^ ' -< #} is finite, because it is contained in the compact set I n U . . . U XXj^xy Consequently, for any 7 E Fx, the set of elements x E 7 is a sequence with respect to the ordering introduced in X; we denote by z(7» x) the number of elements x'^y in this sequence. The map (7, x) -> /(% x) so constructed is a measurable indexing (see [17]). It is not difficult to prove that it is also admissible. 3. Convolution of measures. DEFINITION. The convolution Mi * M2 (see, for example, [17]) of two measures Mi and ji2 on the space of all configurations Ax = F ^ U Bx is defined as the image of the product measure Mi x M2 o n &x x ^x u n d e r the map ( 7 i , y2) ^ Ti u 72REMARK. This definition agrees with the usual definition for the convolution of two measures in the space $F (X) of generalized functions on X(AX is embedded in 2F{X) by 7 -* S 5 X ), because the union of (disjoint) configurations corresponds to the sum of their images in IF (X). It is obvious that ^(Mi* M2) = ^Mi * ^M2 for any \jj E Diff X, where \j/[i is the image of M under the diffeomorphism \p. Hence the convolution of quasi-invariant {under Diff X) measures is itself quasi-invariant. Note that / / Mi and \i2 are Poisson measures with parameters \x and X2, respectively, then their convolution Mi * M2 is the Poisson measure with parameter Xj + X2. (This fact follows easily from the definition of the Poisson measure). Later on we shall be interested in the case when one of the factors is a quasi-invariant measure concentrated on Fx, and the second is a smooth positive measure mn concentrated on B^x\n = 1, 2, . . . ). Since all smooth positive measures mn on B ^ } are equivalent, the type of the measure H*mn depends only on the type of M and on n. Let us agree to call the type of JJL * mn on Yx the n-point augmentation of M and to denote it by n o M- Thus, with each measure JJL on Tx there is associated a sequence of measures 0 o M ~ M> 1 °M> • • • »n ° M> • • • defined up to equivalence. Note that nxo (« 2 ° M) ^ («i + ^ 2 ) ° M f ° r a n Y ^i a n ( i Here we establish the following properties of the operation °. 1) For any quasi-invariant measure M on Tx there exists a quasi-invariant measure (if such that 1 ° M' ~ M2) / / (2 measure M 0/7 F ^ w ergo die, then 1 ° M W 0&O ergodic.
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To prove this we give an admissible indexing / on Fx and let s: Fx -> X°° be the cross section defined by this indexing (see §2.2). We denote by JTS the image of Fx under s and by A^ the minimal ^-invariant subset of X00 containing Y^ obviously, As is the disjoint union As = U Yso. Since Ys C X°° is measurable and S^ countable, As is a measurable subset of X°°. Since the indexing i is admissible, it follows that As is invariant under Diff X. Let jit be a measure on Fx that is quasi-invariant under Diff X. Let c be an arbitrary positive function on S^ such that 2 c(o) = 1; we introduce o£5M
a measure M on X°° by the formula: ? = S c(a)(sfi)o, where (5/x)a is the image of // under the map y -* (.?7)a. In other words, for any measurable subset A C Z°° (1) ? ( 4 ) = 2 c(a)fx[p(^n^scr)] where p is the projection X°° -^ Fx. Obviously, M(A S ) = 1. Note that the choice of the positive function c on S^ does not play a role in defining £, because the measures on X°° constructed from two such functions are equivalent. From the definition of M it follows easily that: a) pfx = ix where pjl is^ the projection of £ onto Fx. b) the measure £ on X°° is quasi-invariant under both Diff X and S^. We cite without proof two further simple assertions. PROPOSITION 3. If a normalized measure fxx on X°° is quasi-invariant under Diff X and if p\x.\ = M and Mi(A5) = 1, then Mi ~ M-
PROPOSITION A. If a measure ii in Fx is ergodic, then ]1 is also ergodic with respect to Diff X. oo
Let us decompose the space X°° = U Xi9 where Xt = X, into the direct OO
i
=
1
r^s
^/
product X" = X X n Xj and consider the induced map h = X X X°° -* X°° / =2
(that is, /i(x; {xft} ^ =1 ) = ( {xfh} %=l;x[j=jc, x'k =xk_x when k > 1)). PROPOSITION 5. /*(m X JLI) ~ m * jit, where m is an arbitrary smooth positive measure on X. PROOF. Consider the diagram
x x rY — - > rY
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A. M. Vershik, I. M. Gel*fand andM. I. Graev
where px = Id X p, h(x,^y) = 7 U {x}. Obviously, this is commutative, and (p o h) (m X /I) = (h ° px) (m X £) = m * fi. Further, the measure him X p) is quasi-invariant and concentrated on As (since the indexing / is admissible, the point h(x, 57) belongs to the same 5^-orbit^as 5(7 U {x} )). Consequently, by Proposition 3, him X pt) ~ m * /x. The proof of the following assertion is similar to that of Lemma 1 in §2.1. PROPOSITION 6. Every measure /x in X°° that is quasi-invariant under Diff X is equivalent\Jo the product mn X \xn of its projections in the factorization X°° = Xn X X™+1; moreover, mn is equivalent to a positive smooth measure on Xn, and fxn is quasi-invariant under Diff X. COROLLARY. A quasi-invariant measure ]x in X°° is ergodic if and only if it is regular (that is, satisfies the 0 — 1 law). PROOF OF PROPERTY 1). Let n be a quasi-invariant measure on Fx. By Proposition 6, JJL ~ him X p^), where m is a smooth positive measure on X, /*! is a quasi-invariant measure on X°°, and h: X X X°° ->• X°° is the map induced by the direct product (see above). Since / is admissible, /xl5 like JJL, is concentrated on As; consequently, by Proposition 3, /xi ~ /x' is a quasiinvariant measure in P^. By Proposition 5, /I ~ /z(m X JU') ~ m * JU'; consequently, /x ~ m * ju'> as required. PROOF OF PROPERTY 2). Ifjhe measure /x in F^ is ergodic, then by Proposition 4, the measure /x in X°° is ergodic; consequently, by the corollary to Proposition 6, £ is regular. Obviously, m X H is then also regular and therefore ergodic. Consequently, the measure m * ju ~ him X /I) is also ergodic and hence, so is its projection m * ju. DEFINITION. We say that a quasi-invariant measure \x is saturated if 1 o /x ~ jLt (and consequently, « o /x ~ /x for any n). It is not difficult to verify that the Poisson measure is saturated (this follows from the property of being infinitely decomposable). We now give a criterion for a measure fx to be saturated. The map T: X00 -• X00, defined by (Tx)t = xi+l(i ^ 1 , 2, . . .) is called left translation in X°°. Obviously, the subset X°° is ^-invariant. PROPOSITION 7. For a quasi-invariant measure on^Tx to be saturated it is necessary and sufficient that the measure £ on X00 corresponding to it (defined by means of a fixed admissible indexing) is quasi-invariant under the left translation T. PROOF. From the definition of the left translation T it^follows that pt ~ h(m X J/x). On the other hand, by Proposition 3, 1 ° JJL ~ h(m X /x). Hence it is obvious that the condition 1 ° ju- ~ /x is equivalent to ju ~ T]x. An example of a non-saturated measure JU will be given in Appendix 1. 4. The space Yx n and Campbell's measure on P j n . We consider the Cartesian product Yx X Xn (n = 1, 2, . . . ) and denote by Tx n the set of elements (7; xu . . . , xn) G F x X Xn, where 7 G r j ( x,- G X, such that xt G 7 (/ = ! , . . . , « ) and xz- =£ x;- when / =£ /. Further, we put Tx 0 = Fy.
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Obviously, Tx n is closed in Tx X Xn. Now Fx n can be regarded as a fibre space, TT: VX n -> F ^ , whose fibre over a point 7 E F ^ is the collection of all ordered «-point subsets in 7. Let us denote by Wn the cr-algebra of all Borel sets in Tx n. We associate with each subset C G I n a function on F ^ : ^ ( 7 ) = {the number of points (x l 9 . . . , xn) E Xn such that (7: xl9 . . . , *„) E C } . From the continuity of IT it follows that *>c is a Borel function. DEFINITION. Let pi be a measure on F ^ . The Campbell measure on TXn associated with /x is the measure # on 2In defined by
= rJ A Campbell measure /I induces on the fibres of the fibration TT: Tx n -* Fx a uniform measure, which is 1 at each point of the fibre. We define in F ^ n the actions of the groups Diff X and Sn: a: (Y; xr,
. . ., xn) - * (y; xo{1),
. . .,
xo(n)).
Obviously, \p and a are continuous and \// o
c=(v; xi, . . ., xn)
Now let / be a measurable indexing in Fx. We denote by Nn the set of all ^-tuples of natural numbers (il9 . . . , /„), where ip =£ iq when p ^ q (p, q = 1, . . . , « ) . We define a map (2) by (Y; This map is bijective, measurable in both directions, and carries^ the Campbell measure M £ n Fx n into the measure /z X v on F ^ X Nn, where v is the measure on Nn 9 that is equal to 1 at each point on Nn. Thus, the space (Tx n, JU) can be identified with ( F ^ X TV", pi X v). Under this identification the actions of Sn and Diff X go over from ^x,n t 0 rx x ^ 1 : Tt i s n o t difficult to verify that the action of these groups on Tx X Nn are given by: a: (Y, a) H^ (Y, aa), op: (Y, a) »-> (tp-^, a(a|),
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A. M. Vershik, I. M. Gel'fand and M. I. Graev
where a = (il9
. . . , / „ ) ; aa = ( i a ( 1 ) , . . . , ia(n))
a G
S n;
aa = (aO'i), . . . , o(in)) o G S°°; and a(#, 7) is the function on Diff X X Fx with values in S°° defined by i (see §2.2). 5. The map Fx X Xn -> Fx n. Let us consider the spaces Yx X Xn and Fx n together with their a-algebras of Borel subsets (see §2.4). Let M be a quasi-invariant measure on Fx, and mn a smooth, positive measure on Xn. In Tx X Xn we specify the measure /i X mn and in Fx n the Campbell measure fT°~ix corresponding to n o ^ ~ M * mn on Yx. A map a: F^ X X" -> Fx n is given by the following formula: a(v; * l t . . ., *„) == (v U {«i, • • ., *n}; *i, • • ., xn); a is taken to be defined on the subset of elements (7: * ! , . . . , x rt ) G Tx X Xn for which 7 O {;cl3 . . . , xn } = 0; it is not difficult to verify that this subset and its image in Tx n are sets of full measure. Nor is it difficult to check that a is measurable in both directions and commutes with the action of Diff X on both Tx X Xn and rx n. THEOREM. The image a(ju X mn) of the measure \x X mn on Tx X Xn under a is equivalent to the Campbell measure n © JU. PROOF. We carry out the proof for the case n = 1; the arguments for arbitrary n are similar. We define maps 0^ : Tx X X -+ Tx and a2: Tx x -> Fx by «i(7» *) = 7 u {^} , ^2(7. ^) = 7- It is obvious that the following diagram commutes:
\ T The image of the measure fxXmonTxXX under e^ = OL2OL is 1 ° /xHence it follows that a(/x X w) •< 1 © pi. It remains to prove that a(ju X m) > 1 © JU. First we construct a measurable indexing in Tx in the following way. We fix a continuous metric p in X and a point x0 G X. We consider the subset of IV {Y £ IY, p(x0, x) =^=p(^0, a;') for any x ¥= x' in 7} (3) and the preimage of (3) under a 2 in F^ x. As is easy to see, these subsets are of full measure in F^ and F^ x, respectively, and it is to be understood in what follows that it is these subsets which are meant by Tx and We prescribe an ordering on each configuration 7 G Fx, putting x < x' for any x, x G 7, if p(x 0, x) < p(x 0 , x'). For any (7, x) G Fx x we denote by i(y, x) the number of the element x G 7, as given by the
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ordering on 7. It is not difficult to see that / is a measurable indexing. We now introduce a sequence of measurable maps ak: Tx -+ X (i = 1 , 2 , . . . ) , where ak(>y) is the A>th element in the configuration 7. Now let C C Tx x be an arbitrary measurable set of positive Campbell measure: f^jxiQ > 0; we have to prove that then (ju X m){a~l C) > 0. Since Tx ± splits into the countable union of subsets {(7, ak(y)); 7 £ Tx} (k = 1, 2, . . . ), we may assume without loss of generality that C C {(7, ak(y))} for some k. We introduce the notation Cn = {(y, x) 6 Tx X X;y^U {x} e a2C, an(y U {*})) = x} (n = 1, 2, . . . Note that the condition 1 o /i(C) > 0 is equivalent to (4)
(\JL X m){(7', x)£Tx
X X; y' [j {x) £ a2C} > 0.
In its turn (4) is equivalent to the existence of a natural number /, for w h i c h (ju X m)
{ ( 7 ' , x)
G Tx
X X,
7 ' U {x}
E a2C,
a^y'U
{x})=x}>0,
that is, (M X m) (C/) > 0. On the other hand, since C C {(7, ^ ( 7 ) ) } , it follows that OL-^C = {(y\ x) 6 Tx x X; 7' U W 6 a a C, afc(v' U W ) = ^ } , that is, a~lC = C^. Thus, the proof of the lemma reduces to proving the following assertion: for any natural numbers k and I the conditions (ju X m) {Ck) > 0 and (ju X m) (Cz) > 0 are equivalent. Let us prove this assertion. We write Xr - {x E X\ p(x0, x) < r}, where r > 0 is an arbitrary rational number (T o = 0). We fix a positive integer n > max(A:, /) and introduce the following subsets in Tx X X: Vl)
| = 1 when i ^ p ,
where rx, . . . , rn are rational numbers such that 0 = r0 < rx < . . . < rn (p = 1, . . . , « ) . It is obvious that the sets U, , cover C ; consequently, the condition (fx X m) (C,) > 0 amounts to the existence of some U such that 1» • • • '
n
(5) (^ x m) {(/, a;) 6 ^ l f ..., rn; 7' U W 6 «2C, a, (7' U {x}) = We n o t e t h a t Ulr M»
• • • >rn
C (B ( F W ~ 1} X L A r
) X I ^n
V x x
a n d m a k e use of t h e
fact
that by Lemma 1 of §2.1 the restriction /zw-i o f M to B^~1} X Txxx r
is equivalent to the product m X . . . X m X projection of nn_l onto ^X\X following condition:
JLIJ2'_1
, where
n IJL'^
r
n
is the
. So we obtain that (5) is equivalent to the
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A. M. Vershik, I. M. Gel'fand and M. I. Graev
(6)
(m x . . . X m X \in-i) {(*!, • • ., xn\ y') 6 l n X
Thus, (JU X m) (C/) > 0 amounts to the condition that (6) is satisfied for some collection of rational numbers 0 = r0 < rx < . . . < rn^ But from the same arguments it follows that the condition (JU X m) (Ck)> 0 also is equivalent to (6). Consequently, the conditions (/x X m) (C{) > 0 and (JU X m) (Ck) > 0 are equivalent, as required. §3. Representations of Diff X defined by quasi-invariant measures in the space of infinite configurations (elementary representations)
1. Definition of elementary representations. Let fx be a quasi-invariant measure in the space of infinite configurations Tx. We introduce a series of unitary representations of Diff X associated with JU. First we consider the space L^(TX). In it a unitary representation U^ of Diff X is defined by1
We do not study the properties of U^ separately, but examine straightaway a wider class — the elementary representations. For the Poisson measure JU these representations are additive generators in the representation ring determined by U^ (see §4). Although the proof of the irreducibility and other properties of U^ are simpler than in the general case, we prefer to study all the elementary representations simultaneously. DEFINITION. A representation of Diff X is called elementary if it is of the form U^ 0 Vp, where U^ is the representation in L^(TX) given by (1), and Vp is the representation defined in §1. Thus, each elementary representation is given by a quasi-invariant measure ix on F^ and a representation p of the symmetric group Sn (n = 0, 1 , 2 , . . . ) . THEOREM \. If ix is an ergodic measure on Tx and p is an irreducible representation of Sn, then the elementary representation U^ ® Vp of Diff X is irreducible. REMARK 1. The converse assertion is obvious. REMARK 2. Another convenient formulation of Theorem 1 is: When ix is ergodic, then U^ is absolutely irreducible, that is, remains irreducible after taking the tensor product with any irreducible representation Vp. Essentially, the whole of §3 is devoted to a proof of Theorem 1. But first we construct some other useful realizations of elementary representations. If ju is concentrated not on rx, but on B^\ then (1) gives the representation VPn (see § 1), where p° is the unit representation of Sn.
Representations of the group of diffeomorphisms
83
2. The representations Up. Let p be a unitary representation of Sn in a space W (n = 0, 1, . . . ). We consider the space L^(TX „, W) of functions F on Tx n with values in W such that 1
I* 11= J
\F(c)\\lrdi!i(c)
x
X,n
ft is the Campbell measure on Tx n corresponding to the measure ix on Tx (see §2.4). A unitary representation U of Diff X is given in L^{TX n, W) by
We denote by H^ n p the subspace of functions F G L~ (F^ w, JV) such that F(7, x a(1) , . . . , x a(w) ) = p" 1 (a)F(7, x x , . . . , xn) for any o E Sn. Obviously, H^ n is invariant under Diff X DEFINITION. The restriction of the representation U of Diff X from *
(Tx X Xn, fi X mn) -> (Tx,n,
/
7^1)
(n o ju is the Campbell measure on F^ rt corresponding to /? ° JU), which commutes with the action of Diff X. Let us consider the isomorphism of Hilbert spaces Ll^{YXn, W) -> L^(F X ) ® ^mn(^"> ^ ) induced by (2), where W is the space of the representation p of Sn. It is easy to verify that the image of #M „ p C L ? _ ( p ^ ^ , ^ ) i s L ^ ( F x ) ® # WfP , where //w p C L^ (Xn, W) is the subspace of the representation Vp (see §1) and that the operators Uno^) in H^np go over to U^)
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A. M. Vershik, I. M. Gel'fand andM. I. Graev
up s u^
II^ll2^ 2 We consider in l2(Nn, W) the subspace Hp of all functions
Xi c= . . . c= Xh a . . .
be an increasing sequence of open connected subsets with compact closures such that X = U L . it
We fix an admissible indexing / (with respect to (5)); let a(i//, 7) be the
Representations of the group of diffeomorphisms
85
map Diff XX Tx -> S^ defined by it. By Lemma 1, the representation
o€iSn. The representation operators are given by (3). We decompose L^(TX) <S> H into a direct sum of subspaces that are primary with respect to the subgroups Diff Xk C Diff X (that is, those that are the identity on X \ Xk) (k = 1, 2, . . . ). First we decompose Fx into a countable union of spaces that are invariant under Diff Xk: (6)
Tx = U B
where B$ is the space of r-point subsets in Xk. It follows from (6) that
LI (Tx) 0 H = J>Q (L ^ ( B ^ x r^NZfe) 0 #), where /xr is the restriction of JU to the subset B ^ X T]CKX C T^. It remains to decompose each term in this sum into a direct sum of invariant subspaces that are primary with respect to Diff Xk. Next we split H into the direct sum of subspaces that are primary with respect to the symmetric group Sr C S^ . This decomposition can be presented in the following way: H = ® (Wlr 0 C}), where Wlr are the spaces in which the irreducible and pairwise inequivalent representations p\ of Sr act; C\ is the space on which Sr acts trivially. As a result, we obtain a decomposition into the direct sum:
All the terms of this decomposition are invariant under Diff X. For since the indexing is admissible, it follows from \p E Diff Xk and ) G B ^ X 1 ] ^ ^ that cr(tf/, 7) G ^ . We claim that these subspaces are primary and disjoint. We denote by \ir and JJL" the projections of nr onto B ^ and Fxxx , (\ respectively. By Lemma 1 of §2, the measure \xr on B^} X Fx^x is equivalent to the product (i'r X fi" of \xr and 11". Consequently, there is an isomorphism xr: defined by
L\T
(B^ x
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A. M. Vershik, L M. Gel'fand and M. I. Graev
We denote by the same letter the trivial extension of rr to an isomorphism We denote the elements of B*J? by 7 (r) and define a map A
k
or\ Diff J ^ X B<£} -> S, by or(\p, y(r)) = o(\jj, y), where K
y G B<£} X r ^ ^ , 7 O Xfc = 7 (r) . This is well defined, because if 7 O I f c = 7' D I f c , then o(\p, y) = o(\p9 y'). Immediately from the definition of r r we derive the next result. LEMMA 2. Under the isomorphism Tr the operators U(\l/) = tf£(\(0, ^ G Diff Xfc ^o over to operators TrU(\l/)T^ of the following form: TrU(\p)r~1 = Ulr(\jj) (8) /, where I is the unit operator in L\ (Txxx) <8> C\ and £//(0) is an operator in L% (B£> ) 0 W// in the space of functions on B ^ with values in Wlr defined by The representation Uj. of Diff X^ is equivalent to. VPr defined in §1.1. By Proposition 3 of §1.2, all the representations VPr of Diff Xk are irreducible and mutually pairwise inequivalent. Therefore Lemma 2 has the following corollaries. COROLLARY 1. The representations of Diff Xk in the subspaces L x r lr^xk * \ ; ^ ® Wr ® C'r (r = 0, 1, 2, . . . \ i = 1, 2, . . .) are primary and disjoint. COROLLARY 2. Any invariant subspace under Diff Xk X\, r a (L^(B^) (g) Wl) 0 (Ll;(Tx^xk)
0 C*)
is of the form XlK T = (L2^ (B^ ) 0 Wlr) 0 D/, w/z^e COROLLARY 3. >4w^ subspace X C I ^ ( F Z ) 0 ^ that is invariant under Diff Xk splits into the direct sum X = © ^ i , r , vv/zere r, i
(7) Xlr = X() (Llr ( B ^ x rZNxA)
IT
K
(Bx ) the space of essentially bounded functions of B^ with respect
v-Xfr
k
k
Representations of the group of diffeomorphisms
to the measure \ix
k
87
(k = 1, 2, . . . ). We identify each space L~
»xk
(Bx ) k
with its image under the natural map L~ (Bx ) -* L~(TX) (that is, the space of essentially bounded functions on Fx that are constant on the fibres of the fibration FxY -+ BxY ). We consider the union U L~ v(BxY ). k
k
»xk
k
It is obvious that for any / G ^ ( r x )
**k
sufficient to check that X is invariant under multiplication by elements o f l ^ (BXk)(k= 1,2,...). We fix k and denote by L~>n (B^ } ) the subspace of functions in L
uv
($xJ
x
that are
k
k
concentrated on B^ } x TY k
A A
Y
~ k
, (here B(Fw) is the subA
k
space of w-point subsets in Xk) (n = 0, 1, . . . ). Obviously, for any / G L2(TX) (8) H and i p G T (B^ ) the product \pf is approximated in X
L r
H b
finite s u m s of
k
k
l( x) ® y elements (prt/, where yn G £~» (B^ } ). Thus, the proof of the lemma reduces to the following assertion: If X C Ll(Fx)
where (8)
45i, r = ^ fl (^^r ( B ^ x Tx^Xh) 0 Wj 0 C\).
It is sufficient to prove the assertion for each subspace X\y r separately. Note that when r ¥= n, the supports of the functions in U \ir ( B x x rx\x ) a n d i n L M' n ( B x } ) d o n o t i n t e r s e c t - Therefore, it is only necessary to consider the case r = n. Let Xk, n be the image of Xl, n under rn. By Corollary 2 to Lemma 2, Xln has the form Xi,n = ^ ; ( B ^ ) <8) Wlr (g) i)J. Hence it is clear that J£ft,n is invariant under multiplication by functions in L< ( B ^ ) . Since the corresponding elements in X\ n a n d Xln differ only by the factor (—^—-)
j
the space <^ft, n is also invariant under
multiplication by elements of U"> ( B ^ ) , and the lemma is proved. A We consider the space H - HMp, a factor in the tensor product ) ® H. A representation R of ^ is defined in H by n
k
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A. M. Vershik, I. M. Gel'fand and M. I. Graev
(R(o)ip)(a) = Mo-1 a). Note that R = Ind|~ x s n (p X I), where SZ is the subgroup of finite permutations leaving 1, . . . , n fixed and / is the unit representation of SZ • Hence the next result follows easily. LEMMA 4. The representation R of Sx is irreducible. LEMMA 5. Every subspace X C L^(TX) 0 H that is invariant under Diff X is also invariant under the operators I 0 R(o), o E S^, where I is the unit operator in L^(TX). PROOF. Since S^ = lim Sp, it is sufficient to prove that X is invariant under the operators / 0 R(o), o £ Sp (p = 1, 2, . . . ). We use the notation and results of §3.4. Let p be any fixed positive integer. We consider the subspace
where Xh, r is defined by (8). Clearly the union (J XhtVi$ everywhere dense in X. Therefore, it is sufficient to prove that each space Xl, r, r > p, is invariant under /
We consider the image X\, r of X\,r under Tr. By Corollary 2 to Lemma 2 Xl, r and hence its preimage X\,r is invariant under /.(g) R(o), a £ Sr. Since p < r, we have S^ C ^ , and so X\%T is also invariant under the operators / 0 R(o), o £ Sp. The lemma is now proved. LEMMA 6. Every subspace X C L£(T X ) 0 # invariant under Diff X w o/ /7ze form X = L^{A)
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89
the restriction of the Vp to G O Diff Y. REMARK 2. The proof of Theorem 3 can be simplified considerably in the case n = 0, that is, for the representation U^ in L2l(Tx). In this case it reduces to proving Lemma 1 of §2.1 and establishing a functional version of the 0 - 1 law. In the simplest case when JU ~ ixx (k = 1, 2,. . . ) (V 9 ^i" are the projections of/i onto Bx and Tx_x , respectively), this law If
fc
^
consists of the following. Let X be a subspace of L2t(Tx). If in terms of the decomposition L2(TX) s L2> {Bx ) 0 Lj. ( r z - ^ , ) t h e subspace x
X C L^(TX)
k
k
x
k
k
for any k has the form # s LJL (Bz ) (g) Cfc,
Cft c L*. (r z -z k ),
*fe
h
x
1
then either ^ = 0 or X = Ll(Tx). An extra difficulty comes from the fact that there is no equivalence /i ~ ii'x X nx , generally speaking, and only the weaker relation \i ~ 2 Mr x M" is true (for the definition of n'r r=0 and /x" see p. 25). §4. Representations of Diff X generated by the Poisson measure
1. Properties of the Poisson measure. Let X be a non-compact manifold with a smooth positive measure m, m(X) - °°, and let \i - JJLX be the Poisson measure on Fx with parameter X corresponding to the measure m on X (for the definition of the Poisson measure, see §0.2). Some basic properties of Poisson measure were stated in §0.6. LEMMA 1. If dim X > 1, then for any two [i-measurable sets Ai, A2 C Fjr with positive measure there exists a diffeomorphism \P e Diff(AT, m) such that2 fi(Al O $A2)>±ii(A1MA2).(Diff(X, m) is the subgroup of diffeomorphisms preserving m.) PROOF. First we recall some definitions and facts. By a cyclindrical set in Tx we mean a set A C Tx, of positive measure, of the form A = ir^A', where Y is a compact set in X, Ar C BY is a measurable subset and TTY is the natural map Tx -* BY i^y 7 = 7 ^ Y)\ Y is called the carrier3 of A Since the Poisson measure JU is infinitely decomposable, it follows that if the carriers of two cylindrical sets A1 and A2 intersect in a set of measure 0, then [x(Ax n A2) = JU(^1)JU(^42). We recall that any /x-measurable set C CTX can be approximated by cylindrical sets (that is, for any e > 0 there is a cylindrical set A such that
H(CAA)< e). We recall that in these terms the usual 0 - 1 law would be formulated as follows. Let / e L2 (Tx). If / = l fe 0 /fe for any k, where l fe is the constant in L2> (B^- ) and fk e L^» ( r x _ x ), then fe k / = const. For X = R1 the lemma is false, because in this case Diff(Jf, m) is trivial. Of course, the carrier of a cylindrical set is not uniquely defined.
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A. M. Vershik, I. M. Gel'fand andM. I. Graev
Hence it is clear that it is sufficient to prove the assertion of the lemma for cylindrical sets Ax and A2. Without loss of generality we may suppose further that X = Rn, n > 1. Let us establish the following property of Diff(X, m): if Yx and Y2 are two compact sets in X, then there is a diffeomorphism \p E Diff(X, m) such that Yx n \p Y2 = 0. For if Ylt Y2 are compact in X = Rn and m is the Lebesgue measure in R", then there exists a disc containing Yx and Y2 and a rotation ^ of the disc (this preserves rn) such that Y1 n i// F 2 = 0; this rotation i// can be extended beyond the boundary of the disc to a finite diffeomorphism of R" preserving rn. If now m is an arbitrary smooth positive measure in R", then it is sufficient to use a lemma (see [21]), which states that any open ball in Rn, n > 1, with a smooth measure m can be mapped diffeomorphically onto itself so that m goes over to the Lebesgue measure. Let Yx and Y2 be the carriers of Ax and A2. By what has just been proved, we can find a diffeomorphism \p E Diff(Z, m) such that Yx O \//72 = 0. But then / i ^ n \J>A2) = M ( ^ I ) M ( ^ 2 ) = M(^I)M(^2) ? and hence /x(^4 j O \jjA2) > y MW I )/x(^42 )• The lemma is now proved. THEOREM l . / / d i m X > 1, then the Poisson measure /x in Yx is ergodic with respect to Diff(X, m). PROOF. Let A C Tx, ix(A) > 0, be a subset that is invariant mod 0 under Diff(X, m); we must prove that fi(A) = 1. Suppose the contrary: that JJL(TX \A) > 0. Then by Lemma 1 there is a \jj E Diff(X, m) such that ju(r x \i4) O \//^) > 0; hence, since A is invariant, ix((Tx \A)C\A)>09 which is false. This proves the theorem. 2. The representation of Diff X generated by the Poisson measure. Let
(i = /xx be the Poisson measure on Tx with parameter X > 0. In §3 we have associated with each quasi-invariant measure /x on Tx a unitary representation U^ of Diff X in L^(TX) defined by
and also a set of elementary representations £/jj). For the Poisson measure JUX the theory of such representations can be advanced considerably further than in the general case. In particular, it is possible to describe the corresponding representation ring. Furthermore, for jix the representations Up can be realized in the form EXP^ T (see [1] and [2]). In what follows we write Ux (instead of U^ ) for representation generated by the Poisson measure nx. Since nx is ergodic, by Theorem 1 of §3, Ux is irreducible. 3. The spherical function of the representation Ux. Let us assume that dim X > 1. We consider the subgroup Diff(X, m) C Diff X of diffeomorphisms preserving the measure m; for us this subgroup will play a role similar to that of maximal compact subgroups in the theory of representations
Representations of the group of diffeomorphisms
of semisimple Lie groups. Since /x is invariant under Diff(Z, m) and Ux is infinitely decomposable, the restriction of Ux to Diff(X, m) is given by
In view of Theorem 1 there is in L^ (Px) one, and up to a multiplicative factor, only one vector that is invariant under Diff(X m), namely, /o = 1DEFINITION. The following function on Diff X is called the spherical function of Ux: where the brackets denote the inner product in L2^ (Tx). Let us find an explicit form for the spherical function. Let supp \p C y, m(Y) < oo. We denote by /Ix the projection of JUX onto B r and by Jxx the restriction of /Ix to B ^ . Then
)= J B
So we obtain (2)
i*x W =
where J . (x) = ^
ex
. Since f0 is defined invariantly in the repres
entation space L2^ (Tx) and since, by (2), ux ¥= ux^ when \x ^ X2, we obtain the following theorem. THEOREM 2. The representations UXi and UX2 of Diff X (dim X > 1) are HO^ equivalent when \ l ^ X2. 4. The Gaussian form of the representation Ux of Diff X. Let us consider the real Hilbert space H - Lm (X), where m is a smooth positive measure on X. A unitary representation T of Diff X is given in H by ), where J ^ (X) = ^ /
}
. A 1-cocycle
91
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A. M. Vershik, I. M. Gel'fand andM. I. Graev
j3: Diff X -> H is given by [M)]{x) = J^2(x) - 1. In accordance with [1] and [2] this is a way of constructing a new representation & = EXP^ T of Diff X. We denote by jl a measure in the space $F (X) of generalized functions on X given by its characteristic functional: (3)
where ||*|| is the norm in L2m(X). We call # the standard Gaussian measure in t F (X). The representation U = EXP^ 71 is given in Z,~( jF (X)) by (£/ (i|>) (D) (F) = ei{F> m)®
(T* (yjp) F),
where the operator T*(\p) is defined by
w
o ^ f i <*•,/«>,
where / l 5 . . . , / „ are smooth finite functions on I (n = 0, 1, 2, . . . ) form a total set in L2( ^ ( X ) ) ; therefore, it is sufficient to prove that X contains all functional of the form (4). Let / be any smooth finite function on X satisfying j f(x)dm(x) = 0, let T be any real number such that 1 - rf(x) > 0 for all x G X. A measure mT in X is given by dmT(x) = (1 - Tf(x))dm(x). The measures m and mT coincide outside a compact set Y D supp /, and m(Y) = mT(Y). Therefore, by a theorem of Moser [21], there exists a diffeomorphism \jj e Diff X carrying m to mT, that is, J ^ (JC) = 1 - r fix). But then M) = VO ~ r / W ) ~ *> a n d h e n c e t h e functional e»<^ Vd-r/lx))-i> belongs to <£ for any sufficiently small r. Hence all the terms in the expansion of el
terms of the form II (F, /}>, k < n. Therefore, by induction on n, we /= l
can verify that X contains all functional of the form (F, f)n,
where
\ f{x)dm{x) = 0, and hence those of the form
Representations of the group of diffeomorphisms
93
as linear combinations of the (F, f)n, they also belong to X, and the lemma is proved. ^ If in the definition of U we replace the 1-cocycle j3 by sj3, where s is any real number, we obtain a one-parameter family of unitary representations of Diff X: Us = EXPs/3r. It is obvious that the assertion of lemma 2 remains true for all representations Us, s ¥= 0. ^ THEOREM 3. If s ^ 0, then Us = Usr, where Ue is the representation of Diff X generated by the Poisson measure with parameter s2 (see ^ 4.2). PROOF. We compute the matrix element (Us()p)$0, <£>0>, where <J>0 = 1. From formula (3) for the characteristic functional of Jt we immediately obtain (Us (i|>) O0, Oc> = exp (s2 j ( j ; / 2 (x) -1) dm (*)) - as2 (if), where wj2 is the spherical function of US2. The assertion of the theorem follows from this and the fact that 4>0 is cyclic. ^ Let us now consider the special case 5 = 0. Then (see [ 1 ]) Uo splits into the direct sum: Uo = T° © T1 © . . . © Tn © . . . , where T° is the unit representation and Tn (n > 1) is the ft-th symmetrized power of T intro-o duced at the beginning of §3.4. In the notation of § 1 we have Tn = Vp°n, where pJJ is the unit representation of Sn. Thus, Uo = © VPn. «=o 5. Elementary representations of Diff X associated with the Poisson measure. According to §3, the tensor product £/£ = Ux
the restriction to Diff(A", m)). Consequently,
(5)
vi^vp®
S Fpop°.
From the decomposition (5) it follows easily on the basis of Theorem 2 in §2 that U* i U^ when p x i p 2 . Hence, a fortiori, U^ i Up* when Pi ^ p 2 - It only remains to discuss the representations £/£ and ^ A '
Let (7? f [/{. We denote by A the vacuum vector in the representation space of Ux. (i = 1, 2) and by F an arbitrary vector in the
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A. M. Vershik, I. M. Gel'fand and M. I. Graev
representation space of Vp. From (5) it follows easily that under an isomorphism of the representation spaces of Up and £/? the vector fx (g> F goes over into the vector fx
for any \p E Diff X. Hence ux = ux , where ux is the spherical function for £/x (see §4.3), and therefore, \ = X2. §5. The ring of elementary representations generated by the Poisson measure
1. The decomposition of the tensor product Ux ® Ux of representations of Diff X into irreducible representations. First we prove a general theorem about representations EXP^T. Let G be an arbitrary group and T a unitary representation of G in a real space H\ let j3: G -> i/ be a 1-cocycle. Then a new unitary representation Us = EXPSj3r can be defined as in [1] and [2], where s is an arbitrary real number. Let H' be the dual space to H and /z a measure in any nuclear completion H of H\ defined by the characteristic functional:
The operators of the representation Us = EXPSj3 T act in the complex Hilbert space Ll(H) according to the formula: (Us (g) O) (F) = e*s
THEOREM 1. If s] + s\ = s[2 + S22, ^ ^ &s ® 6^ - ^ s ; ® ^ ; PROOF. We define operators At, t e R, in Ll(H) ® I ^ ( ^ ) by the formula: F2) = O(cos t Fx + sin ^ F2, —sin ^ ^ + cos t F2). From the definition of the Gaussian measure j2 it follows t h a t ^ f for any t G R is a unitary operator. Further, from the definition of Us it follows easily that A? (USl (g)
® f/s2 = ^ S l C O 3r + sa sinr ® 5L S i sinf + s2 cost
for
an
Y
f G R ; hence the assertion of the theorem follows immediately. COROLLARY. USi 0 US2 = Uj(S] + s]) ® ^0Now let £/x be the representation of Diff X generated by the Poisson measure with parameter X (see §4), let Vp be the representation of Diff X defined in § 1 (p runs over the representations of Sn).
Representations of the group of diffeomorphisms
95
THEOREM 2. (1)
Uu 0 UK2 -
© (Uk 0 FP°), n=0
where X = Xx •/• X2, flwc? P° & ^ wmY representation of Sn. Every term in (1) is an irreducible representation of Diff X. PROOF. Let T be a representation of Diff X in Z^ (X) defined by = 3 J^OO/dJT 1 *). A 1-cocycle 0: Diff X -> Z^ (X) is given by (*) = J J/ 2 (x) - 1. We consider the representation Us = EXP^T of Diff X. By Theorem 1 we have, for any Xj > 0, X2 > 0 (2)
^ y r , ® ^Vr 2 = ^yxi+x; ® ^o-
On the other hand, it was proved in §4.4 that U^x X > 0, and Uo = © F «=o
Pn
C/x 0 C/x ^ 0 (t/ x
0 F p »), where X = Xx + X2.
2
w= 0
= Ux for any
. Consequently, (2) implies that
The irreducibility of Ux 0 F n follows from the main theorem of §3. 2. The decomposition of the tensor product of two elementary representations of Diff X associated with the Poisson measure. THEOREM 3. C/?» 0 Xl
W* = © Up^ ° ^°Pn K
n=0
** + ** .
'
(For the definition of the operation pj o p 2 , see § 1.) 0 Vp^, U** = Ux^ 0 Vp\ Consequently,
PROOF. By definition, Ufr = UK U
x\
® ux] -Wxt =
® ^x,) 1 ® (^ P l 0
KP
®
FP2
Further
)- ^
Theorem 2,
FPl
^ ® ^ ® (^,+Xa ^' ® F P 2 - F P l ° P 2 f orany see Pi>P2 ( § 0 - Hence the required result is obtained straightaway. COROLLARY. The set of representations of Diff X that split into the direct sum of irreducible representations of the form Ux 0 F p is closed under the operation of taking the tensor product. §6. Representations of Diff X associated with infinitely divisible measures The group Diff X acts naturally in the space $F (X) of generalized functions on X. Therefore, representations of Diff X can be constructed for any quasi-invariant measure /I on JF(X). We have already noted earlier that the configuration space Tx has a natural embedding in jF (X), and, therefore, the representations considered earlier are part of a considerably
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A. M. Vershik, I. M. Gel'fand andM. I. Graev
wider class of representations. Here we consider a special class of measures in $F (X) (infinitely decomposable measures), which generate the same stock of representations as the representations Ux
( j (e*a«—l where r is a non-negative finite measure on R (not necessarily normalized). (Xr(O is the Fourier transform of a certain infinitely divisible measure on R.) Let m be a fixed smooth positive measure on X, ra(X) = °°. A measure /I = piT is given on jF(X) by the characteristic functional (1)
L% (/) = exp ( j In xt (/ (*)) <M*)) =exp ( j j (***'<*> - l)dro(*)<*T (oc)) . X
R X
We list some basic properties of M, which follow easily from this definition. 1) If X = Xx U • • • U Xn is a finite partitioning of X and J^(X) = ^ ( A ^ ) © . . . © ^r(Arw) the corresponding decomposition of / ( I ) into a direct sum, then pi =/z1X . . . X jurt, where pt^- is the projection of /x onto the subspace jF(^/), / = 1, . . . , « (infinite decomposability). 2) The measure pt = p[T is concentrated on the set Jo (X) of generalized oo
functions of the form X ak8 fc= l
fe
, ak =^ 0, where o:^ E supp r, and {^fe} is a
set without accumulation points in X (that is, a configuration in X)\ 5X denotes the delta-function on X concentrated at x G I 1 3) The measure /x is quasi-invariant under Diff X, and
dm(\l)x) .(Since \jj is finite, only finitely many factors dm(x) 3 ^(xk) are distinct from 1.) Let us note the particular case when r is concentrated at one point, a - 1. Then t f 0 W = (S5 X .; {xj G F x } , where Tx is the configuration ^ 0 ( X ) -> F x in which each generalized function 25^. is associated with a configuration {xt} G F ^ . It is not difficult to verify that the image of n under this map is the Poisson measure on Fx with parameter X = r(R). where J^ (x) =
1 The converse is also true: any infinitely decomposable measure in &(X) concentrated on a set of the type indicated is a measure MT for a certain T; when X = R 1 , this fact is very well known (see, for example, J.L. Doob, Stochastic processes, Wiley & Sons, New York 1953.
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2. The representation of Diff X associated with j2T. A unitary representation of Diff X is given in Ll(^(X)) by the formula (2)
In the particular case when the measure r on R is concentrated at a single point and is equal to X at this point, the representation VT so constructed is equivalent to the representation Ux corresponding to the Poisson measure with parameter X (see the remark above). Here we shall obtain the decomposition of VT into elementary representations. We denote by ,$FQ (X) the set of all generalized functions of the form 2 <^jcdx , ak =£ 0, where {xk} G Tx; it was mentioned above that JfQ (X) oo
is a subset of full measure in & (X). We introduce the space R°° = n R , /= l
Rz- = R with measure v = r 0 X . . . X r 0 X . . . , where r 0 is the normalized measure on R: r 0 = -r , X = r(R). Next, let i(y, x) be an admissible indexing (for the definition see §2) in X and consider the sequence of maps ak: Tx -+ X (k = 1, 2, . . . ) defined by ak(y) G y, i(y, ak(y)) = k. We define a map JI:
by
Standard arguments establish the following result. LEMMA 1. The map TT is measurable in both directions; the image of ^Q{X) is a subset of full measure in (Tx X R°°;/xX v)\ the image o//I T under IT is the product measure JJL X v, where ji is the Poisson measure on Fx with parameter X = r(R). By means of IT the action of Diff X can be carried over from ,iF0 C^O to I\r X R°°. It is not difficult to see that the action of Diff X on Tx X^ R°° is given by (3)
ip: (Y, a) ^
(ty'^f
a0
(^> Y))»
where a = (OLX , . . . , an,...), and ao = (ce a(1) , . . . , a a ( n ) , . . . ) ; here o(\p, y) is the map Diff X X Tx ^ S^ defined by / (see §2). Lemma 1 and (3) imply the next result.
LEMMA 2. The representation VT of Diff X is equivalent to the representation acting on L2(TX)
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Let us now consider the space Z^(R°°) = Z,J (R)® . . . ®Lj (R)®. . . with a given unitary representation T of S^ : (T(O)
(nl9
. . ., nh;
il9
. . .,
ih),
where nx > . . . > nk, ip ¥= iq when p ¥= q, and ip > ip+1 if np = np+1; k = 0, 1, . . . (k = 0 corresponds to the empty set). With each collection (5) we associate a basis vector in A: (6) e^ ® . . . ® eii ® . .. ® eife <8). .. ® eih
® K\\'.'.'.'X
a n d in
We denote by KTWl"-"rtfe the restriction of FT to Z ^ O V ® K\\'.\'.\nk' LEMMA 4. v"l""*nk = (/xp"i°'"°P"fe w/zere X = r(R), and p^ is the unit representation of Sn. The assertion of the lemma is easily established if we use the realization of elementary representations introduced in §3.3. Lemmas 3 and 4 give the next result.
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THEOREM 1. The representation VT of Diff X in L2^ (JF(X)) defined by (2) splits into a discrete direct sum of irreducible elementary representations of the form UPK, where X = r(R). REMARK 1. If r is concentrated at two points on R, and if the measures of these points are \x and X2, respectively, then, as is easy to show, V~ = UXi (g> JJX . In this way we obtain from Lemmas 3 and 4, in particular, the decomposition of the tensor product Ux 0 Ux into irreducible representations. This was obtained by another method in §5. REMARK 2. The representation VT can be treated as a continual tensor product of Poisson representations Ux\ Theorem 1 then gives a decomposition of the continual tensor product of Ux into irreducible representations. REMARK 3. The representation so constructed is a cyclic subrepresentation in EXP^ T, where T is the representation of Diff X in the real space L2{X X R, m X r) and j3 is the 1-cocycle: [j3(i//)](x, a) = J^2(x) - 1, see §7. If r = 5^ , x0 =£ 0, then we obtain the Gaussian form for the Poisson representation Ux (see §4.4). 3. Criteria for representations VT of Diff X to be equivalent. By Lemma 3, the multiplicity with which y"itmm''nk occurs in VT depends only on the numbers nl9 . . . , nk and on the dimension of L2 (R). Therefore, Lemmas 3 and 4 also imply the following result. THEOREM 2. Let T and r" be two non-negative finite measures on R such that 1) r'(R) = r"(R); 2) the supports of r' and T" are either both infinite or consistent of the same finite number of points. Then the representations VT> and VT» of Diff X are equivalent. By Theorem 2, each representation VT is given, up to equivalence, by a pair of numbers: the parameter X = r(R) of the Poisson measure (0 < X < oo) and the index h, which is equal to n if r is concentrated on n points, and is °o if supp r is infinite. It is convenient, therefore, to denote these representations by Vx h (instead of the previous notation VT). 4. The tensor product of representations VT = Vxh. THEOREM 3. v \lthx ® Vx2,h2 - Vxi+x2,h^h2'^ thus> the set of representations VT = Vxh is closed under the operation of tensor multiplication. PROOF. We have Vx.h. = FT., where rt is any non-negative finite measure on R scuh that r^R) = Xz- and |suppr z | = hf if ht < «>, and supp Tf is any infinite set if ht - °° (i = 1, 2). The measures T1 and r 2 can always be chosen so that supp TX C\ supp r 2 = 0. Let us consider r = T! + r 2 on R. Obviously, VT = Vx +x h +h . Therefore, it is sufficient for us to check that VT ^ VTi ® VT\ We denote by 1il9 /z2) a n d M the measures on & (X) corresponding to T 1? r 2 , and r on R, respectively, and by LT (/), LT (/), and LT(f) their
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characteristic functionals. Since supp Tt O supp r2 = <j>, we have LT(f) = LTi(f) LTi(f). Hence 2£ & (X)) ~ L*~ (& (X)) ® L i ( ^ (X)), and so VT = VT ® VT , as required. §7. Representations of the cross product 3 = C°°(X)*Diff X
1. Definition of & and the construction of representations. Let us introduce the (additive) group C°°(X) of all real finite functions / on X of class C°°. Now Diff X acts on C°°(X) as a group of automorphisms / -+ f o i//"1. In this way we can define the cross product y = C°°(X)-Diff X of C°°(X) with the multiplication: (A, *iXf 2 , * 2 ) = (A + / 2 ° r 1 , * i * 2 ) Let pi be an arbitrary quasi-invariant (under Diff X) measure in the space jf(X) of generalized functions on X. We consider Ll(,lF (X)) and associate with each element (f, \jj) & the following operator V(f, \jj) in (1) It is easy to check that the V(f, \jj) are unitary and form a representation of 3. This representation of # is cyclic with respect to C°°(X) (the constant is a cyclic vector). It is irreducible if and only if the measure ju on $F (X) is ergodic with respect to Diff X. 2. Representations associated with infinitely decomposable measures. From now on we restrict ourselves to the measure jut on :f {X) introduced in §6, that is, measures with characteristic functionals of the form Lx (/) = exp ( \ f (eia^>— 1) dm { where m is a smooth positive measure on X, m(X) = «>, and r is a nonnegative finite measure on R. The representation of 3 corresponding to this measure is now denoted by VT (the measure m on X is assumed to be fixed). It is not difficult to prove that these measures are ergodic with respect to Diff X\ consequently, the representations VT of & =C°°(Z)*Diff X are irreducible. LEMMA I. If dim X > 1, then Ji is ergodic with respect to the subgroup Diff (X, m) C Diff X of diffeomorphisms preserving m. The proof goes as for Poisson measures (see §4). COROLLARY 1. The restriction of VT to the subgroup C°°(X)-Diff(X, m) is irreducible.
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COROLLARY 2. The only vectors in Ll(JF(X)) that are invariant under Diff (X, m) are the constants. Let 4>0 be the function in L~(JF(X)) that is identically equal to 1. The following function on IS is called the spherical function of VT: *c(/,*) = (^(/,^)O 0 , O0>,
where the brackets denote the scalar product in Ll (,f(X)). Since
uT(f, n
M
By a simple calculation we obtain (2)
ux (/, *) = exp ( j j (J^ 2 Or) ete**> - 1) dm (x) dx (a)) ,
rf/nOIT*) where 3 Ax) ^ — — - . Obviously, if r t = r 2 , then MT ^= wT . Since 4>0 is defined invariantly in Z,£(jF(X)), we have the following result. LEMMA 2. / / T J ^ r 2 , ^ ^ representations VT and VT of & = C°°(Z)- Diff(X) (dim X > 1) are inequivalent. 3. The Gaussian form of the representations VT. Let us consider the complex Hilbert space H o f functions o n l X R with t h e n o r m ||cp|| 2 = J j|q>(*.
*)\*dm(z)dx{a).
A unitary representation T of § is given in H by (T(f, *) cp) (a:, a) = cto/(x)j^( a; ) ( p W -i a;f a ) t We define map |3: ^ ->• H by
It is easy to verify that for any glr g2 £ ^ we have: ^ 1 ^ 2 ) = 0tei) + ^ 1 ) ^ 2 ) , so that P is a 1-cocycle. Let us construct from T and the 1-cocycle |3 a new representation FT = EXP^T of S (see [1], [2]). We denote the dual space of H by H\ The standard Gaussian measure on the completion H' of H' is the measure H with the characteristic functional
The representation FT of ^ is given on L^(Hr) by (V, (g) O) (F) = eJ Re
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where the operator T*(g) is given by (T*(g)F, /> =
where uT is the spherical function of Vr (see §7.2). Hence the assertion of the theorem follows immediately. REMARK 1. A representation of C°°{X)* Diff X was constructed in [15] by means of the N/V limit. This representation coincides with that constructed here for the Poisson measure (the connection with the Poisson measure was apparently not noticed), and the transition to a Fock model in [15] is equivalent to the realization of this representation as EXP^T (see above). We emphasize that a representation of the cross product can be constructed for any measure in .ff (X) that is quasi-invariant under Diff X. However, only those that are constructed from an infinitely decomposable measure have the structure EXP^T, because it is only in this case that there is a vacuum vector. REMARK 2. Instead of C°°(X) we can consider an arbitrary group of smooth functions C°°(X, G) = Gx on X with values in a Lie group G and the cross product C°°(T, G)- Diff X. If a unitary representation IT of G is given on a space H, then the representation T of this cross product acts naturally in the space SB = \ ®Hxdm(x), Hx = H. This is irreducible if IT is irreducible. If p: C°°(X, G)-Diff X-+3B is a non-trivial cocycle (see [1]), then in EXP SB we get a representation EXP^T of C°°(X, G)- Diff X. APPENDIX 1 On the methods of defining measures on the configuration space Tx
1. Let Xx C . . . C Xn C . . . be a sequence of open submanifolds in X with compact closures such that X = U Xn. The projections pk: Tx -* Bx n
k
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103
are given by putting pk 7 = 7 0 1 ^ . Let M be a measure on Tx and [ik - pkn its projection on Bx(k = 1, 2, . . . )• Then the measures txk are mutually compatible, that is, for any k > 1 we have P\k[ik = M/> where plk\ Bx -> Bx is the natural projection. A well known theorem of Kolmogorov about the extension of measures enables us to establish the converse: if {\ik} is any compatible sequence of measures on {BXfe}, then there is a unique measure on Tx such that pk\x = fxk(k = 1, 2, . . . ). We can abandon the compatibility conditions and consider sequences of measures nk on Bx for which lim Pikixk - M(/) exists (in the weak sense) for all /. In this case the measures JU(/) on B x are compatible and define a measure /i on Fx. We also recall that Fx is naturally embedded in the space of generalized functions (7 -• 2 5X), therefore, the methods for defining measures in X<Ey
linear spaces are applicable here (by means of the characteristic functional and so on); see, for example, [5]. 2. A fundamentally different method describing measure on Fx has received attention in statistical physics [7]. It generalizes the method of specifying Markov measures (by transition probabilities). It consists in giving conditional measures on By (or their densities with respect to Poisson measure) as functions on Txx Y f° r aH compact domains Y C X by means of a single function (the potential) on B^. The question of existence and uniqueness of the measure on Tx with a given system of conditional measures is, as a rule, very difficult. Curiously enough, in this case the condition for a measure ju on Tx to be quasi-invariant under Diff X can be formulated very simply: all the conditional measures on By, where Y is any open set with compact closure, must be equivalent to a quasi-invariant (under Diff Y) measure on BY • By now there are many such measures known in statistical physics that are not equivalent to the Poisson measure (Gibbs measures). A measure on Tx can also be given with the help of so-called correlation functions on B^; a correlation function defines uniquely an initial measure on Fx (see, for example, [12]). 3. Let us introduce yet another method of defining measures on Tx. We say that^a normalized Borel measure JJL on X°° (see §0.3) is admissible if: 1) fJL(X°°) = 1, that is, X°° is a subset of full measure in X°°; 2) /! is quasi-invariant under Diff X. If /x is an admissible measure on X°°, then its projection JU = pix on Tx (that is, ju(C) = ix(p~lC) for any measurable set C C F j ) is a quasi-invariant measure on Fx. This method of defining a measure on Tx is of limited interest, however, it is convenient for constructing various examples. It is easy to show that an admissible measure M on X°° is ergodic if and only if it is regular (regularity means that it satisfies the 0 - 1 law). In particular, any admissible product measure fx = mx X . . . X mn X . . . is
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ergodic. The following lemma is analogous to the Borel-Cantelli lemma. LEMMA. The product measure y =ml X .. . X mn X . . .on X°° is admissible if and only if f mt(Y) < <» for any compact set Y C X. EXAMPLE OF AN ADMISSIBLE MEASURE. Let X = Rn. We denote by ma the Gaussian measure with centre a t f l G R and with unit correlation matrix: dma(x) = (7-n)-~nl2e~u~a^ 2 dx, where dx is the Lebesgue measure on R". By a direct computation it is not difficult to check that ju = maX . . .X ma X . . . is admissible if, for example, \\an\\>c log n, n = 1, 2, . . .. If \\an\\ -• °° sufficiently quickly, then it is easy to verify that /i is concentrated on the set As = n (sTx)o, where s: Tx -• X°° is the CrOSSsection corresponding to a certain admissible indexing / in Tx. However, ju is not invariant under left translations in X°°\ (Tx\ = xi+l. Hence its projection /x = pfx is a non-saturated measure in Tx (see Proposition 7 in §2.3). Another example refers to the group Diff X, where X is a compact manifold. In this case, let X°° denote the^set of all sequences in X that converge in X. It is easy to verify that X^/S00 = Tx is the union of all countable subsets of X with a unique limit point (one for each subset). Let x0 6 J , let p be a continuous metric in X, and let {mn} be a sequence of smooth measures in X such that lim \ p(x, xo)dmn(x)
= 0. It is clear
x
that the product measure m ° = II mn is concentrated on X°°. We intron duce the measure in = f m*°
According to the standard definition, a 1-cocycle on Diff X with values in the group S^(YX) of measurable maps Yx -> S^ is a map a: Diff X -* ^ ( P ^ ) satisfying the following condition: (l)
Two 1-cocycles ox and o2 are said to be cohomologous if there exists a measurable map o0: Tx -• 5^ such that
Represenrations of the group of diffeomorph isms
(2)
a2(o|), y) =
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vWiV^)-
Let / be a measurable indexing in F^, ak = Tx -+ X (k = 1 , 2 , . . . ) ^ the sequence of measurable maps defined by i (see §2.2); and 5: Tx -> Z°° the cross-section of the fibration1 X°° -> Tx defined by i\ Further, let As = II We say that a measurable indexing / is correct if for any 7, 7' E F^ the conditions I7 n 7T| = | 7' n # | and 7 n (X \ 7Q = 7' O (X \ JO for a certain compact set K C X imply that ak(y) = 0^(7') for all indices k except finitely many.2 A cross-section s: Fx -> X00 defined for a correct indexing / is also called correct. If / is correct, then the set A^ is invariant under Diff X. _ To each correct cross-section s: Tx -> X°° there corresponds a 1-cocycle os defined by the following relation (see §2.2): REMARK. There are examples of cocycles that are not generated by correct cross-sections. Cocycles os generated by correct cross-sections s are also called correct. We give, without proof, some properties of correct cocycles os. 1) The cross-section s is uniquely determined by the correct cocycle os corresponding to it. 2) Any two correct cocycles os and as are cohomologous as cocycles with values in S ^ F ^ ) , that is, (3)
crS2 (of, 7) = oQ(y)a8i (i|>,
YK1^"1^
where a 0 is a measurable map F x -> S°°. 3) No correct 1-cocycle is cohomologous to the trivial cocycle. 4) Let os be a correct cocycle, o0: Tx ^ S°° a measurable map, and o a 1-cocycle defined by afo, 7) = tfo(Y)tfs(^ 7) ffo1^"1?)For a to be correct it is necessary and sufficient that o0(y)<Jol(\lj-ly) e S^ for any \p G Diff X and 7 e Tx. 5) Two correct cocycles os and as are cohomologous as cocycles with values in S^ (Tx) if and only if the cross-sections Sx and s2 are cofinal, that is, s2 = a o ° 5X where o0 G ^ ^ ( r ^ ) . 1 Note that, generally speaking, the fibration X°° -* Vx has no continuous cross-sections. It can be shown that thisfibrationhas no continuous quotientfibrationswith fibre Z 2 . The condition of correctness is weaker than the condition of admissibility introduced in § 2.2
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For each 1-cocycle a: Diff X -+ 5^(1^) we define a corresponding 1-cocycle aa\ Diff X -* Z2{VX) by (4)
a a (f, y) = sign a(ip, v)
(sign o is the parity of a E ^ ) . It is obvious that when ox and a 2 are cohomologous, then the correspondcocycles aa and aa are cohomologous. However, there exist cocycles ox and o2 (even correct ones) that are not cohomologous, although the corresponding cocycles aa and aa are. EXAMPLE. The correct cocycles Oi(\p9 y) and o2(\jj, 7) = o0o(\p, 7)ao 1 , where a 0 ^ S^. Since sign (OQOOZ1) = sign a, we have a a = aa . However, the cocycles ox and o2 are themselves not cohomologous (see property 4). A SUFFICIENT CONDITION FOR COCYCLES TO BE COHOMOLOGOUS. Let a2(i//, 7) = a o (7)o r i(^,7)^o 1 (^" 1 7)- If cr0(7) = cr o a o (7), where 0^0 ^•S'oo(r^) and a 0 is an arbitrary element of S°°, then aa ~ aa^. Note that each Z2-cocycle a: defines a Z2-covering of Tx with a given action of Diff X on it. The elements of this covering are pairs (e, 7), 7 e r x , 8 = ± 1; and Diff X acts by ^(e, 7) = (ea(ij/, 7), ty~ly). Z2-cocycles of the form (4), where a is a correct cocycle, are called correct Z2-cocycles, and the Z2-coverings of Tx defined by them are also called correct. LEMMA. Correct Z2-cocycles are non-trivial. ^ Let / be a correct indexing in Tx, s: Tx -> X°° the cross-section defined by /, ix a quasi-invariant measure in Tx, and T a unitary representation of S^ acting on a Hilbert space H. We associate with the triple (/, jtz, T) a unitary representation V of Diff X in the space 3$ C L^(AS, H) of functions / : As ->- H such that
for every a G S^ ; the representation operators are defined by (5) where p is the projection As ->• F ^ . ALTERNATIVE DEFINITION: 7 is given in the space Ll(Tx, H) of functions f: Tx -* H for which ||/|| 2 = f 11/(7) 11^*(7) <°°- the operators F(\//)have the form (6) where a5 is the 1-cocycle generated by a correct cross-section s: F^ -> X°°. It is obvious that the representations of Diff X so defined are equivalent. Note that V= Ind^ i f f x (7o TT), where (7 is the subgroup of all diffeo-
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107
morphisms \p £ Diff X for which ^7 = 7, and TT is the projection Gy -* Gy\ G° = S^. (G° is the subgroup leaving every point x G 7 fixed). Two 1-cocycles a and a' are said to be equivalent with respect to a measure pt if a(\//, 7) = o'(\p, 7) mod 0 with respect to /z for any \p £ Diff X. Note that the right-hand side of (6) does not change if the 1-cocycle is replaced by any equivalent cocycle. Therefore, the 1-cocycles o need only be defined up to an equivalence. The properties 1) - 5) of 1-cocycles can be reformulated without difficulty for equivalence classes of 1-cocycles. Moreover, 2) can be made more precise as follows: the map a 0 in (3) is uniquely determined mod 0 if n is ergodic. Let us consider the particular case when T = Ind^°°x sn (p X /), where p is a unitary representation of Sn and / is the unit representation of S" OS" is the group of finite permutations of n + 1, n + 2, . . . ) . Comparing the definition of V with that of the elementary representations Up (see §3) it is easy to check that V = £/£. Hence, in particular, in this example all the correct cocycles lead to equivalent representations.1 Quite a different example is the Fermi representation. Let us consider a Z2-cocycle <x(\p, 7) (see above) and define a representation of Diff X in L2(TX) by This is called a Fermi representation of Diff X. When the Z2-cocycle is generated by a correct cross-section s: Tx -* X°°, that is, when a(\p, 7) = sign OS(\JJ9 7), where os: Diff X -+ S^ (Vx) is a correct cocycle, then there is another convenient realization of V: the representation is given in SB by a function fix) on A s c r such that 1) f(xa) = sign of(x) for any a G S^ (an "odd" function). 2) il/H 2 = J \f(sy) |2 ^i(y) < 00. The representation operators are defined by (5). In this case it can be shown by the same arguments as in §3 that // JJL is ergodic and the crosssection s: Tx -+ X00 is generated by an admissible indexing i (in the sense of §2.2), then the Fermi representation V(\jj) is irreducible. REMARK 1. Apparently, there exist non-equivalent Fermi representations of Diff X constructed from the same measure n on Tx (for the construction of a Fermi representation by means of an N/V limit, see [19] and [20]). REMARK 2. The group Diff X has factor representations of type II it is sufficient to take a representation of S^ of type II in H (for example, 1 There is a more general fact: if a representation T of 5» can be extended to a representation of 5°°, then the representation V of Diff X corresponding to T is uniquely determined, up to equivalence, by T and a measure ju on Tx..
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the regular representation) and to construct a representation of Diff X in
Ll{Vx, H) from it. APPENDIX 3 Representations of Diff X associated with measures in the tangent bundle of the space of infinite configurations
The representations of Diff X discussed above are of "zero order", that is, they do not depend on the differentials of the diffeomorphisms. In this context let us note that these representations can be extended to representations of the group of measurable finite transformations of a manifold X with a quasi-invariant measure. However, one can construct representations of Diff X of positive order (by a representation of order k we mean one depending essentially on the A:-jet of the diffeomorphisms). A number of representations of Diff X in spaces of finite functional dimension were defined in [9]; in the terminology of this paper these representations are connected with the space Bx of finite configurations. But there are also representations of positive order connected with the space Tx of infinite configurations. Let us take as an example a representation of order 1 connected with this space. Let (i be a measure in Tx that is quasi-invariant under the action of Diff X. We consider the "tangent bundle" TTX over Tx, that is a fibre bundle over Tx, where the fibre over 7 = {xt} is the direct product II T x . I of the tangent spaces at the points xt £ 7. i= 1
i
__
__
The space JTX can be regarded as the factor space TX^/S00, where TX°° is the subset of (TX)°° = U TXt, Xt = X, consisting of points 1=1
{(xi9 uf); u,- G Tx.Xt, {xt} e X^yjhe topology in TTX and the a-algebra of Borel sets are induced from TX°°. Let XT be a normalized measure in TX.X that is equivalent to the 00
00
Lebesgue measure, and let X7 = II XT be the product measure in II TY X. /= 1
l
/ =1
x
i
In this way a measure X is introduced in TYX such that its projection 00
onto Tx is M and that the conditional measures in TyFx =11 Tx. X are X7. The action of Diff X on (TTX, X) is defined by i//(% a) = (1//7, d\jja), 00
where a = II ax, ax G TXX and d\l/ is the natural action on TTX. The measure X is quasi-invariant under this action. This leads to a unitary representation of Diff X in L^(TTX). A proof that this is irreducible for ergodic measures n can be modelled on §3. The parameters of the representations just constructed are the measure JJL in Tx and the measures
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109
X? in r ^ I V It is easy to see how to construct representations of order 1 analogous to the elementary representations. We do not say much about representations of higher order, because difficulties in describing them arise even in the case of a finite number of particles (see [9]). REMARK. The representations of Diff X listed in this appendix can be extended to representations of the cross-product C°°(Xy Diff X. References
[1] A. M. Vershik, I. M. Gel'fand and M.I. Graev, Representations of the group S£(2, R), where R is a ring of functions, Uspekhi Mat. Nauk 28:5 (1973), 83-128. = Russian Math. Surveys 28:5 (1973), 87-132. [2] A. M. Vershik, I. M. Gel'fand and M. I. Graev, Irreducible representations of the group Gx and cohomology, Functsional. Anal, i Prilozhen. 8:2 (1974), 67-69. MR 50 # 530. = Functional Anal. Appl. 8 (1974), 151-153. [3] D. B. Anosov and A. B. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms, Trudy Moskov. Mat. Obshch. 23 (1970), 3-36. [4] A. Weil, L'integration dans les groupes topologiques et ses applications, Actual. Sci. Ind. 869, Hermann & Cie, Paris 1940. MR 3 # 198. Translation: Integrirovanie v topologicheskikh gruppakh i ego prilozheniya, Izdat. Inost. Lit., Moscow 1950. [5] I. M. Gel'fand and N.Ya. Vilenkin, Obobshchennye funktsii, vypA. Nekotorye primeneniya garmonicheskogo analiza. Oskashchennye giVbertovy prostranstva Gos. Izdat. Fiz.-Mat. Lit., Moscow 1961. MR 26 #4173. Translation: Generalized functions, vol.4, Some applications of harmonic analysis, Equipped Hilbert spaces, Academic Press, New York-London 1964. [6] A. M. Vershik, Description of invariant measures for the actions of some infinitedimensional groups, Dokl. Akad. Nauk SSSR 218 (1974), 749-752. = Soviet Math. Dokl. 15 (1974), 1396-1400. [7] R. L. Dobrushin, R. A. Minlos and Yu. M. Sukhov,Prilozhenie k knige Ryuelya: Statisticheskaya mekhanika (Supplement to Ruelle's book Statistical mechanics). Mir, Moscow 1971. [8] R. S. Ismagilov, Unitary representations of the group of diffeomorphisms of the circle, Funktsional Anal, i Prilozhen. 5:3 (1971), 45-53. = Functional. Anal. Appl. 5 (1971), 209-216. [9] A. A. Kirillov, Unitary representations of the group of diffeomorphisms and some of its subgroups, Preprint IPM, No.82 (1974). [10] A. A. Kirillov, Dynamical systems, factors, and group representations, Uspekhi Mat. Nauk 22:5 (1967), 67-80. = Russian Math. Surveys 22:5 (1967), 63-75. [11] V. A. Rokhlin, On the fundamental ideal of measure theory, Mat. Sb. 25 (1949), 107-150. MR 11 #18 [12] D. Ruelle, Statistical mechanics. Rigorous results, W. A. Benjamin Inc., Amsterdam 1969. Translation: Statisticheskaya mekhanika. Strogie rezul'taty, Mir, Moscow 1971.
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[13] S. V. Fomin, On measures invariant under a certain group of transformations, Izv. Akad. Nauk SSSR Ser. Mat. 14 (1950), 261-274. MR 12 # 3 3 . [14] G. Goldin, Non-relativistic current algebras as unitary representations of groups, J. Mathematical Phys. 12 (1971), 462-488. MR 44 # 1330. [15] G. Goldin, K. J. Grodnik, R. Powers and D. Sharp, Non-relativistic current algebra in the A/K limit, J. Mathematical Phys. 15 (1974), 88-100. [16] A. Guichardet, Symmetric Hilbert spaces and related topics, Lecture Notes in Math. 261, Springer-Verlag, Berlin-Heidelberg-New York 1972. [17] J. Kerstan, K. Mattes and J. Mecke, Unbegrenzt teilbare Punktprozesse, Berlin 1974. [18] D. Knutson, X-rings and the representation theory of the symmetric group, Lecture Notes in Math. 308, Springer-Verlag, Berlin-Heidelberg-New York, 1973. [19] R. Menikoff, The hamiltonian and generating functional for a non-relativistic local current algebra, J. Mathematical Phys. 15 (1974), 1138-1152. MR 49 # 10285. [20] R. Menikoff, Generating functional determining representations of a non-relativistic local current algebra in the iV/K limit, J. Mathematical Phys. 15 (1974), 1394-1408. [21] J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286-294. MR 32 #409 [22] R. S. Ismagilov, Unitary representations of the group of diffeomorphisms of the space R", n>2, Funktsional. Anal, i Prilozhen. 9:2 (1975), 71-72. = Functional Anal. Appl. 9 (1975), 144-145. Received by the Editors, 15 May 1975 Translated by A. West
AN INTRODUCTION TO THE PAPER 'SCHUBERT CELLS AND COHOMOLOGY OF THE SPACES GIF Graeme Segal It is well known that a generic invertible matrix can be factorized as the product of an upper triangular and a lower triangular matrix. A more precise statement is that any invertible n X n matrix g can be Written in the form b1wb2, where bx and b2 are upper triangular and w is a permutation matrix. Here w is uniquely determined by g, though bx and b2 are not. The matrices g for which w is the order-reversing permutation i*-*n —i + 1 form a dense open setinGL n (C). This double-coset decomposition GLW (C) = U BwB, where B denotes the w upper triangular matrices and w runs through the permutation matrices, has an analogue for any connected affine algebraic group G. The role of B is played by a Borel subgroup (i.e. a maximal soluble subgroup), and the role of the permutation matrices by the Weyl group W = N(H)/H, where H s (C x ) ; is a maximal algebraic torus in G and N(H) is its normalizes The decomposition is nowadays called the Bruhat decomposition; but Gelfand had earlier recognized its importance in his work on the representations of the classical groups. It is best to think of the decomposition as the decomposition of the homogeneous space X = G/B into the orbits of the left action of B. The space X plays a central role in representation theory. It turns out that it is a complex projective algebraic variety, and that the orbits of B are algebraic affine spaces Cm of various dimensions, the "Bruhat cells". The closures of the cells are algebraic subvarieties which in general have singularities. It is important that the maximal compact subgroup K of G acts transitively on X, so that X has an alternative description as K/T, where T = K O B is a maximal torus of K. If G = Ghn (C) then X is the flag manifold: a flag in Cn is an increasing sequence of subspaces F=(F1
CF2
C...CFn
-C")
with dim(Fk) = k. For GL n (C) acts transitively on the set of all flags, and B is the isotropy group of the standard flag C C C 2 C . . . C Cn . In this case the cells are indexed by permutations w of {1, 2, . >. ,, n}, and we can take as a representative point in the cell Xw the flag Fw such that F™ is spanned by {euKi)>euKi)> ' • ->ew{k)}> w h e r e {*i> • • •>*/!> is the standard basis of C". Z ^ can be defined by "Schubert conditions": it consists precisely of the flags F such that dimCFfc ^Cm) = vkm, where
vkm - card {/: i < k, w(i) < m }, The dimension of Xw is the length l(w) of w, defined by l(w)=
£
| M;(0-/I. in
112
Schubert cells and cohomology of the spaces G/P
Alternatively l(w) is the number of pairs (/, /) such that / < / but w(i) > w(j). In fact if Nw is the subgroup of B consisting of matrices (ai7-) with diagonal elements aH = 1 and such that
Schubert cells and cohomology of the spaces G/P
113
(i) the centre of the cell Xw maps to the point of P(VX) represented by the weightj/ector/^ G Vx of weight wX, (ii) Xw is precisely Jthe intersection ofZ with a linear subspace of P(VX), and (iii) the boundary Xw ~XW of Xw is Xw D Uw, where 11^ is the hyperplane perpendicular of fw. Now let us recall that the Weyljgroup W - which we are regarding as a group of automorphisms of the lattice T — is generated by the reflections oy in the hyperplanes of T perpendicular to the roots 7 of G. If w G W has length k it turns out that the (k - 1 )-dimensional cells in the boundary of Xw are precisely the Xwa such that l(woy) = k — 1. Thus the cap-product cx E [Xw ] is necessarily of the form 2 n [Xwa ], where n is a positive integer. To deter7
T
mine ny one must calculate the order to which the linear form (fw,), when regarded as a function on Xw, vanishes on the cell XUXJ . That is easy to do because the formula where E_y is the standard element of & in the (- 7) root-space, defines a holomorphic curve in Xw which passes through the centre fwa = woy fe of Xwa when t = 0, and is transversal to Xwa . We calculate
(fW9 woy exp(tE_y)fe )=(oyfe,exp(tE_y)fe
) = 0(A),
where ny = < X, Hy ), Hy being the co-root associated to 7, i.e. the element of the dual lattice to T characterized by the property for all x £ T. The formula
gives us the pairing between homology and cohomology in the form (cXicX2...cXnAXw])=X(\1,Hyi)..A\k,Hyi),
where the sum is over all strings 7 j , . . ., yk of positive roots such that
I shall not describe here the elegant algebraic formulations the authors derive from this. It ought, however, to be mentioned that the methods apply equally well not only to the space G/B, but to G/P for every parabolic subgroup P of G. The most obvious case of this is the Grassmannian Gr^ n of /^-dimensional subspaces of C", which is GLW (C)/P, where P is the appropriate group of echelon matrices.
114
Schubert cells and cohomology of the spaces G/P
(Iri terms of compact groups Gr^ n = Un/Uk. X Un_k.) The analogue of Gr^. „ for the orthogonal groups is the Grassmannian of isotropic ^-dimensional subspaces of Cn for some non-degenerate quadratic form on C" : this space can be identified with On/Uk X On^2k- When k = 1 it is a complex projective quadric hypersurface.
SCHUBERT CELLS AND COHOMOLOGY OF THE SPACES G/P I. N. Bernstein, I. M. Gel'fand, S. I. Gel'fand We study the homological properties of the factor space G/P, where G is a complex semisimple Lie group and P a parabolic subgroup of G. To this end we compare two descriptions of the cohomology of such spaces. One of these makes use of the partition of G/P into cells (Schubert cells), while the other consists in identifying the cohomology of G/P with certain polynomials on the Lie algebra of the Cartan subgroup// of G. The results obtained are used to describe the algebraic action of the Weyl group W of G on the cohomology of G/P.
Contents
Introduction § 1. Notation, preliminaries, and statement of the main results . . § 2. The ordering on the Weyl group and the mutual disposition of the Schubert cells §3. Discussion of the ring of polynomials on f) §4. Schubert cells §5. Generalizations and supplements References
115 .117 120 124 133 136 139
Introduction Let G be a linear semisimple algebraic group over the field C of complex numbers and assume that G is connected and simply-connected. Let B be a Borel subgroup of G and X = G/B the fundamental projective space of G. The study of the topology of X occurs, explicitly or otherwise, in a large number of different situations. Among these are the representation theory of semisimple complex and real groups, integral geometry and a number of problems in algebraic topology and algebraic geometry, in which analogous spaces figure as important and useful examples. The study of the homological properties of G/P can be carried out by two well-known methods. The first of these methods is due to A. Borel [ 1 ] and involves the identification of the cohomology ring of X with the quotient ring of the ring of polynomials on the Lie algebra I) of the Cartan subgroup 115
116
/. N. Bernstein, I. M. Gel'fand, S. I. Gel'fand
H c G by the ideal generated by the ^-invariant polynomials (where W is the Weyl group of G). An account of the second method, which goes back to the classical work of Schubert, is in Borel's note [2] (see also [3]); it is based on the calculation of the homology with the aid of the partition of X into cells (the so-called Schubert cells). Sometimes one of these approaches turns out to be more convenient and sometimes the other, so naturally we try to establish a connection between them. Namely, we must know how to compute the correspondence between the polynomials figuring in Borel's model of the cohomology and the Schubert cells. Furthermore, it is an interesting problem to find in the quotient ring of the polynomial ring a symmetrical basis dual to the Schubert cells. These problems are solved in this article. The techniques developed for this purpose are applied to two other problems. The first of these is the calculation of the action of the Weyl group on the homology of X in a basis of Schuberts cells, which turns out to be very useful in the study of the representations of the Chevalley groups. We also study the action of W on X. This action is not algebraic (it depends on the choice of a compact subgroup of G). The corresponding action of W on the homology of X can, however, be specified in algebraic terms. For this purpose we use the trajectories of G in X X X, and we construct explicitly the correspondences on X (that is, cycles in X X X) that specify the action of W on H^ (X, Z). The study of such correspondences forms the basis of many problems in integral geometry. At the end of the article, we generalize our results, to the case when B is replaced by an arbitrary parabolic subgroup P c G. When G = GL(n) and G/P is the Grassmann variety, analogous results are to be found in [4]. B. Kostant has previously found other formulae for a basis of H*(X, Z), X = G/B9 dual to the Schubert cells. We would like to express our deep appreciation to him for drawing our attention to this series of problems and for making his own results known to us. The main results of this article have already been announced in [13]. We give a brief account of the structure of this article. At the beginning of § 1 we introduce our notation and state the known results on the homology of X = G/B that are used repeatedly in the paper. The rest of § 1 is devoted to a statement of our main results. In § 2 we introduce an ordering on the Weyl group W of G that arises naturally in connection with the geometry of X, and we investigate its properties. § 3 is concerned with the ring R of polynomials on the Lie algebra f) of the Cartan subgroup He G. In this section we introduce the functional Dw on R and the elements Pw in R and discuss their properties. In § 4 we prove that the elements Dw introduced in § 3 correspond to the Schubert cells of X.
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§ 5 contains generalizations and applications of the results obtained, in particular, to the case of manifolds X(P) = G/P, where P is an arbitrary parabolic subgroup of G. We also study in § 5 the correspondences on X and in particular, we describe explicitly those correspondences that specify the action of the Weyl group W on the cohomology of X. Finally, in this section some of our results are put in the form in which they were earlier obtained by B. Kostant, and we also interpret some of them in terms of differential forms on X. §1. Notation, preliminaries, and statement of the main results We introduce the notation that is used throughout the article. G is a complex semisimple Lie group, which is assumed to be connected and simply-connected; B is a fixed Borel subgroup of G; X - GIB is a fundamental projective space of G\ N is the unipotent radical of B\ H is a fixed maximal torus of G, H c B; <$ is the Lie algebra of G; rj and %l are the subalgebras of <$ corresponding to H and N; Ij* is the space dual to t); A C £)* is the root system of t) in @ ; A+ is the set of positive roots, that is, the set of roots of I) in 9i, A_ = -A+, £ C A+ is the system of simple roots; W is the Weyl group of G; if y £ A, then aY : I)*-> f)* is an element of W, a reflection in the hyperplane orthogonal to y. For each element1 w 6 W = Norm(//)/77, the same letter is used to denote a representative of w in Norm (H) C G. l(w) is the length of an element w G W relative to the set of generators {a a , a 6 2} of W, that is, the least number of factors in the decomposition (1)
w = oaioa2...oai,
a* 6 2.
A decomposition (1), with / = l(w), is called reduced; s G W is the unique element of maximal length, r = l(s); N_ - sNs~1 is the subgroup of G "opposite" to N. For any w G W we put Nw = w N_wl n N. HOMOLOGY AND COHOMOLOGY OF THE SPACE X. We give at this point two descriptions of the homological structure of X. The first of these (Proposition 1.2) makes use of the decomposition of X into cells, while the second (Proposition 1.3) involves the realization of two-dimensional cohomology classes as the Chern classes of one-dimensional bundles. We recall (see [5]) that ^ = w A L w ^ n i V i s a unipotent subgroup of Norm H is the normalizer of H in G.
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G of (complex) dimension l(w). 1.1. PROPOSITION (see [5]), Let o 6 X be the image of B in X. The locally closed subvarieties Xw = Nwo C X, w E W, yield a decomposition of X into N-orbits. The natural mapping Nw -* Xw (n H* nwo) is an isomorphism of algebraic varieties. Let Xw be the closure1 of Xw in X, [Xw] £ H2i±w) (XW,Z) the fundamental cycle of the complex_ algebraic variety Xw and sw £ H2t(w) (X,Z) the image of [Xw ] under the mapping induced by the embedding Xw C-+ X. 1.2. PROPOSITION (see [ 2 ] ) . The elements sw form a free basis of #* (*,Z). We now turn to the other approach to the description of the cohomology of X. For this purpose we introduce in Ij the root system {Hy, y £ A} dual to A. (This means that oy\ = x ~ X (Hy)y for all X 6 £) *, 7 6 A). We denote by !)Q CZ ^ the vector space over Q spanned by the Hy. We also set f)£ = {x 6 ^ * I x (^ 7 ) 6 Z for all 7 6 A} and Let R = S i(^) be the algebra of polynomial functions on f)Q with rational coefficients. We extend the natural action of W on ^* to R. We denote by / the subring of PV-invariant elements in R and set / + = { / e / | / ( 0 ) - o } , J = I+R. We construct a homomorphism a: /?-> M*(X9 Q) in the following way. First let x 6 *)z- Since G is simply-connected, there is a character 0 € Mor (7/, C*) such that 6 (exp /z) = exp xW, h G t). We extend 0 to a character of B by setting 0(n) = 1 for « 6 N. Since G->-Ar is a principal fibre space with structure group B, this 0 defines a onedimensional vector bundle Ex on X We set c*i(x) = c x , where cx G iPiX, Z) is the first Chern class of Ex. Then ax is a homomorphism of tjz into ^ ( Z , Z), which extends naturally to a homomorphism of rings a: R -> #*(X, Q). Note that W acts on the homology and cohomology of X. Namely, let K c G be a maximal compact subgroup such that T = K. n / / is a maximal torus in A". Then the natural mapping K/T-+X is a homeomorphism (see [ 1 ]). Now W acts on the homology and cohomology of X in the same way as on K/T. 1.3. PROPOSITION ([1], [8]). (i) The homomorphism a commutes with the action of W on R and H* (X, Q). (ii) Ker a = J, and the natural mapping a: R/J -> H* (X, Q) is an isomorphism. In the remainder of this section we state the main results of this article. The integration formula. We have given two methods of describing the As Xw is a locally closed variety, its closure in the Zariski topology is the same as in the ordinary topology.
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cohomological structure of X. One of the basic aims of this article is to establish a connection between these two approaches. By this we understand the following. Each Schubert cell sw £ #* (X, Z) gives rise to a linear functional Dw on R according to the formula Dw(f)
= <*w, a (/))
(where < , > is the natural pairing of homology and cohomology). We indicate an explicit form for Dw. For each root 7 6 A, we define an operator AY : R -* R by the formula f
(that is, Ayf(h) = [/(h) - f(o7h)]/y(h) for all h 6 \ )• Then we have the following proposition. PROPOSITKDN. Let w = a ttj . . . oav at £ S. If/(">) < /, then A a i • • • A a i = 0 . / / l{w) - /, r/ze/^ /7ze operator A a ^ . . . A a i depends only on w and not on the representation of w in the form w = aai . . . o a i ; we put A w = A a i . . . A a y
This proposition is proved in §3 (Theorem 3.4). The functional Dw is easily described in terms of the Aw : we define for each w E W another functional Dw on R by the formula Dw f = Aw /(0). The following theorem is proved in §4 (Theorem 4.1). THEOREM .Dw = Dw for all w G W. We can give another more explicit description of Dw (and thus of Dw). To do this, we write u)x ^ w2, M>I, w2 £ W, y £ A+, to express.the fact that ifi = oyw2 and / (w;2) ~ I (^1) + 1. THEOREM. Let w 6 W, /(w) = /.
(i) / / / E /? w <2 homogeneous polynomial of degree k ^ ^ ( / ) = 0. (ii) If xi, - - -, Xi €M ,then 4,(xi • • - -Xi) = S X i ( ^ ) where the sum is taken over all chains of the form w0
ry ->
y W i -Z
. . .
y 4
(see Theorem 3.12 (i), (v)). The next theorem describes the basis of H*(X, Q) dual to the basis {sw I w 6 W7} of H*(X, Z). We identify the ring R = R/J with H*(X, Q) by means of the isomorphism a of Proposition 1.3. Let {Pw \ w 6 W} be the basis of R dual to the basis {sw \ w 6 ^F} of ^(A', Z). To specify Pw, we note that the operators Aw: R -+ R preserve the ideal J C R (lemma 3.3 (v)), and so the operators Aw : R -* R are well-defined. THEOREM, (i) Ler s £ W be the element of maximal length, r = /(s) Ps = p r /r! (mod /) = IWI"1 [] 7 (mod / ) , {where P 6 % VGA+
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/• M Bernstein, I. M. Gel'fand, S. I. Gel'fand
the sum of the positive rootsjind \W\ is the order of W) (ii) If w e W, then Pw = AW.UP8 (see Theorem 3.15, Corollary 3.16, Theorem 3.14(i)). Another expression for the Pw has been obtained earlier by B. Kostant (see Theorem 5.9). The following theorem gives a couple of important properties of the Pw THEOREM (i). Let X € %, w£W. Then x-Pw - 2 w x W pw (see Theorem 3.14 (ii)). (ii) Let $ : H^Xf Q) -> H*(X, Q) be the Poincare duality. Then & (sw) = a (Pw8) (see Corollary 3.19). THE ACTION OF THE WEYL GROUP. The action of W on H* (X, Q) can easily be described using the isomorphism a: R/J -+ H*(X9 Q), but we are interested in the problem of describing the action of W on the basis {sw) of H^X, Q). THEOREM. Let a 6 2 , w 6 W. Then oasw = - sw if l(woa) = l(w) - 1 and oasw == — sw +
2J v
w'a(Hy) sw., if l(woa) = /(w) + 1 (see
Wf—>WG
Theorem 3.12 (iv)). In § 5 we consider some applications of the results obtained. To avoid overburdening the presentation, we do not make precise statements at this point. We merely mention that Theorem 5.5 appears important to us, in which a number of results is generalized to the case of the varieties X{P) = G/P (P being an arbitrary parabolic subgroup of G), and also Theorem 5.7, in which we investigate certain correspondences on X. §2. The ordering on the Weyl group and the mutual disposition of the Schubert cells
2.1 DEFINITION (i) Let wx, w2 6 W, y 6 A+. Then wx ^ w2 indicates the fact that OyWr = w2 and l(w2) = l(u)\) + 1. (ii) We put w < w'- if there is a chain w =
Wi->
w2 -*•
. . . - > • wk =
w'.
It is helpful to picture W in the form of a directed graph with edges drawn in accordance with Definition 2.1 (i). Here are some properties of this ordering. 2.2 LEMMA. Let w = o a j . . . oai be the reduced decomposition element w 6 W. We put yt = a t t i . . . a a . _ i ( a , ) . Then the roots 7 i » • • ., Ifi are distinct
and the set {Yi> • • •» Y*} coincides
with
of an A+ n n>A_.
This lemma is proved in [ 6 ] . 2.3 COROLLARY, (i) Let w = oai . . . o^ be the reduced decomposition and let y G A+ be a root such that w~1y G A-. Then for some i (2)
OyOa,i
. . . cra. = a
a i
. . .
Gaii.
Schubert cells and cohomology of the spaces G/P
121
(ii) Letw 6 PF, Y £ &+-Then I (w) < / (oyw)Jf and only if w^y £ A+. P R O O F (i) F r o m L e m m a 2 . 2 w e d e d u c e t h a t y = a t t j . . . o^.^ ( a f ) for some z, a n d ( 2 ) follows. (ii) If w^y £ A_, t h e n b y ( 2 ) oyw = o0Ci . . . aa._t a a . +i . . . oap t h a t is l(oyw) < l{w). I n t e r c h a n g i n g w a n d oyw, w e see t h a t if w~xy g A + , t h e n
l(oyw). 2.4 LEMMA. Letwu
w2 6 W, a 6 2, y 6 A+,and y ± a. Let y = o^y. If
(T a ^ 2 .
Conversely, (3) follows from (4). PROOF. Since a £ 2 and y =£ a, we have 7' = a a 7 6 A+. It is therefore sufficient to show that l(oaw2) > l(w2) = l(wi). This follows from Corollary 2.3, because aau;2 = oy'Wx and (ffa^)"1 ?' = w^cTaV' = ^2*7 6: A_ by (3). The second assertion of the lemma is proved similarly. 2.5 LEMMA. Let w, w' 6 W, a 6 2 and assume that w < w . Then a) either oaw < w or oaw < oaw\ b) either w < oaw' or oaw < oaw'. PROOF a) Let We proceed by induction on A;. If aaiy < w or a aw = w2, the assertion is obvious. Let w < o^w, oaw ¥^ w2 • Then aaw; < oaw2 by Lemma 2.4. We obtain a) by applying the inductive hypothesis to the pair (w2, w'). b) is proved in a similar fashion. 2.6. COROLLARY. Let a 6 2 , ^ J!> u?;, w2 _^> M£. If one of the elements Wi, wi z'5 smaller (in the sense of the above ordering) than one of w2, w2, then Wi < w2 < w2 and Wi < w\ < w2. The property in Lemma 2.5 characterizes the ordering <. More precisely, we have the following proposition: 2.7 PROPOSITION. Suppose that we are given a partial ordering w H w on W with the following properties: a) / / a f 2 , w 6 W with l(oaw) = l(w) + 1, then w -\ oaw. b) If w H wf, a £ 2 , then either oaw -) w' or oaw —| oaw'. Then w -\ w' if and only if w < w'.
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PROOF. Let s be the element of maximal length in W. It follows from a) that e —j w -\ s tor all w 6 W. I. We prove that w < w implies that w -\ if'. We proceed by induction on /(it/). If l{w) = 0, then w = e, w = e and so w H H/. Let /(u/) > 0 and let a 6 2 be a root such that l(oaw') = l{w) - 1. Then by Lemma 2.5 a), either aaw < oaw' or w < oaw'. (i) w < o a if' => if -\ oaw' (by the inductive hypothesis), =» if H u^'. (using a)). (ii) O^M; < a a u;' => a a u; - | aait>' (by the inductive hypothesis), => either w -\ oaw' or w -\ w' (applying b) to the pair (oaw, o^w')), => w -\ w'. II. We now show that w -| w implies that w < wr. We proceed by backward induction on l(w). If t(w) = r = /(s), then w; = 5 , u;' = s, and so u; < w . Let /(u;) < r and let a be an element of 2 such that l(o7w) = /(w) + 1. By b) either oaw -) u;' or o a it; H aait>'. (i) aaw; H M;' =*• oaw < u;' (by the inductive hypothesis) => if < w'. (ii) aaii> H oaw' => a a it; < aait>' => if < H/ (by Corollary 2.6). Proposition 2.7 is now proved. 2.8 PROPOSITION. Let w 6 W arid let w = aax . . . oai be the reduced decomposition of w. a) / / 1 < I'I < h < • • • < k < / ^ « ^ (5)
i*f = a B i i . . . a o V
f^en u;' < w. b) / / a;' < w, then w can be represented in the form (5) for some indexing set{ij). c) / / w -* if, ^ e « there is a unique index /, 1 < / < /, 5wc/z ^/zar (6)
a?' = a t t l . . . ^ a ._ l a a . + i . . . a « r
PROOF. Let us prove c). Let if' -^ if. Then by Lemma 2.2 there is at least one index / for which (6) holds. Now suppose that (6) holds for two indices /, /, i < j . Then oa.+i • • • %• = Oat • • • °aHl • T n u s 5 a a . . . . oa. = a a . +i . . . a ^ , , which contradicts the assumption that the decomposition if = a a j . . . oai is reduced. b) follows at once from c) if we take into account the fact that the decomposition (6) is reduced. We now prove a) by induction on /. We treat two cases separately. (i) it > 1. Then by the inductive hypothesis w' < oaj . . . oav that is, w < oai if < if. (ii) ix = 1. Then, by the inductive hypothesis, °ax w' = oa. . . . oa. < oaiw = oa2 . . . oav By Corollary 2.6, if' < if. Proposition 2.8 yields an alternative definition of the ordering on W (see [7J). The geometrical interpretation of this ordering is very interesting
Schubert cells and cohomology of the spaces G/P
123
and useful in what follows. 2.9 THEOREM. Let V be a finite-dimensional representation of a Lie algebra (& with dominant weight X. Assume that all the weights w\ w 6 W, are distinct and select for each w a non-zero vector fw £ V of weight w\. Then w'^w<=>fw, £U («R) fw {where U (91) is the enveloping algebra of the Lie algebra %l). PROOF. For each root y £ A we fix a root vector £ V 6@ in such a way that [Ey, E.y] - Hy. Denote by 2fv the subalgebra of @, generated by Ey, E-y, and Hy. 21Y is isomorphic to the Lie algebra sl2(C). Let w' -^> w and let V be the smallest 2IY -invariant subspace of V containing 2.10 LEMMA. Let n = w'\(Hy) G Z, n > 0. The elements {ElyU- \i --- 0, 1, . . ., n)form a basis of V. Put f= E^fw>. Then E_~f = 0, Etf = c%. (cf ^ 0) and fw = cf (c =£ 0). PROOF. By Lemma 2.2, w'~l y e A + , hence Eyfw> = ~cEyw'fe = cw'Ew>-iyfe = 0, that is, fw> is a vector of dominant weight relative to $ v . All the assertions of the lemma, except the last, follow from standard facts about the representations of the algebra Wv ^ sl2 (C). Furthermore, / and fw are two non-zero vectors of weight w\ in V, and since the multiplicity of w\ in V is equal to 1, these vectors are proportional. The lemma is now proved. To prove Theorem 2.9 we introduce a partial ordering on W by putting w -\ w if fw> G U(W)fw. Since all the weights w\ are distinct, the relation -\ is indeed an ordering; we show that it satisfies conditions a) and b) of Proposition 2.7. a) Let a 6 2 and l(oaw) = l(w) + 1. Then w ^ oaw, and by Lemma 2.10, fw e U (31) foaW, that is, w H oaw. b) Let w -\ w . We choose an a G 2 such that w ^ oaw. Replacing w by oaw , if necessary, we may assume that oaw' -> it;'. We prove that aau; -| w\ that is, faaw^U(%l)fw>. It follows from Lemma 2.10 that E-afw ~ 0 and fOaW = cEn_afw. Let $« be the subalgebra of @ generated by 31, f) and5Ta. Since w H w', fw£UW)fw> and so /OaU, = ci??a/«; = XL'* where X^U(^a). Any element X of £/ ($ a ) can be represented in the formx= 3\YiYi+YE-a, where t=i
Therefore, /aau> = 2 ^^7™- = S c^i/u,' 6 # (s^) /«- and Theorem 2.9 is proved. We use Theorem 2.9 to describe the mutual disposition of the Schubert cells. 2.1J. THEOREM (Steinberg [7]_). Let w 6 W, Xw c X a Schubert cell, and Xw its closure. Then Xw> c Xw if and only if w' < w.
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To prove this theorem, we give a geometric description of the variety
x w. Let V be a finite-dimensional representation of G with regular dominant weight X (that is, all the weights w\ distinct). As above, we choose for each w E W a non-zero vector /„, 6 F of weight w\. We consider the space P(V) of lines in V; if / 6 K, / i= 0, then we denote by [f] £ A*0 a line passing through /. Since X is regular, the stabilizer of the point [fe] £ P(V) under the natural action of G on P(V) is B. The G-orbit of L/g] in P(V) is therefore naturally isomorphic to X = G/B. In what follows, we regard X as a subvariety of P(V). For each u> 6 W we denote by <£„, the linear function on V given by
In this section we study the rings R and R. For each w 6 W we define an element / ^ 6 ^ and a functional -Dw on R and investigate their properties. In the next section we shall show that the Dw correspond to Schubert cells, and that the Pw yield a basis, dual to the Schubert cell basis, for the cohomology of X. 3.1 DEFINITION, (i) R = © Rt is the graded ring of polynomial fixations on 1)Q with rational coefficients. W acts on R according to the rule wf(h) = Aw* h). (ii) / is the subring of W-invariant elements in R,
Schubert cells and cohomology of the spaces G/P
125
(iii) J is the ideal of R generated by 1+. (iv) R = R/J. 3.2 DEFINITION. Let y £ A. We specify an operator Ay on R by the rule
Ayf lies in R, since f — oyf = 0 on the hyperplane y = 0 in 1)QThe simplest properties of the Ay are described in the following lemma. 3.3 LEMMA.(i) A_y = -Ay, A$ = 0. (ii) wAyw~x = Awy. (iii) oyAy = - Ayoy= Ay, oy = —yAy + 1 = Ayy — 1. (iv) Ayf = 0 o ayf = f (v) AyJ C / . (vi) Let x 6 £)Q- Then the commutator of Ay with the operator of multiplication by X has the form [Ay, xl = x(Hy)oy PROOF, (i) — (iv) are clear. To prove (v), let f - fif2, where /i 6 /+, / 2 € i?.. It is then clear that Ayf = fx.Ayf2 6 J As to (vi), since oyX = X - xCtfyh, we have
y
*>
The following property of t h e Ay is fundamental in what follows. 3.4 T H E O R E M . Let a h . . ., a, G S , and put w = a a j . . . a a j ; a) Ifl(w) < /, then A(Oii = 0. tt|) b) / / l(w) = /, then A{Qti a / ) depends only on w and not on the set « ! , . . . , az. In this case we put Aw = A^^ a/). The proof is by induction on /, the result being obvious when / - 1. For the proof of a), we may assume by the inductive hypothesis that /(a tti . . . oail) = / — l 5 consequently l(oai . . . o^^o^^ = 1 — 2. Then o^ a a . +1 . . . a a w = aaH1 . . . oail oai for some / ( we have applied Corollary 2.3 to the case w = o a H . . . a tti , 7 = aj). We show that A =0 Since / - / < / , the inductive hypothesis shows that AatAai+l . . . Aal_i = Aa.+i . . . A^^A^, and so by lemma 3.3 (i) To prove b), we introduce auxiliary operators Bi0li
a/),
by setting
We put wt = oai . . . oa.. Then in view of Lemma 3.3 (ii, iii) we have
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/• N. Bernstein, I. M. Gel'fand, S. I. Gel'fand
(where A% stands for wAyw'1). 3.5 LEMMA. Let %et)q. The commutator of Bi0li ai)with the operator of multiplication by x is given by the following formula:1 i
(8)
[#«».,
«.), xl =
PROOF. We have
By Lemma 3.3 (ii, vi), [A™*1,
xl
= x(wi+1Hai)
ow.+lOL..
Since o w . + i a . = w^wf* , we have r , = x (^i+i^«4) < 2 • • • 4 ^ 1 + ^ r
1
^ • ^ • Aat.
We want to move the term wi+1Wil to the left. To do this we note that for ;' < /
Therefore,
^;4;;
A,,
By (7), applied to the sequence or roots (a l5 . . ., 6th . . ., az), we have Tt =
X
(^I+I^BJ) ^ i + ^ i
1
^
£
aj)f
a n d L e m m a 3 . 5 is p r o v e d . If / ( a ^ . . . a a . . . . a a / ) < / - 1, t h e n 7J = 0 b y t h e i n d u c t i v e h y p o t h e s i s . I f / ( a t t i . . . a a . . . . a a { ) = / - 1, t h e n , p u t t i n g w' = oUi . . . a a . . . . oai a n d 7 = a t t j . . . a a . . 1 ( a f ) , w e s e e f r o m L e m m a 2 . 2 t h a t M ; ' ^ u;, a n d a l s o
X (wi+iHa.) = w'x (w'wi+lHai) = W'X (^aj • • • Oa^Hat) = and
A
indicates that the corresponding term must be omitted.
Schubert cells and cohomology of the spaces G/P
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Using Proposition 2.8 c) and the inductive hypothesis, (8) can be rewritten in the following form: [£ (ai
a,), Xl =
2
*>'% (Hy) if* A*.
w' —> w
The right-hand side of this formula does not depend on the representation of w in the form of a product oUi . . . oav The proof of theorem 3.4 is thus completed by the following obvious lemma. 3.6. LEMMA. Let B be an operator in R such that B(\) = 0 and [B, X ] = 0 for all X 6 %. Then B = 0. 3.7. COROLLARY. The operators Aw satisfy the following commutator relation: v We put St = Rf (where Rt c R is the space of homogeneous polynomials of degree /) and S =@St. We denote by ( , ) the natural pairing S X R -> Q. Then W acts naturally on S. 3.8 DEFINITION, (i) For any X 6 ^ Q we let x* denote the transformation of S adjoint to the operator of multiplication by \ in jR(ii) We denote by Fy: S -• S the linear transformation adjoint to Ay: R -* R. The next lemma gives an explicit description of the Fy. 3.9 LEMMA. Let 7 6 A. For any D 6 S there is a D £ S such that 7* 0) = D. If D is any such operator, then D - oyD = Fy(D), (in particular, the left-hand side of this equation does not depend on the choice
of S). PROOF. The existence of D follows from the fact that multiplication by 7 is a monomorphism of R. Furthermore, for any / 6 R we have (D-OyD, /) = (S, /-cr Y /) = (5, V - Y ) = (Y'(5), ^v/) = (A ^v/), hence D - oyD = Fy. REMARK. It is often convenient to interpret S as a ring of differential operators on f) with constant rational coefficients. Then the pairing ( , ) is given by the formula (Z), /) = (Df) (0), D £S, f £R. Also, it is easy to check that x*(^) = W, xL where X 6 ^Q and D 6 S are regarded as operators on R. Theorem 3.4 and Corollary 3.7 can be restated in terms of the operators Fy 3.10 THEOREM. Let au . . ., a, 6 2 , w = a o . . . aa. (i) / / /(w) < /, then Fai . . . Fai = 0. (ii) / / l(w) = I, then Fttl . . . Fai depends only on w and not on a,, . . ., a,. In this case the transformation F a; . . . Fa_ ,-5 denoted by Fw. {Note that Fw = A*).
128
/. N. Bernstein, L M. Gel'fand, S. I. GeVfand
(iii)
l%*,Fww]=
S
w'x(Hy)Fw>w.
v w' —> w
3.11. DEFINITION. We set Dw = Fw (1). As we shall show in §4, the functionals Dw correspond to the Schubert cells in H+ (X, Q) in the sense that (Dw,f) = <sw, <*(/)> for all f € R. The properties of the Dw are listed in the following theorem. 3.12. THEOREM, (i) Dw 6 SUw). (ii) Let we W, a 6 2 . Then [0 if l(woa) = l(w) - 1, (iii) Let%e QQ- r/ze« x* (A*) =
S
i
V w'—> w
(iv) Let a 6 2 . Tfterc
[ -A,, ^a^iy = < —Dw+ (v) £er
M;
if S
^ ' a (^v) ^ '
if
Z (w;aa) = Z (w) + 1.
g W, I (w) = Z, %i> . . ., %z G^Q. T/z^n
(A* > Xi5 • • •, Xi) = 2 Xi(^7i) . . . xi(Hi), where the summation extends over all chains PROOF, (i) and (ii) follow from the definition of Dw and Theorem 3.10 (i). (iii) X* ( A J = X*Fww(\) = [x*, Fww] (1) (since x * (1) = 0), and (iii) follows from Theorem 3.10 (iii). It follows from Lemma 3.3 (iii) that oa = a*Fa - 1. Thus, (iv) follows from (ii) and (iii). (v) We put Dw = Dw -i . Then the 5 ^ satisfy the relation (9)
X*(A.)=
S v xu'a —> = v Since (£>, x/) (x*(^)» /)> ( ) is consequence of (9) by induction on /. Let SB be the subspace of S orthogonal to the ideal / c R. It follows from Lemma 3.3 (vi) thatch is invariant with respect to all the Fym It is also clear that 1 6 SS. Thus, Dw 6 SS for all w 6 AT. 3.13. THEOREM. 77ze functionals Dw, w £ W, form a basis for SB. PROOF, a) We first prove that the Dw are linearly independent. Let s 6 W be the element of maximal length and r = l(s). Then, by Theorem 3.12 (v), Ds(pr) > 0 and so Ds * 0. Now let 2 cwDw = 0 and let w be one of the elements of maximal length for which cw =£ 0. Put / = l(w).
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There is a sequence c^ , . . ., ar.j f° r which woai . . . a a r / = s. Let F = Far.t • • • Fal • ^ follows from Theorem 3.10 that F£>~ = Ds and FZ^ = 0 if /(it;) > /, w # w. Therefore F( $>„,/)„,) = c~D, * 0. b) We now show that the Dw span c$£. It is sufficient to prove that if / € /* and (/>„,,/) = 0 for all w £ W, then / 6 / . We may assume that / is a homogeneous element of degree k. For & = 0 the assertion is clear. Now let k > 0 and assume that the result is true for all polynomials / of degree less than k. Then for all a 6 2 and w 6 W, (Dw, Aaf) = (FaDw, f) = 0, by Theorem 3.10 (i) and (ii). By the inductive hypothesis, Aaf 6 J, that is, / — oaf = aAaf 6 / . Hence for all w 6 W, f=wf (mod / ) . Thus, | W I"1 2 M?/ = / (mod / ) . Since the left-hand side belongs to /+, we see that / 6 J. Theorem 3.13 is now proved. The form ( , ) gives rise to a non-degenerate pairing between R = R/J and(§#. Let {Pw} be the basis of R dual to {Dw}. The following properties of the Pw are immediate consequences of Theorem 3.12. 3.14 THEOREM, (i) Let w 6 W, a 6 2 . Then _( 0 if l(waa) = /(M;) + 1, F
(ii)XPa,=
2J
wx{Hy)Pw>
waa if l(W0a) = 1(W) - 1. for
(iii) Let a 6 2. Pw Pw-
if S
wa{Uy)Pw>
if
From (i) it is clear that all the Pw can be expressed in terms of the Ps. More precisely, let w = oai . . . ooip l(w) - r — I. Then
To find an explicit form for the Pw it therefore suffices to compute the Ps 6 R. 3.15 THEOREM. P s = | W I"1 f] Y (mod/). PROOF. We divide the proof into a number of steps. We fix an element h £ f) such that all the wh, w (: W, are distinct. 1. We first prove that there is a polynomial Q (: R of degree r such that (10)
Q(sh) = 1, £>(">&) - 0 for M; ^ 5.
For each u; G IV we choose in R a homogeneous polynomial Pw of degree
130
/. N. Bernstein, I. M. Gel'fand, S. I. Gel'fand
l(w) whose image in R = R/J is Pw. Since {Pw} is a basis of R, any polynomial fe R can be written in the form / = 2 Pwfw. where j£,e/ (this is easily proved by induction on the degree o f / ) . Now let Q' 6 R be an arbitrary polynomial satisfying (10) and let Q' = 2 ? ^ w , gw£I* It is clear that
_
_
To prove this we consider ASQ. On the one hand ASQ = c8, by Theorem 3.13 (i); on the other hand, A$Q is a constant, since 2 is a polynomial of degree r. Hence, ASQ = cs. We now calculate ASQ, Let s = oai . . . oar be the reduced decomposition. We put wt = oa. . . . oa (in particular, w0 = e), yt = wfl a x , Q, = Aa.+i . . . AarQ. ' LEMMA. Qt is a polynomial of degree i,
and Qi(wh) = 0 if w p wt. PROOF. We prove the lemma by backward induction on /. For / = r we have wr = s, Qr = Q, and the assertion of the lemma follows from the definition of Q. We now assume the lemma proved for Qh i > 0. In the first place, it is clear that (2,-_, = AaiQt is a polynomial of degree i — 1. Furthermore, Qi (wh) — Qt (a„ .wh)
Cu, (wh) = AaiQi
(wh) =
a ^ ' ^ •
an
If w - wiA , then w < wt, oa.w = wt ^ OLi(u>i-ih) = (wf.\at) (h) = - (wr\oLi) (h) = - yt(h). Therefore, using the inductive hypothesis, we have
But if u; ^ M;,-.! , Corollary 2.6 implies that u; ^ w;,- and a a .u; ^ wt. So (2i_! (w;^) = 0, and the lemma is proved. Note that by Lemma 2.2, as / goes from 1 to r, yt ranges over all the positive roots exactly once. Therefore
(
0 YGA+
3. Consider the polynomial Alt (Q) = 2 ( - \)Kw)wQ; Alt (Q) is skewsymmetric, that is, oa Alt(Q)= -Alt(Q) for all y G A.Therefore Alt(Q) is divisible
Schubert cells and cohomology of the spaces G/P
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(in R) by [( 7. Since the degrees of Alt(2) and [I 7 are equal (to r), V£A+
V£A+
Alt(g) = X [I y- Furthermore, A\t(Q) (h) = ( - l) r , so that YGA+
4. We put Alt(fi) = 2 ( - l) /(u;) ^ 5 - By Theorem 3.14 (iii), Alt(P.) = S ( - \)liw) wPs = \W\ P.. Therefore Alt(G) = cs IH/I Ps + terms of smaller degree. Since Alt(C) is a homogeneous polynomial of degree r, we have Alt (Q) =
(12)
cs\W\Ps.
By comparing (11) and (12) we find that P* = \W\-* 0 v(niod^). VGA+
The theorem is now proved. 3.16 COROLLARY. Let p be half the sum of the positive roots. Then P$ = pr/r\ (mod / ) . PROOF. For each % £ I)* we consider the formal power series exp x on lj given by
Then we have (see [ 9 ] )
2(-l)'<w>exp(i£;p)= J [ [ e x p | - e x p ( - | ) ] . Comparing the terms of degree r we see that VGA+
r
r
If p (mod J) = \P S , X e C, then (wp ) (mod / ) = \wPs = X(- l)Z(u;)Ps. Thus, S. ^ ( — 1)/(U3) (^p)r == XPS (mod/). The result now follows from Theorem 3.15. To conclude this section we prove some results on products of the Pw in R.
3.17. THEOREM, (i) Let a 6 2 , M; 6 TF. r/iew PoaPw=
2
%a(Hw-iy)PW>,
w —> w'
where Xa 6 ^z is r/ze fundamental dominant weight corresponding to the root a (that is, Xa (fy) = 0 for a ± p 6 2 , x<* (^4) = 0(ii) Let wl9 w2 6 W, Kwx) + /(w2) = r. Then P^ PW2 = 0 for
132 W2
/• N. Bernstein, I. M. Gel'fand, S. I. Gel'fand *
WXS9 PWlPWl.=
Psj_
v
(iii) Let w £ W, f £ R . Then f Pw = j£wCw'Pw>. (iv) / / wt ^ w2sy then PWlPW2= 0. PROOF, (i) By Theorem 3.12 (v), POQI = Xa (mod / ) . Therefore (i) follows from Theorem 3.14 (ii). (ii) The proof goes by backward induction on l(w2). If l(w2) = r, then w2
= s, wx = e and PWI = 1.
To deal with the general case we find the following simple lemma useful, which is an easy consequence of the definition of the Ay. 3.18 LEMMA. Let 7 £ A, /, g £ R. Then Ay(Ayf-g) = Ayf-Ayg. Thus, let w2 £ W9 l(w2) = I < r, and choose a 6 2 so that ^2 ~* 0 ^ 2 - We consider two cases separately. A) Wi ^ oaWi. We observe that the following equation holds for any w e• w
(13)
Z(i^5) = r — l(w).
Since in our case l(oaw2) = / + 1 and l(oawi) - r — I -\- 1, we see that oawis i= oaw2, and so wxs ¥= w2. On the other hand, PWi = AaPOaw2 and ^ ! = AaPo^i by Theorem 3.14 (i). Therefore, an application of Lemma 3.18 shows that
Since Ko^Wi) + l(w2) = / • - / + Hence PWlPW2 = 0 as well.
1 + / > r, we have P a ^ i ^
=
B) OaWi ^ Wi. In this case, P 0 a U ; i = ^ P u ; ! an<* Pw2 = AaPoaWl, Theorem 3.14 (i). Again applying Lemma 3.18, we have
°by
Since the Pw form a basis of /?, any element / of degree r in R has the form / = XPS, X 6 C. Furthermore, ^ a P s = i>aaS =£ 0. But deg PWIPW2 = deg ^ u ; , - ^aaiy2
= r
- Therefore (14) is equivalent to
Applying the inductive hypothesis to the pair (oaWi, oaw2), we obtain part (ii) of the theorem. (iii) is an immediate consequence of Theorem 3.14 (ii). (iv) follows from (ii) and (iii). We define the operator
Schubert cells and cohomology of the spaces G/P
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§4. Schubert cells
We prove in this section that the functional Dw, w 6 W introduced in § 3 correspond to Schubert cells sw, w 6 W. Let sw 6 H*(X, Q) be a Schubert cell. It gives rise to a linear functional on H*(X, Q), which, by means of the homomorphism a: R -* H*(X, Q) (see Theorem 1.3), can be regarded as a linear functional on R. This functional takes the value 0 on all homogeneous components Pk with k J= l(w), and thus determines an element Dw £ SKw). 4.1. THEOREM. Dw = Dw (cf. Definition 3.11). This theorem is a natural consequence of the next two propositions. PROPOSITION '[.be = 1, and for any X€*)z (15) V —> w
PROPOSITION 2. Suppose that for each w 6 W we are given an element Dw 6 Sl(w), with De = 1, for which (15) holds for any% 6fyz• Then Dw = Dw. Proposition 2 follows at once from Theorem 3.12 (iii) by induction
on l(w). We turn now to the proof of Proposition 1. We recall (see [10]) that for any topological space Y there is a bilinear mapping Hl(Y,
Q ) x / / ; - ( r , Q ) - ^ IIM (F, Q)
(the cap-product). It satisfies the condition: (16)
1. (cny.z)
= (yiC>z)
] l
for all y£Hs(Y, Q), z£H - (Y, Q), cf,IP (Y, Q). 2. Let / : Yx -^ F 2 be a continuous mapping.. Then (17)
U(f*cny) -cnUy
for all y 6 Hj (7i, Q). ^ 6 //* (V2^ Q)By virtue of (17) we have for a n y x 6 n z , 1£R (X* 0w), f) = 0 W , if) = ( ^ , a! (X) a (/)) - ( ^ n a 4 (x), a (/)). Therefore (15) is equivalent to the following geometrical fact. PROPOSITION 3. For all x€fyz (18)
*a;na1(x)=
I
wf%(Hy)sw,
We restrict the fibering Ex to Xw C X and let cx 6 H2 (Xw, Q) be the first Chern class of Ex. By (17) and the definition of the homomorphism al::bfc->H2(X, Q), it is sufficient to prove that
134
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(19) in H2liwy-2(XW,
swncx=
2 V
Q).
To prove (19), we use the following simple lemma, which can be verified by standard arguments involving relative Poincar6 duality. 4.2 LEMMA. Let Y be a compact complex analytic space of dimension n, such that the codimension of the space of singularities of Y is greater than 1. Let E be an analytic linear fibering on Y, and c 6 fP(Y, Q) the first Chern class of E. Let M be a non-zero analytic section of E and ^mtYi = div n the divisor of n. Then [Y] n c = S ^ t ^ l £ H2n-2(Y, Q), where [ Y] and [ Yi are the fundamental classes of Y and Y(. Let w 6 W, and let Xw c X be the corresponding Schubert cell. From Lemma 4.2 and Theorem 2.11 it is clear that to prove Proposition 3 it is sufficient to verify the following facts. 4.3. PROPOSITION. Let w -> w. Then Xw is non-singular at points x € Xw>. 4.4. PROPOSITION. There 4s a section ix of the fibering Ex over Xw such that w'%(Hy)Xw v w' —>
To verify these facts we use the geometrical description of Schubert cells given in 2.9. We consider a finite-dimensional representation of G on a space V with regular dominant weight X, and we realize X as a subvariety of F(V). For each w 6 IV we fix a vector /„, 6 V of weight wX. PROOF OF PROPOSITION 4.3. For a root y 6 A+ we construct a three-dimensional subalgebra WY cz @ (as in the proof of Theorem 2.9). Let i: SL2(C) -> G be the homomorphism corresponding to the embedding a b \1 (/a 0
K
0
H ;r = i\[) ^L a
a LJ)
1
f/1 OM / 0 1\ an and NL = \[ . I \ d the element a = I . ^ 1. We may assume that c H, i(B') c B. Let F be the smallest 2Iv-invariant subspace of V containing fw>. It is clear that V is invariant under i(SL2 (C)), and that the stabilizer of the line [fw>] is B'. This determines a mapping 5: SLaiQlB'- -> ^ . The space SL2(C)/B' is naturally identified with the projective line P 1 . Let o, °° 6 P1 be the images of e, o 6 iSTZ/2(C). We define a mapping J: Nw> X P1 -» I by the rule (a:, z) >—• o:-6(z). 4.5. LEMMA. The mapping J /za5 r/ze following properties: (i) EW,' X {o}) = Xw.f E(^w. X (P 1 \ o)) c: X w .
Schubert cells and cohomology of the spaces G/P
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(ii) The restriction of £ to (Nw> x P 1 \ °°)) is an isomorphism onto a certain open subset of X^. Proposition 4.3 clearly follows from this lemma. PROOF OF LEMMA 4.5. The first assertion of (i) follows at once from the definition of Xw>. Since the cell Xw is invariant under N, the proof of the second assertion of (i) is reduced to showing that 5(z)'€ Xw for z € P1 \ o. Let h 6 SL2(C) be an inverse image of z. Then h can be written in the form h = bxob2, where br, b2 6 B'. It is clear that i{bi)fw' = dfw' and i(o)fw> = c2fw , where cx, c2 are constants. Therefore Kh)fw> = c1c2i(bl)fw, that is, 6(z) 6 Xw. To prove (ii), we consider the mapping w'-iol-.Nv. X ( P x \ 00) -+X.
The space P ^
00
is naturally isomorphic to the one-parameter subgroup
NL c SL2(Q. The mapping f: A^/X AL -• X is given by the rule l(n, rii) = ni(ni) [fw>], n g iV^, wt 6 N\ Thus, i^'"1 ° I (w, wO = ( M ; ' " 1 ^ ' ) (I//"1* (w,) u;') [/,].
We now observe that w'~lNw*w c AL (by definition of A^'). and w~li(N')w 6 AL (since u;'"1 7 6 A+). Furthermore, the intersection of the tangent spaces to these subgroups consists only of 0, because Nw> C N, i(N'_) C 7V_. The mapping AL -> X {n h> n [ / e ] ) is an isomorphism onto an open subset of X. Therefore (ii) follows from the next simple lemma, which is proved in [ 5 ] , for example. 4.6. LEMMA. Let Nx and N2 be two closed algebraic subgroups of a unipotent group N whose tangent spaces at the unit element intersect only in 0. Then the product mapping N{ X N2 -* N gives an isomorphism of N1 X N2 with a closed subvariety of N. This completes the proof of Proposition 4.3. PROOF OF PROPOSITION 4.4. Any element of \)£ has the form X = X - X', where X, X' are regular dominant weights. In this case, E% — Ex
136
/• N. Bernstein, I. M. Gel'fand, S. I. Gel'fard
Since Xv is an irreducible variety, we see that div ju = 2J ayXw', where w —*• w
ay 6 Z, ay > 0. It remains to show that ay = w'x(Hy). In view of Lemma 4.5 (i) and (ii), the coefficient ay is equal to the multiplicity of zero of-the section 5*Gu) of the fibering S*(EX) on P1 at the point o, that is, the multiplicity of zero of the function *K0 = 0u/((exp tE_y)fw') for t = 0. It follows from Lemma 2.10 that \l/(t) = ctn, hence ay = n = w'\(Hy). This completes the proof of Proposition 4.4 and with it of Theorem 4.1. §5. Generalizations and supplements
1. Degenerate flag varieties. We extend the results of the previous sections to spaces X(P) = G/P, where P is an arbitrary parabolic subgroup of G. For this purpose we recall some facts about the structure of parabolic subgroups P c G (see [7]). Let 0 be some subset of 2, and A0 the subset of A+ consisting of linear combinations of elements of 0 . Let G0 be the subgroup of G generated by H together with the subgroups Ny = {exp tEy\ I £ C} for 7 6 A0 u - A 0 , and let NQ be the subgroup of TV generated by the Ny for 7 £ A+\A 0 . Then G0 is a reductive group normalizing NQ, and P& = GQN& is a parabolic subgroup of G containing B. It is well known (see [7], for example) that every parabolic subgroup P c G is conjugate in G to one of the subgroups PQ. We assume in what follows that P = P@, where 0 is a fixed subset of 2 . Let WQ be the Weyl group of G 0 . It is the subgroup of W generated by the reflections oa, a 6 0. We describe the decomposition of X{P) into orbits under the action of B. 5.1. PROPOSITION, (i) X(P) = JJW Bwo, where o 6 X(P) is the image of P in G/P (ii) The orbits Bwxo and Bw2o are identical if WiW2l 6 W© and otherwise are disjoint. (iii) Let WQ be the set of w 6 W such that w® C A+. Then each coset of W/W@ contains exactly one element of W@. Furthermore, the element w 6 WQ is characterized by the fact that its length is less than that of any other element in the coset wWQ. (iv) / / w E W®, then the mapping Nw ->• X(P) (n -» nwo) is an isomorphism of Nw with the sub variety Bwo c X(P). PROOF, (i)—(ii) follow easily from the Bruhat decomposition for G and G 0 . The proof of (iii) can be found in [7], for example, and (iv) follows at once from (iii) and Proposition 1.1. Let w 6 W0, XW(P) = Bwo, let XW(P) be the closure of XW(P) and IXW(P)] 6 H2liw)(Xw(P), Z) its fundamental class. Let sw(P) € H2liw)(X(P), Z) be the image of [XW(P)] under the mapping
Schubert cells and cohomology of the spaces GfP
137
induced by the embedding Xw (P)C-+X(P). The next proposition is an analogue of Proposition 1.2. 5.2. PROPOSITION ([2]). The elements sw(P), w 6 W&, form a free basis in H*(X(P), Z). 5.3. COROLLARY. Let ocP: X -+ X(P) be the natural mapping. Then («p)*sw = 0 if w £ W&, (ctp)^ = sw(P) if w e W&. 5.4. COROLLARY. (a P )*: H^X, Z) -* H*(X(P), Z) w fl« epimorphism, and (aP)*: H*(X(P), Z) -» //* (jr, Z) w a monomorphism. 5.5. THEOREM, (i) Im(a P )* c i/*(X, Z) = R coincides with the set of WQ-invariant elements of R. (ii) Pw e Im(a P )* /or u, € W& and {(ccp)*-1?^^ is the basis in H*(X(P), Z) dual to the basis {sw(P)}w£Wi@ in H*(X(P), Z). PROOF. Let w 6 R/&. Since (Pw, ^ > = 0 for Wi £ H£, />„, is orthogonal to Ker(a P )^, that is, Pw 6 Im(a P )*. Now (ii) follows from the fact that <(aP)*PWi sw'(P)) =
n
(PI)*^
is given by the submanifold Pi (C) c Z. We want to study correspondences in the case Y = X = G/B. 5.6. DEFINITION. Let w £ W. We put Zw = {(gwo, go)} C X X X and denote by zw the correspondence zw = [Zw ] C H*(X X X, Z). 5.7. THEOREM. (zw)* = Fw . PROOF. We calculate (zw)# ( v ) .
138
/. N. Bernstein, I. M. Gel'fand, S. I. Gel'fand
Since the variety Zw is G-invariant and G acts transitively on X, the mapping px: Zw -* X is a fibering. Thus, It is easily verified that p\x (Xw>) = n? (Xw>) n Z ^ . W e put Y = TTI1 (*„,') n Zw c JT X X. Then (20)
Y = {(»M?'O,rcu/fcwo)| rc £ JV, 6 6 #}.
Since the dimension of the fibre of px : Zw -+ X is equal to 2l(w), we see that dim Y = 2l(w) + 2/(u/). It is clear from (20) that 9i(Y) = {nw'bwo \n £N, b £ B} = Bw'Bwo. It is well known (see [6]. Ch. IV, §2.1 Lemma 1) that Bw'Bwo = Bw'wo U (
(J
Bwto).
I (wi)
Thus, two cases can arise. a) l(w'w) < l(w) + l(w). In this case, dim p 2 (7) < 2/(u/) + 2l(w), and so (z w ) # (v) = (Pa)*m = 0. b) /(u;fu;) = /(a;') + /(u;). In this case, p2(Y) = Xw>w + X\ where dim Z' < dim Xw>w = 2/(u;') + 2l(w). Thus, ( p 2 ) j y ] = [Zw>w], that is, (zu;)*(5u;') = sw'w • Comparing the formulae obtained with 3.12 (ii), we see that (zw )* = Fw . 5.8. COROLLARY. zu; = 2 5 u; / s 0 5u;'u;> where the summation extends over those w 6 W for which l(w'w) = l(w) + /(«;'). In § 1 we have defined an action of W on H#(X, Z). This definition depended on the choice of a compact subgroup K. Using Theorem 5.7 we can find explicitly the correspondences giving this action. In fact, it follows from Lemma 3.3 (iii) that oa = a*Fa - 1 for any a E 2. The transformation Fa is given by the correspondence ZOQL. The operator a* can also be given by a correspondence: if Ua = 2^(7, is a divisor in X giving the cycle <^(a) 6 H2r-2(Xi Z) (for example, J7a = 2 a(/^)X afl ), then the cycle £/a = 2c,6^, where f/f = {(x, x) \x£Ut}cz X X X, determines the correspondence that gives the operator a*. The operator oa in H*(X, Z) is therefore given by the correspondence Ua*ZOa - 1 (where * denotes the product of correspondences, as in [11]). Using the geometrical realization of the product of correspondences (see [11]), we can explicitly determine the correspondence Sa that gives the transformation 1 + oa in H*(X, Z), namely, Sa = 2qUi where Ut = {(z, y) 6 X X X \ x 6 Uh ~x~ry 6 P{a}} • I n t h i s expression, %, y E G are arbitrary representatives of x, y, and P{ay is the parabolic subgroup corresponding to the root a. 3. B. Kostant has described the Pw in another way. We state his result.
Schubert cells and cohomology of the spaces G/P
139
Let h e % be an element such that <x(h) > 0 for all a e X. Let Jh ={fe R \ f{wh) = 0 for all w e W} be an ideal of R. 5.9. THEOREM, (i) Let w e W, l(w) = I. There is a polynomial Qw G R of degree I such that (21) Qw(wh) = 1, Qw(w'h) = 0 if l(w')^l(w), w'^w. The Qw are uniquely determined by (21) to within elements of Jh. (ii) Let Q% be the form of highest degree in the polynomial Qw. The image of Q% in R is equal to II The proof is analogous to that of Theorem 3.15. 4. We choose a maximal compact subgroup K cG such that K n B c H (see § 1). The cohomology of X can be described by means of the ^-invariant closed differential forms on X. For let x e §z» and let Ex be the corresponding one-dimensional complex G-fibering on X. Let £5X be the 2-form on X which is the curvature form of the connection associated with the ^-invariant metric on Ex (see [12]). Then the class of the form cox — cox i s cx G IP(X, Z). The mapping x "* <^>x ex*ends to a mapping 6: R -* £2*y(X), where £2*,, is the space of differential forms of even degree on X. One can prove the following theorem, which is a refinement of Proposition 1.3 (ii) and Theorem 3.17. 5.10. THEOREM (i) Ker 6 = J, that is, d induces a homomorphism of rings 0: R -* Sl*v(X). (ii) Let wx, w2 e PV, ^ 4 u;2s- 77*e« r/ze restriction of the form W(PWj ) to XWi is equal to 0. (iii) Let wx, w2 G W, Wi 4 w2s. Then B{PWx) 0 ( P J 3 ) = 0. * References
[1] A. Borel, Sur la cohomologie des espaces fibres principaux et des espaces homogenes des groupes de Lie compacts, Ann. of Math. (2) 57 (1953), 115-207;MR 14 #490. Translation: in 'Rassloennye prostranstva\ Inost. lit., Moscow 1958 [2] A. Borel, Kahlerian coset spaces of semisimple Lie groups, Proc. Nat. Acad, Sci. U.S.A. 40 (1954), 1147-1151; MR 17 # 1108. [3] B. Kostant, Lie algebra cohomology and generalized Schubert cells, Ann. of Math. (2) 77 (1963), 72-144; MR 26 # 266. [4] G. Horrocks, On the relations of S-functions to Schubert varieties, Proc. London Math. Soc. (3) 7 (1957), 265-280; MR 19 #459. [5] A. Borel, Linear algebraic groups, Benjamin, New York, 1969; MR #4273. Translation: Lineinye algebraicheskie gruppy, 'Mir', Moscow 1972. [6] N. Bourbaki, Groupes et algebras de Lie, Ch. 1—6, Elements de mathematique, 26, 34, 36, Hermann & Cie, Paris, 1960-72. Translation: Gruppyialgebry Li, 'Mir', Moscow 1972. [7] R. Steinberg, Lectures on Chevalley groups, Yale University Press, New Haven, Conn. 1967. [8] M. F. Atiyah and F. Hirzebruch, Vector bundles and homogeneous spaces, Proc.
140
[9] [10] [11] [12] [13]
/. N. Bernstein, I. M. Gel'fand, S. I. Gel"fand
Sympos. Pure Math.; Vol. Ill, 7-38, Amer. Math. Soc, Providence, R. I., 1961; MR 25 #2617. = Matematika 6: 2 (1962), 3-39. P. Cartier, On H. Weyl's character formula, Bull. Amer. Math. Soc. 67 (1961), 228-230; MR 26 #3828. = Matematika 6: 5 (1962), 139-141. E. H. Spanier, Algebraic topology, McGraw-Hill, New York 1966; MR 35 # 1007 Translation: Algebraicheskaya topologiya, 'Mir', Moscow 1971. Yu.I. Manin, Correspondences, motives and monoidal transformations, Matem. Sb. 77 (1968), 475-507; MR 41 # 3482 = Math. USSR-Sb. 6 (1968), 439-470. S. S. Chern, Complex manifolds, Instituto de Fisca e Matematica, Recife 1959; MR 22 #1920. Translation: Kompleksnye mnogoobraziya, Inost. Lit., Moscow 1961. I. N. Bernstein, I. M. Gel'fand and S. I. Gel'fand, Schubert cells and the cohomology of flag spaces, Funkts. analiz 7: 1 (1973), 64-65. Received by the Editors 13 March 1973
Translated by D. Johnson
FOUR PAPERS ON PROBLEMS IN LINEAR ALGEBRA Claus Michael Ringel This volume contains four papers on problems in linear algebra. They form part of a general investigation which was started with the famous paper [Q] on the four subspace problem. The r subspace problem asks for the determination of the possible positions of r subspaces in a vector space, or, equivalently, of the indecomposable representations of the following oriented graph
(*) with r + 1 vertices. For r > 5, this problem seems to be rather hard to attack, however one may try to obtain at least partial results dealing with special kinds of representations. Also, the r subspace problem can be used as a test problem for more elaborate problems in linear algebra. This seems to be the case for some of the investigations published in this volume, they have been generalized recently to the case of arbitrary oriented graphs [M, S]. Three of the four papers deal with the r subspace problem. (We should remark that there is a rather large overlap of [F] and [I, II]. However, the main argument of [F], the proof given in section 7, is not repeated in [I, II], whereas [I, II] give the details for the complete irreducibility of the representations pt j which only was announced in [F]. We also recommend the survey given by Dlab [8].) Given r subspaces Ex, . . .,Er of a finite-dimensional vector space V, we obtain a lattice homomorphism p from the free modular lattice Dr with r generators ex, . . ., er into the lattice L(V) of all subspaces of V given by p(ef) = Et. Such a lattice homomorphism is called a representation ofDr. In [F], Gelfand and Ponomarev introduce a set of indecomposable representations ptj with 0 < t < r and / E N , which we will call the preprojective representations (in [F], the representations pt j with 1 < t
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Claus Michael Ringel
[F]: one first defines a finite set At(r, I) (which later we will identify with a set of paths in some oriented graph), considers the vector space with basis the set At(r, /), and also a subspace Zt(r, I) generated by certain sums of the canonical base elements of At(r, I). The residue classes of the canonical base elements of At(r, /) in Vtl = At(r, l)/Zt(r, /) will be denoted by £a (with ocGAt(r, /)). Now, the representation pt z is given by the vector space Vtj together with a certain r tuple of subspaces of Vt /, all being generated by some of the generators £ a . Note that this implies that ptl is defined over the prime field k0 of k. (Gelfand and Ponomarev usually assume that the characteristic oik is zero, thus k0 = Q. However, all results and proofs remain valid in general.) The main result concerning these representations pt / asserts that in case dim Vt i > 2, the representation pt / is completely irreducible. This means that the image of Dr under the lattice homomorphism ptl\ Dr -* L(Vtj) is the set of all subspaces of Vt / defined over the prime field k0, thus pt j(Dr) is a projective geometry over k0. The first essential step in the proof of this result is to show that the subspaces k%a are of the form p(ea) for some ea GDr. (In [F], this is only announced, but it is an immediate consequence of theorem 8.1 in [II].) The second step is to show that any subspace of Vtj which is defined over the prime field, lies in the lattice of subspaces generated by the k%a provided dim Vtl>2. Combining both assertions, we conclude that ptj is completely irreducible unless dim Vt j < 2. The proof of the second step occupies section 9 of [II]. Here, one considers the following situation: there is given a set R = {£a | a } of non-zero vectors of a vector space V (= Vtj), with the following properties: (1) R generates V (2) R is indecomposable (there is no proper direct decomposition V= V © V" withR = (R H V') U(RD V")), and (3) R is defined over the prime field (there exists a basis of V such that any £a 6 R is a linear combination of the base vectors with coefficients in the prime field fr0). Then it is shown that the lattice of subspaces of V generated by the onedimensional subspaces k%a, is isomorphic to the lattice of subspaces offrg> with n = dim V. Perhaps we should add that the representations p: Dr -* L(V) with V being generated by the one-dimensional subspaces of the form p(a), aGDr, seem to be of special interest. In this case, the one-dimensional subspaces of the form p(a), a^Dr determine completely p(Dr). (Namely, let b €Dr, and U the subspace generated by all one-dimensional subspaces of the form p(x), x satisfying p(x) C p(b), and choose xx, . . ., xs such that p(b) C£/ep(jc 1 )0...ep(jc J ) = tf©p(2; x/).Thus,p(Z?)=C/©(p(i: xt) /=1
i=1
Assume, U is a proper subspace of p(b). Then there exists t < s with
143
Four papers on problems in linear algebra t- 1
t
/=l
i= l
p( 2 xA n p(b) = 0, whereas p( 2 *.•) n p(Z>) is non-zero, and therefore onedimensional. This however implies that p ( 2 JC) PI p(^) = p(b 2 xz-) is i-\
/= l
contained in t/, a contradiction. Thus p(b) = t/.). For r > 4, there always are indecomposable representations which do not have this property. In the case r = 4, we may give the complete list of all lattices of the form p(D4), where p is an indecomposable representation. Besides the projective geometries over any prime field, and of arbitrary finite dimension =£ 1, and the lattice
, we obtain all the lattices S{n, 4) introduced by Day,
Herrmann and Wille in [6]. Let us just copy S(14, 4) and note that any interval [cn, cm ] is again of the form S(n - m, 4).
(In fact, in case either p: D4 -* L(V) or its dual is preprojective and dim V> 2, we have seen above that p(D4) is the full projective geometry over the prime field. If neither p nor its dual is preprojective, p is said to be regular. If p is
Claus Michael Ringel
144
regular and non-homogeneous, say of regular length n (see [9]), then p(D 4 ) « S(n, 4), whereas for p homogeneous, we have
Gelfand and Ponomarev use the representations ptj ofDr in order to get some insight into the structure of Dr. The existence of a free modular lattice with a given set of generators is easily established, however the mere existence result does not say anything about the internal structure ofDr. In fact, it has been shown by Freese [14] that for r > 5, the word problem in Dr is unsolvable. The free modular lattice/) 3 in 3 generators elf e2, e3 was first described by Dedekind [7], it looks as follows:
We have shaded two parts of D3, both being Boolean lattices with 2 3 elements. For r > 4, Gelfand and Ponomarev have constructed two countable families of Boolean sublattices B+(l) and B~(l) with 2r elements, where / G N, and such that and
. . . B\l
. . .
called the lower and the upper cubicles, respectively. Let B = U B (/), and
Four papers on problems in linear algebra
145
B*= U B\l). /EN
The elements of these cubicles have an important property: they are perfect. This notion has been introduced by Gelfand and Ponomarev in [F] for the following property: a is said to be perfect if p(a)is either O or V for any indecomposable representation p: Dr -> L{V). This means that for any representation, the image of a is a direct summand. For any perfect element a, let Nk(a) be the set of all indecomposable representations p: Dr -* L(V), with F a finite dimension vector space over the field k and which satisfy p(a) = 0. It is shown in [F] that fora G5 + , the setNk(a) is finite and contains only preprojective representations. Dually, for
146
Oaus Michael Ringel
p'\Dr -+ L(V/U) the induced representation, with p'(e() = (p(et) + U)/U for the generators et, 1 < / < r . We claim that for x E D r , w e have dim p'(x) < c(x) — 1 + dim p(x). [For the proof, we consider instead of p the representation p": Dr -> Z(K) with p"(x) the full inverse image of p'(x) under the projection V ^ V/U, thus dim p"(x) = 1 + dim p'(x), for* eDr. Also note that p(x) C p"(x) for all*. By induction on c(x), we show the formula dim p"(x) — dim p(x) < c(x). Since dim U = 1, this clearly is true for x = et, with p"(ei) = p(et) + £/. Now assume the formula being valid both for x1 and x2. For x = x x + x 2 with c(x) = c(xi) + c(x 2 ), we have dimp"0c) = dim p"{xx + x 2 ) < d i m p ( x ! + x2) + c(xx) + c{x2) = dim p(x) + c(x). Similarly, for x = xtx2
with c(x) = c{xx) + c(x 2 ), we have
dim p"(x) = dim p"(x1x2) = dim p"(Xj) + dim p"(x 2 ) - dim p'^Xj
Four papers on problems in linear algebra
147
thus, the inclusion ft =$-'(*) o . . . o $-ifo>) o p. PQl
^POJ+2
has regular cokernel (extensions of regular representations being regular, again). We now only have to choose i such that dim Voj+2 > c(a). This finishes the proof in case r > 5. (For r = 4, we again take
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Claus Michael Ringel
modulo all exact sequences) with dim being the canonical map (sending a representation to the corresponding residue class). On Ev, there is defined a quadratic form B. In fact, for any orientation A of F, we may consider the (non-symmetric) bilinear form BA on Ev given by
BA(x,y) = 2 xOiya/ e r0
2
xa0)yM
/er,
and B is the corresponding quadratic form B(x) = BA (x, x). Note that B is positive definite if and only if F is the disjoint union of graphs of the form An, Dn, E6, En and E8, and in these cases, the root system for F is by definition just the set of solutions of the equation B(x) = 1. For k algebraically closed and B being positive definite we will outline a direct proof that dim: L(T, A) -* Er induces a bijection between the indecomposable representations of (F, A) and the positive roots. There is the following algebraic-geometric interpretation of B due to Tits [ 15]: The representations of (F, A) of dimension type x may be considered as the algebraic variety ra*(F, A) = and there is an obvious action on it by the algebraic group
Gx = n
c*er0
GL(a, k)/A
with A being the multiplicative group of k diagonally embedded as group of scalars. Clearly B(x) = dim Gx + 1 - dim mx(T, A). Using this interpretation, Gabriel has shown in [16] that it only remains to prove that the endomorphism ring of any indecomposable representation is k. So assume Fis indecomposable, and that there are non-zero nilpotent endomorphisms. Then V contains a subrepresentation (/with End(£/) = k and Ext1 (U, U)¥=0. [Namely, let 0 =£ \p be an endomorphism with image S of r
smallest possible length, thus
with all Wt indecomposable. Now S C W, thus the projection of S into some Wj must be non-zero. Since S was an image of a non-zero endomorphism of smallest length, we see that S embeds into this Wt. We may assume / = 1. Thus there is an inclusion t: S -> Wx. If Wx has non-zero nilpotent endomorphisms, we use induction. Otherwise End(H/1) = k. Also, ExtH^i, ^ i ) ^ 0> since on the one hand Ext 1 ^, Wx) =£ 0 due to the exact sequence i=1
Four papers on problems in linear algebra
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and, on the other hand, the inclusion i gives rise to a surjection Ext1 (t, W). Here we use that L(T, A) is a hereditary category]. The bilinear form BA has the following homological interpretation [25]: £ A (dim V, dim V) = dim* Hom(F, V) - dim^ Ext^K, F'), for all representations V, V. Consequently, the existence of a representation U satisfying End(U) = k, Ext1 (U, U) ¥= 0 would imply that B(dim U) = BA (dim £/, dim U) < 0, contrary to the assumption that 5 is positive definite. This finishes the proof. For any finite connected graph F without loops, Kac [21, 22] gave a purely combinatorial definition of its root system A. Note that A is a subset of ET containing the canonical base vectors 0, for 0 E F o , and being stable under the Weyl group W, the group generated by the reflections a^ along j3 with respect to B. The set A can also be interpreted in terms of root spaces of certain (usually infinite dimensional) Lie algebras [21]. Denote by A+ the set of roots with only non-negative coordinates with respect to the canonical basis. Then A is the union of A+ and A_ = - A+. In case F is of type An, Dn, E6, E1 or E8, the root system is finite and coincides with the set of solutions of B(x) = 1. Otherwise the root system is infinite and will contain besides certain solutions of B(x) = 1 also some solutions of B(x) < 0. The elements x of the root system which satisfy B(x) = 1 are called real roots, they are precisely the elements of the W-orbits of the canonical base elements. The remaining elements of the root system are called imaginary roots, and Kac has determined a fundamental domain for this set, the fundamental chamber. Now, one has the following results (at least if k is either finite or algebraically closed): For any finite graph F without loops, and any orientation A, the set of dimension types of indecomposable modules is precisely the set A+ of positive roots. For any positive real root x, there exists precisely one indecomposable representation V of (F, A) with dim V = x. For any positive imaginary root x, the maximal dimension JJLX of an irreducible component in the set of isomorphism classes of indecomposable representations of dimensions is precisely 1 ~B(x, x). (Note that the subset of indecomposable representations in mx (F, A) is constructive, and Gx -invariant, thus we can decompose it as a finite disjoint union of Gx -invariant subsets each of which admits a geometric quotient. By definition, JJLX is the maximum of the dimensions of these quotients.) In particular, we see that the number of indecomposable representations (or of the maximal dimension of families of indecomposable representations) of (F, A) does not depend on the orientation A. For F of the form An, Dn, E6, E7 or E8, this is Gabriel's theorem (of course, there are no imaginary roots). For F of the form An, Dn, E6, En, or E8, the so called tame cases, these results have been shown by Donovan—Freislich [13] and Nazarova [23], see also [9]; in fact, in these
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cases one obtains a full classification of all indecomposable representations; also, it is possible in these cases to describe completely the rational invariants of the action of Gx on mx (F, A), for any dimension type x, see [27]. Of course, the oriented graphs of finite or tame representation type are rather special ones. It has been known since some time that the remaining (F, A) are wild: there always is a full exact subcategory of Z,(F, A) which is equivalent to the category Mk{X, Y > of k (X, Y >-modules (k (X, Y) being the polynomial ring in two non-commuting indeterminates). In this situation, the results above are due to Kac [21, 22]. Note that this solves all the conjectures of Bernstein—Gelfand—Ponomarev formulated in [BGP]. However, there remain many open questions concerning wild graphs (F, A). One does not expect to obtain a complete classification of the indecomposable representations of such a graph, but one would like to have some more knowledge about certain classes of representations. For example, there does not yet exist a combinatorial description of the set of those roots which are dimension types of representations V with End(F) = k. We have mentioned above that the root system A of F is stable under the Weyl group W and that any W-orbit of A contains either one of the base vectors j3 (with j3 £ F o ) or an element of the fundamental chamber. One therefore tries to find operations which associate to an indecomposable representation V of (F, A) with A an orientation, and a Weyl group element w E W a new indecomposable representation of (F, A'), where A' is a possibly different orientation of F. By now, several such operations are known (see [BGP, 21, 28]), the first one being the reflection functors Ff, F* introduced by Bernstein, Gelfand and Ponomarev in [BGP]. Here, for the definition of Fp, the vertex 0 is supposed to be a sink, thus the simple representation Lp with dimension vector dim Lp = J3 is projective. This concept has been generalized by Auslander, Platzeck and Reiten [ 1 ] dealing with any finite dimensional algebra A (or even an artin algebra) with a simple projective module L. For this, we need the Auslander—Reiten translates T,T~1 . Recall p
that TXA is defined for any A -module XA : let Px -* Po -> XA -* 0 be a minimal projective resolution of XA, then Tr XA is by definition the cokernel of the map Hom(p, AA ) and TX = D Tr X, r ~* X = Tr D X, with D the usual duality with respect to the base field k. So assume L is a simple projective A -module, let P be the direct sum of one copy of each of the indecomposable projective modules different from L, and B = End(P © r" 1 L). The functor considered by Auslander, Platzeck and Reiten is F= Hom^ (P © r" 1 L, —) from the category MA of A -modules to MB . The functor induces an equivalence of the full subcategory T of MA of all modules which do not have L as a direct summand and a certain full subcategory of MB. Note that P © r" 1 L is a tilting module in the sense of [ 18], except in the trivial case of L being, in addition, injective. (A tilting module TA is defined by the following three properties:
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(1) proj. dim. TA < 1, (2) there exists an exact sequence 0 -*AA -+T' ^T" -• 0, with T', T" being direct sums of direct summands of TA , and (3) Ext1 (TA, TA ) = 0. Now, if LA is simple projective and not injective, the middle term Y of the Auslander—Reiten sequence starting with L is projective. This sequence shows, on the one hand, that proj. dim. T~1L = 1. On the other hand, it also gives an exact sequence of the form needed in (2). Finally, Ext^ (P © T~1L, P © r" 1 L) « ^ D Hom(P © T~1L, L) = 0, since any non-zero homomorphism from a module to L is a split epimorphism.) A certain composition of the reflection functors F£ (or Fp, respectively) is of particular interest, the Coxeter functor
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epimorphism [2]). Now, the Auslander—Reiten quiver is a translation quiver with respect to r: if X is indecomposable and not projective, then there exists an irreducible map Y ->• X iff there exists an irreducible map TX -> Y. For the finite dimensional hereditary algebras A, the structure of the Auslander—Reiten quiver is known. We will recall this result in the special case of A = k(F, A). First, we need some notation. Define Z(F, A) as follows: o ] 9 there its vertices are the elements of F o X Z, and for any arrow l o « (a, z) (a*,z) * (/, z) and (/, z) • (/, z + 1), for all z E Z, see are arrows (/, z) [24] and also [ 17, 29]. Note that in case F is a tree, Z(F, A) does not depend on the orientation A and just may be denoted by ZF. If / C Z, let /(F, A) be the full subgraph of all vertices (z, z) with / G /. In particular, we will have to consider N(F, A) and N"(F, A), where N = { 1, 2, 3, . . .} and N ~= { — 1, - 2 , - 3 , . . .}. Also, denote by A^ the following infinite graph
The result is as follows: in case F is of the form An, Dn, E6, En or Es, the Auslander-Reiten quiver of fc(F, A) is a finite full connected subquiver of ZF. (In case Dn with n = 0(2), the Auslander-Reiten quiver of k(F, A) is [ l , r c - l ] (F, A), in case of En o r £ 8 , i t i s [1,9] (F, A) or [1,15] (F, A), respectively, in the remaining cases, it is slightly more difficult to describe, see [17, 29]). In all other cases, the Auslander—Reiten quiver of A:(F, A) has infinitely many components, all but two being quotients of ZA^ (see [26]), the remaining two being of the form N(F, A) and N "(F, A). The component of the form N(F, A) contains the indecomposable projective modules: in fact, the indecomposable projective module Pf corresponding to the vertex / G F o appears as indexed by (/, 1), and the module indexed by (/, z), z EN, is just <£~z + 1 (P.)f this component is called the preprojective component. Similarly, the component of the form N"(F, A) is called the preinjective component, it contains the indecomposable injective module Jt corresponding to / E F o as indexed by (i, —1), and the module indexed by (/, -z), z G N, is just ^>+z-1 (J.). Let us consider in more detail a preprojective component &, and the modules belonging to:^; they will be called preprojective modules. In case F is of type An, Dn, E6, E-j, or E8, we let .^denote the full Auslander—Reiten quiver; in any case, we note that an indecomposable representation of (F, A) is said to be preprojective iff it is of the form "ZP, with P indecomposable projective and z > 0. (A general theory of preprojective modules has been developed by Auslander and Smal0, see [3]). For an indecomposable preprojective representation X, there are only finitely many indecomposable modules Y such that Hom(y, A") =£ 0, all of them are preprojective again, and any noninvertible homomorphism Y -> X is a sum of compositions of irreducible maps. In particular, if X, Y are indecomposable and preprojective and
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Hom(X, Y) =£ 0, then there is an oriented path [X] - • . . . - » [Y] m0>. In fact, the complete categorical structure of the full subcategory of preprojective modules can be read off from the combinatorial description of ^ a s a translation quiver: the category of all preprojective modules is equivalent to the quotient category <} 0* of the path category of ^modulo the so called mesh relations (see [4, 24, 17]). Note that the category ()0> allows to reconstruct all the modules in 0*. Namely, any module XA is isomorphic to H o m ^ ^ , XA ), thus, if AA = 0 P?i9 then XA can be identified with © Hom(Pz., X)nf, and Hom(P/f X) can be calculated inside 0-^, since both Pj, X are preprojective. Starting from the preprojective component 0 of A;(F, A), one may define a (usually infinite-dimensional) algebra II as follows: Take the direct sum of all homomorphism spaces Hom(/\ 1), it, /)) in <> 0* and define the product of two residue classes w, w' of paths w: (/, 1 ) - * . . . - * (f, /) and 10': (/', 1)-* . . . - * ( * ' , / ' ) as follows: in case t=j',letwwf be the residue class of the composed path r" /+1 (">')° w: (/, 1)"*. • • "• (*', / + / ' - l),andO otherwise. There is a purely combinatorial description of II in terms of (F, A) due to Gelfand and Ponomarev, see [R]. Let F be obtained from (F, A) by adding to each arrow a: i -•/ an additional arrow «*:/-> /. We clearly can identify II with the factor algebra of the path algebra kT modulo the ideal generated by the element £ era* + Z a**a. Note that this description is independent of the choice of the orientation A. Also, we see from both descriptions that II contains as a subalgebra &(F, A), thus we may consider II as a right k(F, A)-module, and the first description now shows that the A:(F, A)module n ^ ( r A ) decomposes as the direct sum of all preprojective representations of (F, A) each occurring with multiplicity one, and therefore is called the preprojective algebra of F. (For the proper generalisation to the case of a species, we refer to [11]. We also should note the slight deviation of the preprojective algebra from the model algebra defined in [M], which reduces to the algebra Ar given in [I, II] in the case of the r-subspace situation. Namely, here the constant paths have square zero, whereas they are idempotents in n. Now, in II the sum of the constant paths is the identity element. In order also to have an identity element, Gelfand and Ponomarev add to the direct sum of all preprojective modules an additional one-dimensional space ke. There is a change of definition proposed in [S], using the constant paths as idempotents as in II, but adding again an additional identity element.) Since II is the direct sum of the preprojective representations of (F, A), it follows that II is finite dimensionaHf and only if F is of the form^4w, Dn, E6, Elf or Es. In [12], the tame cases An, Dn, E6, E7 and Es have been characterized by the fact that the Gelfand-Kirillov dimension of II is 1, whereas it is °° for the wild cases.
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Let us return to the special case of the r subspace graph (*), with r > 4. The description above gives that the preprojective component ^is of the form
(0,1)
(0,2)
(0,3)
If we denote the arrows in the following way:
then the mesh relations are as follows: at af = 0 for all /, and 2 ai at = 0. Thus, if we want to determine the total space of the representation labelled (t, /), we have to calculate Hom((0, 1), {t, /)) inside the category 0 ^ , and this amounts to the calculation of all possible paths from (0, 1) to (t, /), taking this as the basis of a vector space and factoring out the mesh relations. However, taking from the beginning into account the relations <xf af = 0, we just as well may work with the vector space generated by the set At{r, I) and factoring out the remaining mesh relations. This shows that we obtain as total space the vector space Vtj. Similarly, the r different subspaces of the representation labelled {t, I) are given by the various Hom((y, 1), (t, /)), 1 < / < r , again calculated in 0 ^ , and therefore coincide with the subspaces ptj(e;-). In this way, we obtain directly the description of the preprojective representations of Dr given by Gelfand and Ponomarev (and a direct proof of Proposition 8.2 in [F]). Finally, let us note in which way the preprojective component of Dr determines the lattice B + of perfect elements belonging to the upper cubicles. For any perfect element a, we have denoted by N(a) the set of indecomposable representation p satisfying p(a) = 0. We claim that for a £B+, the set N(a) is a
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finite, predecessor closed subset of :/(an element x is said to be a predecessor of y in case there is an oriented path x -+ . . . ->y. For the proof, we first note that clearly N(a) H;/is predecessor closed, since for indecomposable representations p, p with Hom(p, p) J= 0, and a perfect, p GN(a) implies p GN(a). Since not all of:/is contained in N(a), it obviously follows that N(a) Hi/is finite. However, any complete slice of
[Q]
Gelfand, Ponomarev: Problems in linear algebra and classification of quadruples in a finite dimensional vector space. Coll. Math. Soc. Bolyai 5, Tihany (1970), 163-237. [BGP] Bernstein, Gelfand, Ponomarev: Coxeter functors and Gabriel's theorem. Uspekhi Mat. Nauk 28 (1973), Russian Math. Surveys 28 (1973), 17-32, also in this volume. [F] Gelfand, Ponomarev: Free modular lattices and their representations. Uspekhi Math. Nauk 29 (1974), 3-58. Russian Math. Surveys 29 (1974), 1-56, also in this volume. [I] Gelfand, Ponomarev: Lattices, representations and algebras connected with them. I.Uspekhi Math. Nauk 31 (1976), 71-88. Russian Math. Surveys 31 (1976), 67—85, also in this volume. [II] Gelfand, Ponomarev: Lattices, representations and algebras connected with them. II. Uspechi Math. Nauk 32 (1977), 85-106. Russian Math. Surveys 32 (1977), 91—114, also in this volume. [M] Gelfand, Ponomarev: Model algebras and representations of graphs. Funkc. Anal, i Pril. 13.3 (1979), 1-12. Funct. Anal. Appl. 13 (1979), 157-166. [R] Rojter: Gelfand-Ponomarev algebra of a quiver. Abstract, 2nd ICRA (Ottawa 1979). [S] Gelfand, Ponomarev: Representations of graphs. Perfect subrepresentations. Funkc. Anal, i Pril. 14.3 (1980), 14-31. Funct. Anal. Appl. 14 (1980), 177-190. [1] [2] [3] [4] [5] [6]
Auslander, Platzek, Reiten: Coxeter functors without diagrams. Trans. Amer. Math. Soc. 250(1979), 1-46. Auslander, Reiten: Representation theory of artin algebras III, IV, V. Comm. Algebra 3 (1975), 239-294; 5 (1977), 443-518; 5 (1977), 519-554. Auslander, Smal0: Preprojective modules over artin algebras. J. Algebra (to appear). Bautista: Irreducible maps and the radical of a category. Preprint. Brenner, Butler: The equivalence of certain functors occurring in the representation theory of artin algebras and species. J. London Math. Soc. 14 (1976), 183-187. Day, Herrmann, Wille: On modular lattices with four generators. Algebra Universal 3 (1972), 317-323.
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[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
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Dedeking: Ober die von drei Moduln erzeugte Dualgruppe. Math. Ann. 53 (1900), 371-403. Dlab: Structure des treillis lineaires libres. Seminaire Dubreil. Springer LNM 795 (1980), 10-34. Dlab, Ringel: Indecomposable representations of graphs and algebras. Mem. Amer. Math. Soc. 173(1976). Dlab, Ringel: Perfect elements in the free modular lattices. Math. Ann. 247 (1980), 95-100. Dlab, Ringel: The preprojective algebra of a modulated graph. Springer LNM 832 (1980), 216-131. Dlab, Ringel: Eigenvalues of Coxeter transformations and the Gelfand-Kirillov dimension of the preprojective algebra. Proc. Amer. Math. Soc. (to appear). Donovan, Freislich: The representation theory of finite graphs and associated algebras. Carleton Lecture Notes 5 (1973). Freese: Free modular lattices. Trans. AMS 261 (1980), 81-91. Gabriel: Unzerlegbare Darstellungen I. Manuscripta Math. 6 (1972), 71-103. Gabriel: Indecomposable representations II. Symposia Math. Inst. Naz. Alta Mat. 11 (1973), 81-104. Gabriel: Auslander-Reiten sequences and representation finite algebras. Springer LNM 831 (1980), 1-71. Happel, Ringel: Tilted algebras. Trans. Amer. Math. Soc. (to appear). Herrmann: Rahmen und erzeugende Quadrupeln in modularen Verbanden. To appear in Algebra Universalis. Hutchinson: Embedding and unsolvability theorems for modular lattices. Algebra Universalis 7 (1977), 47-84. Kac: Infinite root systems, representations of graphs and invariant theory. Inv. Math. 56 (1980), 57-92. part II: preprint. Kac: Some remarks on representations of quivers and infinite root systems. Springer LNM 832 (1980), 311-327. Nazarova: Representations of quivers of infinite type. Izv. Akad. Nauk. SSSR. Ser. Mat. 37 (1973), 752-791. Riedtmann: Algebren, Darstellungskocher, Uberlagerungen und Zuriick. Comment. Math. Helv. 55 (1980), 199-224. Ringel: Representations of A'-species and bimodules. J. Algebra 41 (1976), 269-302. Ringel: Finite dimensional algebras of wild representation type. Math. Z. 161 (1978), 235-255. Ringel: The rational invariants of tame quivers. Inv. Math. 58 (1980), 217-239. Ringel: Reflection functors for hereditary algebras. J. London Math. Soc. (2) 21 (1980), 465-479. Ringel: Tame algebras. Springer LNM 831 (1980), 137-287.
COXETER FUNCTORS AND GABRIEL'S THEOREM I. N. Bernstein, I. M. Gel'fand, and V. A. Ponomarev It has recently become clear that a whole range of problems of linear algebra can be formulated in a uniform way, and in this common formulation there arise general effective methods of investigating such problems. It is interesting that these methods turn out to be connected with such ideas as the Coxeter—Weyl group and the Dynkin diagrams. We explain these connections by means of a very simple problem. We assume no preliminary knowledge. We do not touch on the connections between these questions and the theory of group representations or the theory of infinite—dimensional Lie algebras. For this see [3]—[5]. Let F be a finite connected graph; we denote the set of its vertices by P o and the set of its edges by 1^ (we do not exclude the cases where two vertices are joined by several edges or there are loops joining a vertex to itself). We fix a certain orientation A of the graph F; this means that for each edge / G Vi we distinguish a starting-point a(l) e r 0 and an end-point
«/)e r0.
With each vertex a G F o we associate a finite-dimensional linear space Va over a fixed field K. Furthermore, with each edge / G r 2 we associate a linear mapping / ) : Va0) -+ VpU) (a(l) and <3(/) are the starting-point and end-point of the edge /). We impose no relations on the linear mappings /,. We denote the collection of spaces Va and mappings fx by (V, f). DEFINITION 1. Let ( r , A) be an oriented graph. We define a category X (F, A) in the following way. An object of X(T, A) is any collection (K, f) of spaces Va (a G r 0 ) and mappings // (/ G F j ) . A morphism
K ar
«i
ri commutative, that is, v>^(i)/( ~ 8i<Pau)157
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Many problems of linear algebra can be formulated in these terms. For example, the question of the canonical form of a linear transformation /: V -» V is connected with the diagram
The classification of a pair of linear mappings fx : Vx -> V2 and f2 : Vy -> F2 leads to the graph
A very interesting problem is that of the classification of quadruples of subspaces in a linear space, which corresponds to the graph
This last problem contains several problems of linear algebra.1 Let (F, A) be an oriented graph. The direct sum of the objects (V, f) and (U, g) in g(T, A) is the object (W, h), where Wa = Va ® Uat hi =fi®gi(*e r0, / e r , ) . We call a non-zero object (V, f) € X (r, A) indecomposable if it cannot be represented as the direct sum of two non-zero objects. The simplest indecomposable objects are the irreducible objects La (a £ r 0 ) , whose structure is as follows: (La)y = 0 for y =£ a, (La)a = K, fx = 0 for all / e IV It is clear that each object (F, f) of X (r, A) isisomorphic to the direct sum of finitely many indecomposable objects.2 In many cases indecomposable objects can be classified.3 In his article [1] Gabriel raised and solved the following problem: to find all graphs (r, A) for which there exist only finitely many non-isomorphic indecomposable objects (V, f) e X (r, A). He made the following Let us explain how the problem of the canonical form of a linear operator /: V -* V reduces to that of a quadruple of subspaces. Consider the space W = V 0 V and in it the graph of/, that is, the subspace E4 of pairs (£,/£), where J 6 K . The mapping/is described by a quadruple of subspaces in W, namely Ex = V 0 0, E2 = 0 © V, E3 = {(£, *) I * e V)(E3 is the diagonal) and E4 = {(£,/£) | % e V}- the graph of/. Two mappings/and/' are equivalent if and only if the quadruples corresponding to them are isomorphic. In fact, Ex and E2 define "coordinate planes" in W, E3 establishes an identification between them, and then EA gives the mapping. It can be shown that such a decomposition is unique to within isomorphism (see [6], Chap. II, 14, the Krull-Schmidt theorem). We believe that a study of cases in which an explicit classification is impossible is by no means without interest. However, we should find it difficult to formulate precisely what is meant in this case by a "study" of objects to within isomorphism. Suggestions that are natural at first sight (to consider the subdivision of the space of objects into trajectories, to investigate versal families, to distinguish "stable" objects, and so on) are not, in our view, at all definitive.
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surprising observation. For the existence of finitely many indecomposable objects in X (F, A) it is necessary and sufficient that F should be one of the following graphs: (n vertices, n> 1) (n vertices, 4)
£7
(this fact does not depend on the orientation A).The surprising fact here is that these graphs coincide exactly with the Dynkin diagrams for the simple Lie groups.1 However, this is not all. As Gabriel established, the indecomposable objects of X (F, A) correspond naturally to the positive roots, constructed according to the Dynkin diagram F. In this paper we try to remove to some extent the "mystique" of this correspondence. Whereas in Gabriel's article the connection with the Dynkin diagrams and the roots is established a posteriori, we give a proof of Gabriel's theorem based on exploiting the technique of roots and the Weyl group. We do not assume the reader to be familiar with these ideas, and we give a complete account of the necessary facts. An essential role is played in our proof by the functors defined below, which we call Coxeter functors (the name arises from the connection of these functors with the Coxeter transformations in the Weyl group). For the particular case of a quadruple of subspaces these functors were introduced in [2] (where they were denoted by 4>+ and 4>~). Essentially, our paper is a synthesis of Gabriel's idea on the connection between the categories %{T, A) with the Dynkin diagrams and the ideas of the first part of [2], where with the help of the functors 3>+ and
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We hope that this technique is useful not only for the solution of Gabriel's problem and the classification of quadruples of subspaces, but also for the solution of many other problems (possibly, not only problems of linear algebra). Some arguments on Gabriel's problem, similar to those used in this article, have recently been expressed by Roiter. We should also like to draw the reader's attention to the articles of Roiter, Nazarova, Kleiner, Drozd and others (see [3] and the literature cited there), in which very effective algorithms are developed for the solution of problems in linear algebra. In [3], Roiter and Nazarova consider the problem of classifying representations of ordered sets; their results are similar to those of Gabriel on the representations of graphs. § 1. Image functors and Coxeter functors
To study indecomposable objects in the category X (F, A) we consider "image functors", which construct for each object V & X (F, A) some new object (in another category); here an indecomposable object goes either into an indecomposable object or into the zero object. We construct such a functor for each vertex a at which all the edges have the same direction (that is, they all go in or all go out). Furthermore, we construct the "Coxeter functors" 3>+ and 3>~, which take the category X (F, A) into itself. For each vertex a e F o we denote by Ta the set of edges containing a. If A is some orientation of the graph F, we denote by oaA the orientation obtained from A by changing the directions of all edges / e F a . We say that a vertex a is (—)-accessible (with respect to the orientation A) if 0(/) =£ a for all / G F2 (this means that all the edges containing a start there and that there are no loops in F with vertex at a). Similarly we say that the vertex 0 is (+)-accessible if a(/) =£ 0, for all / e Ti. DEFINITION 1.1 1) Suppose that the vertex 0 of the graph F is (+)-accessible with respect to the orientation A. From an object (F, f) in X(V, A) we construct a new object (W, g) in X (F,a^A). Namely, we put Wy = Vy for y =£ 0. Next we consider all the edges lx, l2, . . ., h that end at 0 (that is, all k
edges of F^). We denote by Wp the subspace in the direct sum © VaOi) consisting of the vectors v = (vx, . . ., vk) (here vt e VocO.)) for which fi.(Pi)
+ . . . + fik
(ufc) = 0. In other words, if we denote by h the
k
mapping h: © Va^t) ->- V$ defined by the formula
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h(uu v2, . . ., vh) = Ux (Vi) + . . . + fih(vh), then Wp = Ker h. We now define the mappings gt. For I $ Fp we put gt = ft. If I = lj G r^, then gi is defined as the composition of the natural embedding of Wp in © VaOi) and the projection of this sum onto the term V^^p = Wa(lj). We note that on all edges / G r^ the orientation has been changed, that is, the resulting object (W, g) belongs to X (r, a^A). We denote the object (W, g) so constructed by Fp(V, f). 2) Suppose that the vertex a e r 0 is (—)-accessible with respect to the orientation A. From the object (V, f) G X (r, A) we construct a new object F-(F, /) = (R/, g) e X (r, a a A). Namely, we put Wy = Vy for 7 =^ a gl = U for Z $ r« ^
= © F^(/.)/Im /z, where {lu . . . , Zft} = r a , and the mapping
h'Va-+ © ^p(^) is defined by the formula h (u) = (fti (y), . . ., /Zfc (L>)). If / G r a , then the mapping gt: Wp0) -> H^a is defined as the composition h
of the natural embedding of WP0)= VPil) in 0 V^i^ and the projection i=l
of this direct sum onto Wa. It is easy to verify that F£ (and similarly F~) is a functor from X (r, A) into X(F, a^A)(or ^ ( r , a a A), respectively). The following property of these functors is basic for us. THEOREM 1.1 1) Let (T, A) be an oriented graph and let 0 G r 0 be a vertex that is {-^-accessible with respect to A. Let V G X (F, A) be an indecomposable object. Then two cases are possible: a) V « Lp and F^V = 0 (we reca// that Lp is an irreducible object, defined by the condition (Lp)y = 0 for y * p, (Lfi)p = K, fx = 0 for all
/er,). b) F*(V) is an indecomposable object, F^F^(V) = V, and the dimensions of the spaces F^(V)y can be calculated by the formula (1.1.1) dim F$(V)y = dim F v for y =^= p, dimF a ( 0 .
2) // the vertex a is (-)-accessible with respect to A and if V G X (r, A) is an indecomposable object, then two cases are possible: a) V * La, F~(V) = 0.
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b) F~{V) is an indecomposable object, F*F~{V) = V, (1.1.2) dim F~(V)V = dim Vy for y =/= a,
+ 2 di PROOF. If the vertex 0 is (+)-accessible with respect to A, then it is (-)-accessible with respect to o^A, and so the functor F^F^'X(T, A)-+X{T, A) is defined. For each object V e%(r, A) we construct a morphism fiy: F~^F*{V) -> V in the following way. If 7 * P, then F~F+(F) 7 = Vy, and we put (i* ) 7 = Id, the identity mapping. For the definition of (fv)p we note that in the sequence of mappings > 0 Va(i)--> V3 (see definition 1.1) Ker h = Im /z ; we take for zerP the natural mapping It is easy to verify that fy is a morphism. Similarly, for each (—)-accessible vertex a we construct a morphism pav: F -• F^F~(F). Now we state the basic properties of the functors F~, Fp and the morphisms p°y, /^. LEMMA 1 . 1 . 1 ) ^ ( ^ 1 © ^ ) = ^ (Vi) 0 ^a (^2) • 2) p y is an epimorphism and i$ is a monomorphism. 3) If fv is an isomorphism, then the dimensions of the spaces Fp(V)y can be calculated from (1.1.1). / / P y is an isomorphism, then the dimensions of the spaces FZ(Y)ycan be calculated from (1.1.2). 4) The object Ker p ^ is concentrated at a {that is, (Ker p
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V « Fp F$ (V) © F/Im iy and F is indecomposable, F coincides with one of the terms. CASE I). F = F/Im i%. Then F 7 = 0 for y ± 0 and, because F is indecomposable, V ** L&. CASE II). F = FJFp{V), that is, £ is an isomorphism. Then (1.1.1) is satisfied by Lemma 1.1. We show that the object W = Fp{V) is indecomposable. For suppose that W = Wt © W2.ThenV = F$ (Wx) © F$ {W2) and so one of the terms (for example, F^{W2)) is 0. By 5) of Lemma 1.1, the morphism p%\ W -> FpFJ{W) is an isomorphism, but P0V{W2)
c
F;F;{W2)
= o, that is, w2 = o.
So we have shown that the object F${V) is indecomposable. We can similarly prove 2) of Theorem 1.1. We say that a sequence of vertices ax, a2, . . . , afe is (+)-accessible with respect to A if ax is (+)-accessible with respect to A, a2 is (+)-accessible with respect to a a j A , a3 is (+)-accessible with respect to a a 2 a t t i A, and so on. We define a (—)-accessible sequence similarly. COROLLARY 1.1. Let ( F , A ) be an oriented graph and aXi a2, . . ., afe a {-^-accessible sequence. 1) For any i (1 < i < k\ F ^ • . . . •F« f _ l ( L a . ) w ezY/zer 0 o r aw indecomposable object in X (F, A) {here La. £ X {T, oot._i oa._2 . . . oai A)) 2) Let V e X {r, A) be an indecomposable object, and F+ a F+ a fe
ft-i
•
" *
•JF+ a v(V) ; = 0 i
T^en /or some i
We illustrate the application of the functors Fp and F^ by the following theorem. THEOREM 1.2. LeJ F ^^ <2 grap/z without cycles {in particular, without loops), and A, A' two orientations of it. 1) There exists a sequence of vertices ati . . ., afe, {-^-accessible with respect to A, swc/z //zatf °afeffah t* • • • #cra 1 A = A ' . 2) Le£ <M, S' be the sets of classes {to within isomorphism) of indecomposable objects in X (F, A) and X (F, A'), <Jl a o/ft — the set of classes of objects F^F^ . . . -F^.^La.) ( K i < f c ) , and<JCr a oM'the set of classes of objects FZk-...'FZUi(Lai) (l
\Q/K>
QM\QM and
.
Where it cannot lead to misunderstanding, we denote by the same symbol La irreducible objects in all categories X(r, A), omitting the indication of the orientation A.
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This theorem shows that, knowing the classification of indecomposable objects for A, we can easily carry it over to A'; in other words, problems that can be obtained from one another by reversing some of the arrows are equivalent in a certain sense. Examples show that the same is true for graphs with cycles, but we are unable to prove it. PROOF OF THEOREM 1.2. It is clear that 2) follows at once from 1) and Corollary 1.1. Let us prove 1). It is sufficient to consider the case when the orientations A and A' differ in only one edge /. The graph V \ I splits into two connected components. Let r ' be the one that contains the vertex 0(7) (0(7) is taken with the orientation of A). Let ax, . . ., ak be a numbering of the vertices of r ' such that for any edge /' e r\ the index of the vertex a(l') is greater than that of j3(/'). (Such a numbering exists because r ' is a graph without cycles.) It is easy to see that the sequence of vertices a{, . . ., ak is the one required (that is, it is (+)-accessible and aafe • . . . • aOiA = A'). This proves Theorem 1.2. It is often convenient to use a certain combination of functors F~ that takes the category X (r, A) into itself. DEFINITION 1.2. Let (I\ A) be an oriented graph without oriented cycles. We choose a numbering ax, . . ., an of the vertices of T such that for any edge / G Fj the index of the vertex a(/) is greater than that of 0(/). We p u t * + =Kn ' . . . • < < , < * > " = / X ' F:2- ... ' FZn. We call
• • •
^ a ;-
t
oc
n
•••
^ , -
The proof is similar for the functor $ . Following [2] we can introduce the following definition.
Coxeter Functors and Gabriel's Theorem
165
DEFINITION 1.3. Let ( I \ A) be an oriented graph without oriented cycles. We say that an object V E X ( I \ A) is (+)-(respectively, (-)-) irregular if {<&)kV = 0 ((<$~)kV = 0) for some k. We say that an object V is regular if V * (<£"")* (
V{ = K, FZ7' . . . 'F~fH (La.) e X ( r , A), h =F:n-...
-F a + . +i ( I a . ) e 2 ( r , A )
(here ] < i < n). Then $+(F,-) = 0 and any indecomposable object V E <£ ( r , A) for which 3>+(F) = 0 is isomorphic to one of the objects Vt. Similarly, 3>~(F,-) = 0, and if V is indecomposable and 3>~(F) = 0, then V «» Vt for some i. 3) Each (^-(respectively, (—)-) irregular indecomposable object V has the form (3>"~)feFi (respectively\ ($+)fei^) for some i, k. Theorem 1.3 follows immediately from Corollary 1.1. With the help of this theorem it is possible, as was done in [2] for the classification of quadruples of subspaces, to distinguish "simple" (irregular) objects from more "complicated" (regular) objects; other methods are necessary for the investigation of regular objects. § 2. Graphs, Weyl groups and Coxeter transformations
In this section we define Weyl groups, roots, and Coxeter transformations, and we prove results that are needed subsequently. We mention two differences between our account and the conventional one. a) We have only Dynkin diagrams with single arrows. b) In the case of graphs with multiple edges we obtain a wider class of groups than, for example, in [ 7 ] . DEFINITION 2.1. Let r be a graph without loops. (1) We denote by %v the linear space over Q consisting of sequences x = (xa) of rational numbers x a (oc G F o ) . For each 0 E Fo we denote by 0 the vector in %v such that (fi )a - 0 for a * 0 and (0^ = 1. We call a vector x = (xa) integral if xa E Z for all a E Fo. We call a vector x = (xa) positive (written x > 0) if x ¥= 0 and
166
/. TV. Bernstein, I. M. Gel'fand, and V. A. Ponontarev
xa > 0 for all a G r 0 . 2) We denote by B the quadratic form on the space %v defined by the formula B (x) = 2 xa — 2 xviii)'xV2(i)y where x = (x a ), and 71 (/) and y2 (0 are the ends of the edge /. We denote by <, > the corresponding symmetric bilinear form. 3) For each 0 G r 0 we denote by op the linear transformation in %v defined by the formula (o$x\ = xy for 7 =£ 0,
{O$X)Q
- - Xp + 2 a:v(/>5
where 7(0 is the end-point of the edge / other than 0. We denote by W the semigroup of transformations of £r generated by the
°P $
e r
° )•
L E M M A 2 . 1 . 1 ) If a , P e r 0 , a ^ j3, r / i e « < a a > = 1 fl«cf 2 ( a j S ) i s //ze negative of the number of edges joining a and p. 2) Let 0 G P o . 7%e« ffpOO = x - 2 <0, x>0, ag = 1. / « particular,
W is a group.
3) T/ze ^rowp W
preserves the integral lattice in %? and preserves the quadratic form B. 4) / / the form B is positive definite {that is, B(x) > 0 for x =£ 0), then the group W is finite. PROOF. 1), 2) and 3) are verified immediately; 4) follows from 3). For the proof of Gabriel's theorem the case where B is positive definite is interesting. Dn,
PROPOSITION 2.1. The form B is positive definite for the graphs An, E6, Zs7, E* and only for them (see [7], Chap. VI).
We give an outline of the proof of this proposition. 1. If F contains a subgraph of the form I f f
I
f
f
2 2 2 2 then the form B is not positive definite, because when we complete the numbers at the vertices in Fig. (*) by zeros, we obtain a vector x 6 ^ r for which B(x) < 0. Hence, if B is positive definite, then F has the form (**)
Zr
Zr-f
Zr-2
Zj
Z2
Z{
Zj xz Xj x^Xj, a fy y^ % % yf where p, q, r are non-negative integers. 2 For each non-negative integer p we consider the quadratic form in (p + 1) variables xlf . . ., xp + l Cp
Coxeter Functors and Gabriel's Theorem
167
This form is non-negative definite, and the dimension of its null space is 1. Moreover, any vector x =£ 0 for which Cp{x) = 0 has all its coordinates non-zero. To prove these facts it is sufficient to rewrite Cp{x) in the form
3. We place the numbers xl9 . . ., xp, yl9 . . ., yq, z l 5 . . ., zr a at the vertices of r in accordance with Fig. (**). Then B(xt,
y
u
zt, a) = Cp(xlt
...,xp,
a) + Cq(y{,
...,yq,
a) +
2{p + l)
Hence it is clear that B is positive definite if and only if +
+ 2(r+l) < l j t h a t ^ + + 4. We may suppose that p < q < r. We examine possihle cases. a) p = 0, q and r arbitrary. A= positive definite (series An). b) p = \, q - \, r arbitrary. c) p = 1, q = 2, r = 2, 3, 4. d) p = 1, 5. p = 1, q = 3, r > 3. p > 2,^r > 2 , r > 2.
-|—XJ-+ A ^ ^ 4
> > < <
I
> 1>
that is, B is
1 (series Dn), 1 (£*, £ 7 , £ 8 ), 1, 1,
>4 < 1.
Thus i? is positive definite for the graphs An, Dn, E6, Elf Eg and only for them. DEFINITION 2.2 A vector x E %v is called a roo/1 if for some 0 G r 0 , w G H / w e have x = w^. The vectors j8 (j3 G r 0 ) are called simple roots. A root x is called positive if x > 0 ( see Definition 2.1). LEMMA 2.2 \) If x is a root, then x is an integral vector and B{x) = 1. 2) If x is a root, then (~x) is a root. 3) If x is a root, then either x> 0 or (-JC) > 0. PROOF. 1) follows from Lemma 2.1; 2) follows from the fact that oa (<*) = ~<x f ° r a ^ a E F o • 3) is needed only when 5 is positive definite and we prove it only in this case. We can write the root x in the form attj oaj • . . . • aafej3, where a l5 . . ., afc, j3 G r 0 . It is therefore sufficient to show that if y > 0 and a e r 0 , then either a j > O o r y = a (and -oay = + a > 0). Since ILyll = II aII = 1, we have |
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/. N. Bernstein, I. M. Gel'fand, and V. A. Ponomarev
b) 2
is
the other end-point
Z6ra
of the edge /), we have ya > 0, that is, ya > 1. Hence oay = y - a > 0. This proves Lemma 2.2. DEFINITION 2.3. Let F be a graph without loops, and let ai9 . . ., an be a numbering of its vertices. An element c = aan • . . . • aai (c depends on the choice of numbering) of the group W is called a Coxeter transformation. LEMMA 2.3. Suppose that the form B for the graph F is positive definite: 1) the transformation c in %v has non non-zero invariant vectors; 2) if x 6 %v, x -^ 0, then for some i the vector clx is not positive. PROOF. 1) Suppose that */€$r> y¥=0 and cy = y. Since the transformations oan, <*<*„_,, • • ., 0<*, do not change the coordinate corresponding to ai (that is, for any z6§r (ffa*z)ai = zai for i=^l), we have (oaiy)otl ~ toOa, = yOtl • Hence aai>> = j> Similarly we can prove that Oa3y = ^, then oasy = ^, and so on. For all a. E r 0 , a ^ = ^ - 2
Let (F, A) be an oriented graph. For each object V e X (F, A) we regard the set of dimensions dim Va as a vector in %v and denote it by dim V. THEOREM 3.1 (Gabriel [1]). 1) If in X{T, A) there are only finitely many non-isomorphic indecomposable objects, then F coincides with one of the graphs An, Dn, E6, £ 7 , Es. 2) Let F be a graph of one of the types An, Dn, E6, E7, Es, and A some orientation of it. Then in X(T, A) there are only finitely many nonisomprphic indecomposable objects. In addition, the mapping V h* dim V sets up a one-to-one correspondence between classes of isomorphic indecomposable objects and positive roots in %vWe start with a proof due to Tits of the first part of the theorem. TITS'S PROOF. Consider the objects (V, f) e X(T, A) with a fixed dimension dim V = m = (ma). If we fix a basis in each of the spaces J&, then the object (F, f) is completely defined by the set of matrices Ax (/ € Fj), where A{ is the matrix of the mapping fx\ F a ( 0 -> Vp0). In each space Va we change the basis by means of a non-singular (ma X ma) matrix ga. Then the matrices A i are replaced by the matrices
Coxeter Functors and Gabriel's Theorem
169
Let A be the manifold of all sets of matrices Ax (/ G r1) and G the group of all sets of non-singular matrices ga (a G r 0 ) . Then G acts on A according to (*); clearly, two objects of X (I\ A) with given dimension m are isomorphic if and only if the sets of matrices {At} corresponding to them lie in one orbit of G. If in %(V, A) there are only finitely many indecomposable objects, then there are only finitely many non-isomorphic objects of dimension m. Therefore the manifold A splits into a finite number of orbits of G. It follows1 that dim A < dim G - 1 (the -1 is explained by the fact that G has a 1-dimensional subgroup Go = {g{k)\k £&*}, g(X)a = X'lv T which acts on A identically). Clearly, dim G = 5 ^a* dim ,4= 2 Therefore the condition dim A < dim G - 1 can be rewritten in the form2 B(m) > 0 (if m =£ 0). In addition, it is easy to verify that B((xa)) > B((\xa\)) for all x = (xa) G g r . So we have shown that if in X(T, A) there are finitely many indecomposable objects, then the form B in £r is positive definite. As we have shown in Proposition 2.1, this holds only for the graphs Ani Dn,
E6,
En,
E8.
We now prove the second part of Gabriel's theorem. LEMMA 3.1. Suppose that (r, A) is an oriented graph, 0 G r 0 a (+)accessible vertex with respect to A, #rcd F E <^(F, A) an indecomposable object. Then either Fp(V) is an indecomposable object and dim Ffi(V) = a^dim F), or V = L0, FfrV) = 0, dim FfcVy ± a^dim V) < 0. A similar statement holds for a (-)-accessible vertex a and the functor F^. This lemma is a reformulation of Theorem 1.1. COROLLARY 3.1. Suppose that the sequence of vertices otXi . . ., ak is {^-accessible with respect to A and that V G X (F, A) is an indecomposable object. Put Vj = Ft/ij^' • • • -KXV, mj =
2
This argument the number of the number of We can clearly
is suitable only for an infinite field K. If K = ¥q is a finite field, we must use the fact that non-isomorphic objects of dimension m increases no faster than a polynomial in m, and orbits of G on the manifold A is not less than C - ^ ^ " 1 ^ ~ < d i m G~l). restrict ourselves to graphs without loops.
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/• N. Bernstein, I. M. Gel'fand, and V. A. Ponomarev
We choose a numbering aif a2i . . ., <*„ of the vertices of r such that for any edge / e Ti the vertex <*(/) has an index greater than that of 0(/). Let c = aa • . . . • aa be the corresponding Coxeter transformation. By Lemma 2.3, for some k the vector cfe(dim V) e %? is not positive. If we consider the (+)-accessible sequence 0i, 0 2 , . . . , 0nfe = (<*!,. . ., an, « ! , . . . , an, . . . , «i, . . . , <*„)(& times),then we have a ^ - . . . • aPi (dim F) = c^(dim V) > 0. From Corollary 3.1 it follows that there is an index / < kn (depending only on dim F) such that V = FJt'Fp% • . . . • F^(L0.+il
dim F = oPi* . . . • aPi0M).
It follows that
dim F is a positive root and F is determined by the vector dim F. b) Let x be a positive root. By Lemma 23, ckx > 0 for some &. Consider the (+)-accessible sequence 0i, &, • . •, ftife = (ai> • • •, <*n, • • •, <*i, . . ., an) (k times). Then <jpnk* . . . • o&x(x) = cfe(x) > 0. Let / be the last index for which opppi-/ • • • * opxW > 0- It is obvious from the proof of 3) in Lemma 2.2 that ofii* . . . • afii(x) = 0 I+1 . It follows that Corollary 3.1 that F = ^ F p , - • • • 'Fh (Lfiui) € # (r, A) is an an indecomposable object and dim V = o^^ . . . • ap.(fii+i) = x. This concludes the proof of Gabriel's theorem. NOTE 1. When B is positive definite, the set of roots coincides with the set of integral vectors x e i r for which B(x) = 1 (this is easy to see from Lemma 2.3 and the proof of Lemma 2.2). NOTE 2. It is interesting to consider categories X(T, A), for which the canonical form of an object of dimension m depends on fewer than C*\m\2 parameters (here |m| = 2 |m a |, a G r 0 ) . From the proof it is obvious that for this it is necessary that B should be non-negative definite. As in Proposition 2.1 we can show that B is non-negative definite for the graphs An9 Dn, E6, Ely E8 and Ao, An, Dn, E6, Elf £ 8 , where
(n + 1 vertices, n > 1) (n + 1 vertices, n > 4)
Coxeter Functors and Gabriel's Theorem A
A
/\
/S
171
^.
(the graphs An, Dn, E6, El9 E8 are extensions of the Dynkin diagrams (see [7])). In a recent article Nazarova has given a classification of indecomposable objects for these graphs. In addition, she has shown there that such a classification for the remaining graphs would contain a classification of pairs of non-commuting operators (that is, in a certain sense it is impossible to give such a classification). § 4 . Some open questions
Let F be a finite connected graph without loops and A an orientation of it. CONJECTURES. 1) Suppose that x G %? is an integral vector, x > 0, B(x) > 0 and x is not a root. Then any object V G %(T, A) for which dim V = x is decomposable. 2) If x is a positive root, then there is exactly one (to within isomorphism) indecomposable object V G X(Y, A), for which dim V - x. 3) If V is an indecomposable object in X(T, A) and i?(dim V) < 0, then there are infinitely many non-isomorphic indecomposable objects V d X(T, A) with dim V = dim V (we suppose that K is an infinite field). 4) If A and A' are two orientations of r and V G X(T, A')[s an indecomposable object, then there is an indecomposable object V G X{T, A') such that dim V = dim V. We illustrate this conjecture by the example of the graph (F, A) ct,
(quadruple of subspaces). For each x G %T we p u t p(x) = - 2 < a 0 , x) (if x = (xOi x l 9 x 2 , x 3 y x 4 ) , then p ( x ) = Xi + x 2 + x 3 + x 4 - 2 x 0 ) . In [2] all the indecomposable objects in the category X{Y, A) are described. They are of the following types. 1. Irregular indecomposable objects (see the end of §1). Such objects are in one-to-one correspondence with positive roots x for which p(x) =£ 0. 2. Regular indecomposable objects V for which #(dim V) # 0. These objects are in one-to-one correspondence with positive roots x for which P(x) = 0. 3) Regular objects V for which i?(dim.F) = 0. In this case dim V has the form dim V - (2«, n, n, n, n), p(dim V) = 0. Indecomposable objects with fixed dimension m = (2n, n, n, n, n) depend on one parameter. If m G %v is an integral vector such that m > 0 and B{m) = 0, then it has the form m = (2n, n, n, n, n) (n > 0) and there are indecomposable objects V for which dim V - m. If / is a linear transformation in ^-dimensional space consisting o f one
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Jordan block then the quadruple of subspaces corresponding to it (see the Introduction) is a quadruple of the third type. References
[1] [2]
[3]
[4] [5]
[6]
[7]
P. Gabriel, Unzerlegbare Darstellungen I, Manuscripta Math. 6 (1972), 71-103. I. M. Gelfand and V. A. Ponomarev, Problems of linear algebra and classification of quadruples of subspaces in afinite-dimensionalvector space, Colloquia Mathematica Societatis Ianos Bolyai, 5, Hilbert space operators, Tihany (Hungary), 1970,163—237 (in English). (For a brief account, see Dokl. Akad. Nauk SSSR 197 (1971), 762-765 = Soviet Math. Doklady 12 (1971), 535-539.) L. A. Nazarova and A. V. Roiter, Representations of partially ordered sets, in the collection "Investigations in the theory of representations", Izdat. Nauka, Leningrad 1972,5-31. I. M Gelfand, The cohomology of infinite-dimensional Lie algebras. Actes CongrSs Internat. Math. Nice 1970, vol. 1. (1970), 95-111 (in English). I. M. Gelfand and V. A. Ponorarev, Indecomposable representations of the Lorentz group, Uspekhi Mat. Nauk 23: 2 (1968), 3-60, MR 37 # 5325. = Russian Math. Surveys 23: 2 (1968), 1-58. C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Interscience, New York-London 1962, MR 26 #2519. Translation: Teoriya predstavlenii konechnykh grupp i assotsiativnykh algebr, Izdat. Nauka, Moscow 1969. N. Bourbaki, Elements de mathematique, XXVI, Groupes et algebres de Lie, Hermann & Co., Paris 1960, MR 24 # A2641. Translation: Gruppy ialgebry Li, Izdat. Mir, Moscow 1972. Received by the Editors, 18 December 1972.
Translated by E. J. F. Primrose.
Dedicated to the memory of Ivan Georgievich Petrovskii
FREE MODULAR LATTICES AND THEIR REPRESENTATIONS I. M. Gel'fand and V. A. Ponomarev Let I be a modular lattice, and V a finite-dimensional vector space over a field k. A representation of L in V is a morphism from L into the lattice J£(V) of all subspaces of V. In this paper we study representations of finitely generated free modular lattices Dr. An element a of a lattice L is called perfect if for every indecomposable representation p: L -+• %(kn) the subspace p(a) of V = kn is such that p(a) = V or p(a) = 0. We construct and study certain important sublattices of Dr, called "cubicles." All elements of the cubicles are perfect. There are indecomposable representations connected with the cubicles. It will be shown that almost all these representations, except the elementary ones, have the important property of complete irreducibility; here a representation p of L is called completely irreducible if the sublattice p(L) cz J£(kn) is isomorphic to the lattice P(Q, n - 1) of linear submanifolds of projective space over the field Q of rational numbers.
Contents
§1. Definitions and statement of results §2. The category of representations §3. Construction and elementary properties of the cubicles B+(l) and B'(l) §4. Representations of the first upper cubicle §5. The functors ®+ and
173 180 184 190 195 204 213 224 228
§ 1. Definitions and statement of results
1.1. Lattices. A lattice L is a set with two operations: intersection and sum. If a, b G L, we denote their intersection1 by ab and their sum by The intersection of elements a and b of L is often denoted a n b. We use the notation aft to avoid clumsy formulae. For a + b the notation 0 u & or a V b, is also used fairly frequently. n We denote the sum of the elements a,,..., an by V # . a n d their intersection fl i==i a, n j 2 n . . . n a n by f] i173
X74
I- M. Gel'fand and V. A. Ponomarev
a+ b. Each of these operations is commutative and associative. Moreover, for any a, b G L we have the identities aa = a,
a -\- a — a,
a(a + b) = a,
a -{- ab = a.
An order relation is defined in a lattice L by (a^ b) <^> (ab = a).
If a, 6, c are arbitrary elements of a lattice L, then it is easy to show that a(b + c) ^ ab + ac. A lattice £ is called distributive if for any a, b, c & L a(b + c) = ab + dc and a + be = (a + b)(a + c). A lattice L is called modular (or Dedekind) if for any a, b, c £ L such that a C fcf &(a + c) = a + &c. This relation is called Dedekind's axiom. EXAMPLE 1. Let V be a finite-dimensional vector space over a field #, and F = kn. The set of all subspaces of V is a lattice in which £ F is the intersection of the subspaces E and F, and £" + F is their sum, that is, (E + J1) == (^ + y I ^r 6 E, y e F). We denote this lattice by ^ (V) oxX (kn). It is well-known that for n > 1 the lattice <£ (/c77) is modular, but not distributive. Let P be the projective space generated by V = kn. Then the lattice X (kn) is isomorphic to the lattice of linear submanifolds of P. We denote the latter lattice by P(fc, n - 1) or P(fc, F), and call it the projective geometry over k. EXAMPLE 2. Let M be an arbitrary module over a commutative ring A. Then the set of all submodules of M is a modular lattice under the operations of intersection and sum. 1.2. Basic definitions. Let I be a modular lattice, and F a finite-dimensional vector space over a field k. A representation of L in F is a morphism from L into the lattice X(V). Thus, a representation p: L-*X(V) associates with each element x G X a subspace p(#) ^ V such that for all x, y G I p(;ry) = p(z)p(y) and p(a: + */) = p Or) + p(y). Let Pi and p2 be representations of a lattice L in spaces Vl and F 2 , respectively. We set p(x) = px(^) 0 p2Oz) for every x G X , where piW ® p2(#) is the subspace of Vx 0 F 2 consisting of all pairs (£, 17) such that I G PiW and rj 6 p2(^)« ^ is not hard to show that this defines a representation 0 in the space V - Vx 0 F 2 . This representation is called the direct sum of px and p2 and is denoted by p = px 0 p2A representation p is decomposable if it is isomorphic to the direct sum pi 0 p2 of two non-zero representations pj and p2. It is easy to see that a representation p in a space F is decomposable if and only if there exist
Free modular lattices and their representations
175
subspaces Ux and U2 such that Ux U2 = 0 and U{ + U2 ~ K, and that p(a) = f/ip(a) + £/2p(a) f° r every a E L. DEFINITION. An element a of a modular lattice Z is called perfect if for every field k and every representation p: L-+X (V) — £ (h*1) the subspace p(a) gz y has the following property: there is a subspace £/ complementary to p(a) (that is, Un{a) = 0 and £/ + p(o) ~ T7) such that the subspaces £/ and p(a) define a decomposition of p into the direct sum of subrepresentations, that is, p(x) — Up(x) + p(&)p(.z) for every x E Z. It is easy to check that this definition is equivalent to the following: an element a E L is called perfect if, for every indecomposable representation p\ L-+X (kn) with K = fc", either p(a) = F o r p(d) = 0. Two elements 0 and & of a modular lattice L are called linearly equivalent if p(a) = p(b) in every representation p:L-*-£ (kn) for any fc and n. In this case we write a = b. It can be shown that if L is the free modular lattice with 4 generators, then in L there are unequal, but linearly equivalent, elements. Such examples are of interest to us in connection with the following problem. A modular lattice L is called linear if for any x, y E L and every representation p: L -> £ (kn) we have x = y if and only if p(^) = p(y). This leads to the following question: can a linear lattice be characterized by adding to the axioms for a modular lattice finitely many identities? 1.3. Cubicles in the lattice Dr. In this paper we study representations of the free modular lattice Dr with r generators eu . . . , er. The key idea in this paper is the construction of an important sublattice B of Dr whose elements are all perfect, for each integer / > 1 we construct sublattices B+(l) and B~(l), each consisting of 2r perfect elements. We call B+(l) the l-th upper cubicle and B~(l) the l-th tower cubicle. It is quite simple to define the upper cubicle B+(\). We set hi (1) = 2 eJ> The sublattice of Dr generated by the elements fti(l), . . . , hr{\) is then the upper cubicle ^ + (1). We shall prove that 1) i? + (l) is a Boolean algebra with 2r elements (see §3), and 2) every element x E B+(\) is perfect (see §4). The lattice # + ( l ) is, thus, isomorphic to the lattice of vertices of an r-dimensional cube with the natural ordering. The element ht{\) corresponds to the point (1, . . . , 1, 0, 1* . , , , 1) with 0 in the z-th place. Now we construct the lattice B+(l) with / > 1. The elements of the cubicle B+(l) are "constructed" from certain important polynomials1) eti . . .fr which are of independent interest. We proceed to define these polynomials. Let / > 1 and / = {1, . . ., r}. We denote by Air, I) the set whose elements are sequences of integers a = (il9 , . „ , /,) with iK E / such that ' The elements of Dr are also called lattice polynomials, or simply polynomials.
176
/. M. GeVfandand V. A. Ponomarev
i\ ^ h> h ^ h> - • - > h-i ^ h- *n Particular, A(r, 1) = /. For fixed oc G Air, I) we construct a set F(a) consisting of elements f$ E A(r, I - 1) in the following way: r(a) = {p = fa, . . ., * M ) 6 4(r, Z - 1) | &x $ {flf *,}, * t £ {j 2 , * 3 }, . . . Note that kx ¥= k2i k2 ¥* k3, . . . , kx_2 =£ kx_x, because 0 G ,40, / - 1). With each a G A(r, I) we now associate an element ea G D ' by the following rule. Let / = 1 and o: = (/j). We set ea = et . For 1=2 and a = (/j, i2) with /j ^ /2 w e s e t In general, for arbitrary / and a G A(r, I) we set by induction (1.1) ea = e{ i =ek 2 ep. Now we introduce the elements ht(l), from which we construct the /-th cubicle, just as B+(l) was constructed from the elements ht{\). We denote by At(r, 1) the subset of Air, I) consisting of all a = (ii9 . . . , //_!, f) whose last index is fixed and equal to Z\ We set
(1.2)
M0
S
(1.3)
ht(l
^
The sublattice of Z)r generated by /*!(/), . . . , /?r(/) is called the l-th upper cubicle B+(l). It is fairly elementary to prove (see §3) that for every / > 1 the sublattice B+il) is a Boolean algebra. It is vastly more complicated to prove that the number of elements in B+(l) is equal to 2r and that every element x G £ + (/) is perfect ( § § 4 - 7 ) . We shall also prove (§3) that the elements of any cubicles B+(l) and B+im) can be ordered in the following way: for every xt G B+(l) and every ym G B+(m), if / < m, then xx D xm . It follows from this that the collection of elements of all of the cubicles B+il) is also a lattice, which we denote by B*. We denote by B~(l) the sublattice of If dual1) to B+il). The sublattice B~{1) is called the l-th lower cubicle. Apparently, the following is true.
) A lattice p o l y n o m i a l ^ * , , . . . , jcr) in the variables x , , . . . , xr is dual to a polynomial/(x,,. . . , x r ) if g is obtained f r o m / b y changing the operation of intersection into addition and addition into intersection. For example, the polynomials x, (x2 + x3 + . . . + JCP) and x, + X 2 JC 3 . . . xr are dual to each other. Let a, and a2 be elements of Dr, and a-x = /,.(
Free modular lattices and their representations
111
PROPOSITION. Let x+ G B+(1) and y~ G B~{m). Then y~ C x+ for every I and m. We are able to prove this proposition only up to linear equivalence, that is, for every representation p: Dr -> %(kn) and any x+ G B+(l) and y~ G B~(m) we have p(y~) s p(x+). MAIN THEOREM. The elements of the sublattices B+(l) and B~{1) are perfect. CONJECTURE. Let a be a perfect element in Dr. Then there exists an I > 1 such that either a G B+(l) or a G B~(l). REMARK 1. The elements ea = eii... ^ used to construct the perfect elements in B+(l) are of considerable independent interest. Below, in §1.4, we construct completely irreducible representations pt, i (t = 1, . . ., r; Z = 1, 2, . . .). For these representations the images Pt, i(ea) of the elements ea G D r are one-dimensional subspaces of Vtl. These pt, u with £ 6 {1» • • • > r}, are called completely irreducible representations of the first kind. REMARK 2. We shall also construct completely irreducible representations pOtl. of the second kind. In these representations the elements r ea, oc G At(r, /), are replaced by elements ft i0 GZ) , where iv G / and iv =£ zy+1. The elements ft / 0 are "constructed" in the following manner. The set F(a) = T(il, . . . , / M , 0) C ^(r, / - 1) consists in this case of all elements 0 = {ku . . . , A:^!) G ^(r, / - 1) such that kx £ {h, i2}, k2f{i2, iH}, . . ., /c/_2 ^ { i / - 2 , ^/-i}» ^;-i $={£M}- We note that all kx in ]3 are different from zero and therefore e& is defined by (1.1). We set fa=:fi1...il_iQ--=eii
]V. gp. For example fi1o = €ii 2J e^.
By analogy to the elements e r (/), t G /, we set
/o(0 = 2 / a »
where
a
a = (/1} . . . , / ^ j , 0), /y G /, and the summation is over all such a. We shall prove later that / 0 ( 0 is linearly equivalent to the smallest element of B+(l - 1), that is, fo(l) ~ r
/o(O=
f] ht(l — 1). Apparently, in Dr we have l
.D ^i(^ — 1). for every / > 2. However, we can prove this only
for / = 2. 1.4. Representations of the first and second kind. We denote by p(L) the sublattice of %{kn) consisting of all elements p(a), a ^ L. DEFINITION. A representation p of a modular lattice I in a space V over a field k of characteristic zero is called completely irreducible if p(L) ^ P(Q, m), where Q is the field of rational numbers and m = (dim^. V) — 1. We first construct the representations pi, i of the first kind in the spaces Vt th t 6 {!> • • •» r}> J = 1, 2, . . . Clearly, a representation p of Dr in a space K is completely determined by the subspaces
178
/. M. Gel'fandand V. A. Ponomarev
p(ej) (/ = 1» • • •» r ) o f v- F o r brevity we set pt, i(ej) = Eu t s V
I) {a E At(r,
We denote by Zt
I) «-> a = (z l5 . . . , il_l, l
the subspace of Wt
l
t)
and t is fixed).
spanned by all possible vectors
Sa.h = 2 ' t\it ..i f c ...i, Au where 1 < k < / - 1 and y '
is summation over
those a = (/j, . . . , z^, . . . , //_!, 0 in which iu . . . , z jt _ 1 , ik + l , . . . , //_! are fixed. Next, we set Vtl - Wtl\Ztl. The images of the vectors r?ain the factor space Vtl are denoted by £ a . Thus, Vtl is the vector space over k spanned by the vectors £a for which . . . i , .1 = 0 for every k. By Ej t we denote the subspace of Vt t spanned by all vectors £a such that a = (il9i2, • • • > ^-i» 0 = (/, «2> • • • » *'z-i> 0 (where the index z't = / is fixed). We define a representation p*. i in F r z by setting pt,i(ej) = ^y,/. Now we define the representations po,i of the second kind (/ = 1, 2, . . . ). For / = 1, Po.z is the representation in the one-dimensional space V = k' such that po.ife) = 0 for all i = 1, . . . , r. For / > 1 we define a set A0(r, I) in the following way: Ao(r, I) = {a = (iit . . ., it-v
0) | iK 6 J = {1, . . ., r } , ix ^ i 2 ,
Clearly, ^ 0 ( ^ 0 — ^( r » ^ ~ 0- The representation p 0 , t is constructed on A0(r, I) in the same way as pt,i is constructed on At{r, I). Namely, we denote by Wo l the vector space over k with the basis {T]a}, where a E A0(r, I). Further, let ZOj be the subspace of Wol spanned by all possible vectors ga,h = S ^- • -'w • -Vi 0 - We denote by Vo
t
the factor space
WOJIZOJ and by £ a the image of r?a under the canonical map WOj -• F o z. The subspace of F o ; spanned by all vectors £a with a = 0i> ^2^ • • • > ? /-i 5 0) = 0"> ?25 • • • 5 //__i, 0) is denoted by Ej 0 . We define a representation pOt z in F o / by setting p 0 , /(e;-) = Ejt0. REMARK. Our definition of the representations pt, i of Dr makes sense for all r > 1. It is known [2] that the lattices Dl, D2, D3 are finite, and each of these lattices has only finitely many non-isomorphic indecomposable representations. It can be shown that the number of such representations is 2, 4, and 9, respectively, and that they coincide with representations pt, i, I ^ r. It can be shown by direct computation that for r 6 {1, 2, 3} and / > r the space Vtl is equal to 0, hence, the corresponding representation pt, i = 0.
Free modular lattices and their representations
179
The second main result of this paper is the following theorem. THEOREM 1.1. (I) For every t £ {0, 1, . . ., r) and all r, I > 1 the Vttl) is indecomposable. representation pt,i' Dr-*X{k, (II) / / the characteristic of k is 0, if r > 3, / > 1, and if (f, r, Z)£{(1, 4, 2), (2, 4, 2), (3, 4, 2), (4, 4, 2)}, r/tert the representation Vt, i) is completely irreducible. Pt, i' Dr-+%(k, Other properties of the representations pt, i are described by the following propositions. PROPOSITION 1.1. Let pt,i be a representation of the first kind (t (i {1, . . ., r}) in the space Vt x. We denote by vt l the element ofB+(l). fl ht(l) (I) For every element x G 5
+
= U B+(s) such that x D vt j we have
pt,i(x) = Vt, i. In particular, this equality holds for all x E B+(m), m (II) For every element y £ B+ such that y C ht(l) we have pt, i(y) In particular, this equality holds for all y E B+(n), n > /. (III) For every a = (il9 . . . , il_l, t) E At{r, I) we have pt, i{ea) = where k(%a) is the one-dimensional subspace of Vt z spanned by the
< I. = 0. k(la), vector
PROPOSITION 1.2. Let po,i be a representation of the second kind in the space Vo> t. We denote by vQ, ^ the element C\ ht(l — \) of B+(l~ 1). (I) For every element x E B* such that x D vd !_{ we have pOi j(x) =V0, i. This means that p 0, i(x) = Vo, z for x E B+(m), m < I - 1. (II) For every element y E B+ such that y C vdl_l we have p0 t(y) = 0. This means that p 0 , i(y) = 0 for every y £B+(n), n > I. We shall now briefly describe the representations p7, i associated with the lower cubicles B~(l). DEFINITION. Let p be a representation of a modular lattice L in a space V over a field k. We denote by V* the space dual to V. A representation p* in F* is called dual to p if p*(x) = (p^))^ for every x G L, where (p(x))^ is the subspace of functional in V* that vanish on p(x). We set pjf i = (pt, i)*' Thus, pjt i is the representation in Vf t such that P7. i(et) = (pt, liei))1- for all / = 1 , ' . . . , r. We do not describe in detail the properties of the representations pjy z, because they are dual to those of Theorem 1.1 and Propositions 1.1 and 1.2. 1.5. As we have already mentioned, the p/, / describe all the indecomposable representations of the lattices D1, D2, D3. For the lattices Dr with r > 4 this is not the case. We describe below how to split off from an arbitrary representation p of Dr, r > 4, indecomposable representations Pt, i and p7? z. By v+Q t = vQ j we denote the smallest element of the cubicle B+(l). It follows easily from the definition of B\l) that v
e,l
= nr
/=l
hi(0> where ht(l) is defined by (1.3). It can be shown
180
/• M. Gel'fandand V. A. Ponomarev
(see §3) that v+dl D i y 2 D • • • D v*eJ D • - \ Dual to the element e,i ^ Dr is ujp which is the largest element in B~{1). Then
v+
r
*>/, i = S &F(0> where hJ(J) is polynomial dual to ht(l). The elements uj z also form a chain i>^ x C uj 2 C • • • C u/-/ C • • • . Let p be any representation of Dr. We write p(i>e, z) = Ve,i a n d p(vjti) = Fj, i. We define F^ ^ = O Fa+; and Fj ^ = U Vjv We shall prove (§6) that Vj „ C K5fOO. Thus, the subspaces Fj-, and V*, form a chain (7 ,1
THEOREM 1.2. Le/^ p be a representation of Dr, r > 4, m a 5/?ace F, a«c? /ef 0 C ^"^ C V*^C V, that is, all the terms of this chain are distinct. Then p is decomposable into a direct sum p = p" 0 px 0 p + , where p~ = p | v is the restriction of p to the subspace Vf^ and (p" 0 P0 = P l v j f00 / / the representations p~ and p + are decomposable, then p+ ^ 0 pt, i and p~ & 0 p7, h where t 6 {0, 1, . . ., r) (Z = 1, 2, . . .)• //a&o r/ie representation p% decomposes into a direct sum Px = 0 T^, 5 o/ indecomX, 8
posable representations rx s, then among the r x s there are no representations isomorphic to pt, i and pT, iWe shall prove this theorem in §6. For the lattice Z)4 the representations p^were studied in [5], where they were called regular. The indecomposable regular representations of D 4 are completely classified. For a specification of these indecomposable representations one needs not only discrete invariants (of the type of a dimension), but also continuous parameters (analogous to the eigenvalues of a linear transformation). Very little is known about the regular representations of the lattices Dr (r > 5). It is clear (see [5], [7]) only that the classification problem (up to similarity) of an arbitrary set of linear transformations Ai, . . . , An (n > 2), Ai : F -> F, reduces to a special case of the classification problem of indecomposable regular representations of Dr (r > 5). §2. The category of representations
In this section, which is of an auxiliary role, we describe some elementary properties of the category of representations. Also, we introduce the notion of an admissible subspace, and we formulate a simple criterion for decomposability of representations. 2.1. The category ^?(£, k). Let px and p2 be representations of a modular lattice L in finite-dimensional spaces Vx and F 2 over one and the same field
Free modular lattices and their representations
181
k. A morphism u: pt -> p2 is a linear mapping u: Vx -> F 2 such that upxix) s P2W f° r every x E Z,, where up^x) is the image of the subspace Pi(#)- When there is no ambiguity, we denote a morphism u: Pi->• P2 by u: pi-^p 2 We denote by Hom(p!, p2) the set of all morphism from pi to p2. Note that Horn (p1? p2) is a vector space over k. It is not hard to verify that we now have a category °rt(L, k) , that of finite-dimensional representations of L over k. REMARK. Let u: p] ->• p2 be a morphism in 3?(L, A;) and w: Fj -• F 2 the corresponding linear transformation. It is not true that the set of all subspaces up^x), x£L, defines a representation of L in F 2 . If x, y £ L, then upifc + y) = w(pi(aO + Pi(*/)) = upi(x) + up^y). However, in general, up^xy) = uipiixfaiy)) =^ ("Pi(«))("Pi(y))« W e c a n o n l y a s s e r t t h a t ( ( For any two objects px and p2 in ^(L, /c) there is the direct sum p! 0 p2. Namely, let pi> p2 6 ^(L, A:) be representations in spaces Fj and V2. We set K = F! © F 2 . For every x £ L we define pW = PiW © P2W ^ Fx 0 F 2 . It can be shown that this defines a representation p in Vx © F 2 . It is not hard to check that fi{L, k) is an additive category. We now return to the category M{Dri k) of representations of a free modular lattice Dr. For brevity, we denote ^(D11, k) by # . It is easy to show fl is additive, and that every morphism u: px ->• p2 has a kernel and a cokernel,1) that is, the category >? is pre-Abelian. However, it is not Abelian, because the canonical mapping2) Coim u -> Im u is not an isomorphism for an arbitrary morphism u . EXAMPLE. Let 91 = #(D 3 , A:). We define representations px and p, in spaces Vx and F 2 in the following way: V1 ^ F2 ^ /J1, Pi(«i) = Pi(«2> = V\> Pi (^3) = 0, pofe) = p2(e2) = p2(e3) = F 2 . Next let u: Vx -> F 2 be any isomorphism (in the category of linear spaces), and let u: px -> p2 be the morphism corresponding to the mapping u. It is not hard to check that Ker u = 0 and Coker u = 0. Consequently, in the canonical decomposition px -> Coim u -> Im iT -> p2 we have
^ We recall that a fern*/ of a morphism M: p x -»- p 2 is a subobject fi: p ' - • p 2 of p x such that for every T of R there is an exact sequence of vector spaces 0 -* Horn (r, p') -> Horn (T, p^-* Horn (r, p 2 ). In other words, /T is a monomorphism such that if u w = 0 with u ) e Horn (r, p,), then there exists a morphism M)' e Horn (r,p') such that u; - jH w'. The cokernel of a morphism u: pl -* p 2 is a factor object TT: p 2 -»• p " such that for every object T of /? there is an exact sequence of vector spaces Horn (PX,T) «- Horn (p2,r)+- Horn (p", T) •- 0. ' We recall that Im u is the kernel of the cokernel of u, and Coim u the cokernel of the kernel of u.
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I' M. Gel'fandand V. A. Ponomarev
Pi ^ Coim u and p2 ^ Im u. Therefore, the mapping Coim u -> Im u is not an isomorphism. 2.2. Decomposable representations and admissible subspaces. DEFINITION. Let p be a representation of a modular lattice L in a linear space V. A subspace U of Fis admissible relative to p if for any x, y E L one of the following conditions is satisfied: (I) (II)
U(p(x) + p(y)) = Up(x) + Up(y); U + p{x)p(y) = (U + p(x))(U + p(y)).
The equivalence of (I) and (II) follows from a more general statement. LEMMA 2.1. Let L be a modular lattice and let x, y, z E L. The sublattice generated by x, yy z is distributive if one of the following conditions is satisfied: 1) z(x + y) = zx + zy and 2) z + xy = (z + x) (z + y). A proof of this assertion can be found, for example, in Birkhoff [2]. Three elements x, y, z of a modular lattice L satisfying the conditions of Lemma 2.1 are called a distributive triple. It follows from Lemma 2.1 that the following equalities are equivalent: z(x + y) = zx + zy, x(y + z) = xy + #z, #(z + z) = i/a: + z/z, z + ^ (z + z)(z + y), x + yz = (x + y){x + z), y + xz = (y + .r)(z/ + z).
=
PROPOSITION 2.1. Le? p be a representation of a lattice L in a space V. Let U be a subspace of V and let U" - V/U the factor space. Let 6 : V -> U" be the canonical mapping. Then the following conditions are equivalent: 1°. The subspace U is admissible relative to p. 2°. The correspondence x -> Up(x) defines a representation in U. 3°. The correspondence x H-> dp(x) defines a representation in U". PROOF. Let us show, say, that 2° follows from 1°. Let U be an admissible subspace. Then Up(z + y) = U(p(x) + p(y)) = Up(x) + Up(y). Moreover, Up(xy) = Up(x)p(y) = (Up(x))(Up(y)). Consequently, the rule x *-+ Up(x) defines a representation in U. This representation is called admissible; it is also called the restriction of p to U and is denoted by pu or p 1^. The proofs of the remaining parts of the proposition are elementary. The representation in V/U, where U is an admissible subspace, is called an admissible factor representation. We say that a representation p 6 M{L, k) is decomposable if it is ison
morphic to a direct sum 0 p., n > 2, of representations pi =£0. i=l
PROPOSITION 2.2. A representation p 6 3^{L, k) in V is decomposable if and only if there exist non-zero subspaces Uu . . . , Un such that V = Ux © • • * © Un, and if for every x E L (2.1)
P(*)=S Utp{x).
Free modular lattices and their representations
183
PROOF. The necessity of (2.1) is clear. To prove the sufficiency we show that every subspace Uf is admissible, that is Uj(p(x) + p(y)) = Ujp(x) + Uj9(y) for any x, y € L. By (2.1) we have p(x) + p(y) = % Utp(x) + 2 Utp(y). t=1 l==1 axiom, we find
Using Dedekind's
Uj (p (x) + p (y)) = Uj(j] (Utp (x) + Utp (y))) = = U, (Uj9
(X) +
UiP (y) + S (Utp (x) + UiP (y))) = = Ujp (X) + Ujp (y) + Uj S (Utp (x) + L^p (y)).
Note that C/; ( 2 ^/P W + k7;P (y)) s ^ ; S Ut = 0. ()
Consequently,
This proves that every subspace U}- is admissible and means that the correspondence x H-> Ujp(x) defines a subrepresentation p^' in Uj. It is easy to check that P = ® p u s so that p is decomposable. We now assume that V = Ux ® * * * © Un, and that each of the subspaces Ut is admissible relative to p. The following example shows that we cannot, in general, assert that p is equal to the sum of its restrictions pUi. EXAMPLE. Let D2 be the free modular lattice with two generators. (Note that D2 consists of four elements el9 e2, exe2, ex + e2.) Let V denote the 8-dimensional vector space over a field k with the basis £ l5 . . . , £ 8 . We define a representation p of D2 in V in the following way. We set
pta) = kit + kib + k(i2 + y ,
p(e2) = ki3 + ki7 + HI, + y ,
where k%t and k(^- + £z) are the one-dimensional subspaces spanned by the 4
vectors £,- and £• + ^, respectively. Let U1 = 2 &£;
8
anc
^ ^2
=
S
i=l
^?f-
i=5
It is easy to check that each of the subspaces Ut is admissible relative to p,but that p ^pUi + pU2In conclusion of this section we state a simple criterion for the decomposability of a representation p of Ef. PROPOSITION 2.3. Let p be a representation of Drin a space V. Then n
p is decomposable
into a direct sum p = © pi of representations
pi if and
i=l
only if there exist non-zero subspaces Ul9 . . . , Un with V = 0 Ut such i=l
n
that p(et) = ^ p(et)Ui r
generators of D .
for every
t 6 {!> • • .» ^}»
where the et are
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/. M. Gel'fandand V. A. Ponomarev
§3. Construction and elementary properties of the cubicles^ + (/) andi?~(/)
3.1. Definition of the cubicles B\l) and B~{1). The sublattices B\l) and B~(l) were defined in §1. To make the text independent, we repeat here the definition and introduce some notation. The sublattice of Dr generated by the elements ht (1) = 2 ei *s' by Hi definition, the upper cubicle B+(\). We define the sublattices B*(l) for / > 2. First we construct important polynomials et ir Let / = {1, . . ., r}. We set A(r, 1) = /, and let A(r, I) be the subset of all sequences {iu . . . , /,), ix E /, with h ^ *2> h ^ h> • • > t n a t is, ^ ( ^ Z) = {a = (it1 . . ., ii)\i£l and VXifc, =^^+1}.From an element a. = (il9 . . . , zz), / > 2, we construct a set F(a) C y4(r, / - 1) in the following way:
Note that from 0 = (fclf . . . , k^) e A(r, / - 1) it follows that kx =£ k2, k2 ^ k3, . . . , kx_2 ^ kx_x. The polynomials ea = ei
t
are
defined b y i n d u c t i o n o n /: if a = ( / j ) G , 4 ( r , 1 ) , t h e n ea
= e(i)=
if a = ( / l 5 / 2 ) G y l ( r , 2 ) , t h e n ^ a =
eix ;
e{i
1 2
= eti
2 *e; P£r(a)
if a = ( / j , . . . , it) G yl(r, / ) , t h e n
J
YJ e 3 . P£r(a)
E X A M P L E . L e t a = ( i l 9 / 2 ) ; t h e n T ( a ) = {ft = {kx) \kl£{i1,
j2}},.
that
is,
ea = eiii2=eu
S PET(a)
«p = ^i1 S
e
J-
^ h ^
Now we define generators ht(l) (t = 1, . . . , r) of i?+(/). We set At(r, I) = {a = (iu • • ., *M» 0 I a 6 ^(r» 0. ^ is fixed }, r, Z)
i=^t
The sublattice of D1" generated by h^l), . . . , /zr(/) is denoted by B\l) and is called the /-th upper cubicle. The sublattice dual to B+(l) is denoted by B~(l) and is called the /-th lower cubicle. 3.2. A structural lemma and its consequences. In this section we prove that B+(l) and B~(l) are Boolean algebras. LEMMA 3.1. Let L be an arbitrary modular lattice, and {ex,. . ., er) a finite set of elements of L. Then the sublattice B generated by the elements hj = 2 *i (/ = 1» • • •» r) is a Boolean algebra. PROOF. Let C be a non-empty subset of / = {1, . . ., r}. We claim that the following identity holds in L:
Free modular lattices and their representations
(3.i)
185
nto= 2*«k«+ 2 «*•
If C consists of a single element, C = {/}, then (3.1) takes the form hj-=ejhj+ 2/*«
(3.2)
By definition, 2 ek = hj. Thus, in the case C = {/} we must prove that hj = ejhj + hj, which is obviously true. Suppose that (3.1) is proved for every subset C of m elements (m < r). We show that then (3.1) holds for every subset C\ containing C and consisting of m + 1 elements. Suppose, for example, that Cx -= C U {s}, where s £ C. Then
( S + It follows from s ^ C that 2 ^ ^ 2 ei ^ S ei^i <#=«
t£C
and
i£C
2
+)
2 et ^ /^s
2 *ft. k£I-Ci
Consequently, by Dedekind's axiom, (3.4) n hi = 2 etht + 2 ^ + ( 2 et)e8 ek+eh=T
2
eihi+
2
eA.
£ J C
We have denoted by B the sublattice of L generated by the elements hj — 2 et- We claim that every element v £ B can be written in the form 2 *j/ij, j£'
where 0 ^ a and a = I - a. Note that in the case u = hj we have proved in (3.2) that hj =
2
*,- + £,-//>. Now let Ui and y2 be two elements of L such that vq= 2 ei+ 2 ejhj
(g=l,2).
It easily follows from the identity e(h + et - e( that 2=
2 ^i+
2
Applying (3.1), we can write in the case a
186
/. M. Gel'fand and V. A. Ponomarev
i£a
j£a'
fea*
In accordance with this identity, in case aa =£ /,
Since a\ U a'2 = (ax O a2)', we have
It is not hard to verify that the formula
= 2 ** + 22 remains true when tfj = / or a2 = I. It follows from the relations we have proved that every element v G B can be written in the form u = 2 ef + 2 ejhj- We denote such an element by va.
i£a
jea
'
We denote by % (I) the set of all subsets of / = {1, . . . , r}. It is well known that $B (I) is a Boolean algebra with 2r elements. We have proved that va + vb = vaUb and vavb = vanb for any a, b£$9(I). This shows that the correspondence a -> va is a morphism of 98 (I) onto B. It is not hard to prove from this that B is a Boolean algebra, and the number of elements of B is 2m , where m < r. COROLLARY 3.1. Each sublattice B\l), B~{1) (I = 1, 2, . . . ) of Dr is a Boolean algebra. COROLLARY 3.2. Every element v of B+(l) can be written in the form
where a is an arbitrary subset of I ( 0 ^ a ^ / ) . / / a =£ /, va.i= n M Z ) . i
minimal element of B\l) is vQtl = S ^{1)^(1) = fl MO, ^ iGJ" iGJ" maximal element is vjtt = >^ e^Z). We shall prove later that each sublattice B+(l) and B~(l) consists of 2r elements, so that B+(l)^&(I) and B'{1) ^ &(I). Occasionally we denote an arbitrary element of B+(l) and B~(l) by v+(l) and v~(l), respectively. 3.3. Ordering of the sublattices B\l) and £"(/). PROPOSITION 3.1. Let v\l) e B\l) and v\m) G B\m). If I < m, then v\l) D v\m). Similarly, if I < m, then iT(/) C v~(m), where v~(i) G B~(i). The proof of this proposition rests on two lemmas. LEMMA 3.2. Let a = (/,, . . . , /,) G A{r, I) for I > 2. We write
187
Free modular lattices and their representations
of Dr can be ordered
ir(ot) = ( * ! , . . . , //_i). Then the elements ea and en^ as follows: e^a) D ea.
T h e p r o o f is b y i n d u c t i o n o n /. L e t 1 = 2, t h a t i s , a = (il9
via) = ii^
and ea = eili2 = e i l ^
e, and en(a)
= et
i2).
Clearly, ea C
Then
en,a).
Suppose that the lemma has been proved for every a E A{r, X) with X < /. We prove it for a = (il, . . . , /,). By definition,
where
Similarly, where
Clearly, for any j3 = (fc1} . . . , ^/_i) ^ F(a) we can find an element ff ^ r(7r(a)) such that 0' = TT(/3) = (ku . . . , kt__2\ By induction on such 0 and |3r = 7r(j3) we have e6 C e .flV Consequently S eP ^ S ^P'» hence LEMMA 3.3. L^r htil - 1) G ,g+(/ - 1) / o r / > 1 and t G /. Then ea Q ht(l - 1) for every a E A(r, I). PROOF. By definition,
S
, l-l)
We consider first the case when a = (i1, . . . , z ^ , /z) with / ^ j =^= Z1. Then 7r(a) = (/ l5 . . . , I ^ J ) E ^ ^ ^ r , / - 1), and so
*««x) s 2
j(r, l - l )
= ht (I — 1).
By the preceding lemma, ea C e ff(a) . Consequently e a C /zr(/ - 1). Now we consider the case a = {iu . . . , il_1, / z ), where ii_x = t. By e P> and A;/-I^{JZ-I» i/} f ° r every definition, e a = e\x S per(a) 0 = (&!, . . . , fy.j) E r ( a ) . In this case, / ^ j = t. Therefore kl_l Consequently, F(a) cz 2 ^;( r » J — 1)» ^ n d so
¥= t.
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I.M. Gel'fand and V. A. Ponomarev
A fortiori ea = eh 2 e$ ^ ht (I —1). P6r(a)
PROOF OF PROPOSITION 3.1. It was proved in Corollary 3.2 that the maximal element of B+(l) is vIti = 2 ^ ( 0 = S e<x., and the minimal i£/ £A( I) element of B+(l - 1) is va , , = n /z,(/ - 1). It is clear from Lemma 3.3 that for any a G ,4(r, /), Therefore, t£I
a£A(r, Z)
Now if u+(/ - 1) and u~(/ - 1) are arbitrary elements of B+(l - 1) and B\l), respectively, then v\l - 1) D vdJ_1 D vIfl 3 u+(/). The corresponding statement for the cubicles B~(l) and B~(m) is obtained by duality. We denote by B+ the subset of Dr that is the union of the B\l), / = 1, 2, . . . . Similarly, B~ = U 5"(/). COROLLARY 3.3. £ + a«c? 5 " flre sublattices of Dr. 3.4. Fundamental properties of cubicles. We have, thus, proved the following theorem. THEOREM 3.1. (1) Every sublattice B+(l) and B~(l) (/ = 1, 2, . . . ) is a Boolean algebra. Any element va t G B+(l) can be written in the following form: i£a
where a is an arbitrary subset of I = {1, . . ., r}, and a' - I - a. If a ¥* I, then va z = mCi h^l). (2) Let v\l) E B\l) and v\m) G B\m). If I < m, then (3.5) v+(l) ^ v+{m). Similarly, if I < m, then (3.6) v~(l) c= v-(m), where v~(j) G B~(J). REMARK. Later we shall prove the stronger statement:
where v+(j) G JB+(/') and iT(/) G J?"(/). CONJECTURE. Let v+(l) G ,8+(/) fl«^ v~(m) G ^"(m u"(m) C u+(/) /or a// / and m. 3.5. In this subsection we introduce another definition of the elements
Free modular lattices and their representations
189
ea E Dr, which appears rather clumsy, but leads to shorter formulae than before. First, however, we examine an example. Let r = 5, and let D5 be generated by the five elements ex, . . . , es. Let a = (2, 1, 5), that is a E ,4(5, 3). Then it is not difficult to compute that F(a) = {(3, 2), (3, 4), (4, 2), (4, 3), (5, 2), (5, 3), (5, 4)} hence *a = e2> 1. 6 = e2
2
^ = *« (e3, 2 + *3, 4 + e4, 2 + *4. 3 + *5. 2 + *5. 3 + *5. 4) •
If in this formula we substitute eitj = ef 2 ^ ,
then we obtain the final
t=hh i
formula for the polynomial ^ 2 1,5 • However, we do not write it down because of its extreme length. It can be shown that in Dr «2.i.5 = ^2(^3 + e5 + eii2 + e4t3) = = ^2^3 + ^5 + ^4(^1 + e3 + ^5)+ ^4^1 + ^2 + O)» and also ^2.1.5 = «a(^4 + e5 + e3>2 + e 3i4 ) = = e2(*4 + ^5 + ez(ex + ei + e5) + ez(ex + e2 + e5))It is clear from this example that our method for writing the formulae for the elements ea, a E ,4(5, 3), as ea = ^ l i 2 i 3 = eii 2 ^P i s n ° t v e r Y |36r(cc) economical. Let a = 0*!, . . . , /7) G ^4(r, /) with / > 3. We describe an inductive method of constructing polynomials e'a from an element aGA(r, I). For every a E A(r, 1) with / > 3 we construct an entire family of polynomials {e«}. We associate with « = ( / ! , . . . , / z ), / > 3, a fixed sequence (fc1? . . . , fc/_2) of numbers A:z- G / such that A:x ^ k2, k2 =£ fc3, . . . , fc/_3 ^ kl_2, and *i?{ii» *2» ':3}» k2f{h^ *s. *4>» • • •» ^z-2?{*/-2. */-i» *'}• W e s e t
= {6 = (A:lf . . ., **_!, / 0 Mx 6 / — {^ = (S = (*i, • • ., kt.2,
/ M ) I ^_x 6 /
-
We denote by H the disjoint union of Hx, . . . , i/j_ j . We set i 2
6£H
where 5 ranges over the whole of //, that is, Hx, . . . , ^ _ j . This definition is ambiguous. It depends on the choice of the numbers kl9 . . . , kt_2, and, of course, on the choice of e'b. For example, e\ t t , depends on the choice of ku that is, there are r - 3 (or r - 2 if il= i3) polynomials e\ i t . The polynomial e\ t f- ^depends on the sequence {ku k2) and on the choice of e'd, where 5 = (ku k2i t), and so on.
190
/. M. Gel'fandand V. A. Ponomarev
We propose the following conjecture. CONJECTURE. For every a E A(r, I) with I > 3, and every e'a
We are able to prove this only for / = 3. For / > 3 we can prove only the weaker assertion that, for every a and every e'a9 the elements ea and e'a are linearly equivalent. §4. Representations of the first upper cubicle By definition, the cubicle B+(J) is the sublattice of Dr generated by the elements /zi(l), . . . , hr(\),
where ht (1) = 2 *i- I n this section we prove that
all elements of B (1) are perfect. 4.1. Atomic representations and their connection with representations of B+(l). We define the most trivial among the indecomposable representations of Dr - the atomic representations pj, i for / 6 {0, 1, . . ., r } . a) The representation po,i in the one-dimensional space V = kl is defined by po.i(^i) = 0 for all i 6 {1, . . ., r } . It follows that p O t l (x) =^ 0 for every x E Dr. b) The representation pt, i for t 6 (1, . . ., r} in V = kl is defined by Pt.ife) = 0 for r = / and p M (e f ) = F. Presently we describe the connection between the atomic representation ptrl and those of i? + (l). In §3 we provide that i? + (l) is a Boolean algebra with the minimal element vdl = O ht(\). Now we prove that u 0 1 is perfect, that is, the restriction of p to p(vQtl) is a direct summand of p. We denote p(^e.i) bY ve,\PROPOSITION 4.1. Let p be a representation of Dr in a space V over k, and let vdl be the minimal element of B+(\). Then p decomposes into a direct sum P= (
© PJ\ i) © *e. i, iGIU{0}
where rQl is the restriction of p to the subspace F e > 1 = p(^ e ,iX ~pjt j w a multiple of the atomic representation pj, ly that is, p}.i = = P J . I © • • • © Pi.i, where m;- > 0.
an<
^
PROOF. We indicate how to choose subspaces Uj such that V = ( © Uj) © F 6 t l , where /° = / y {0} = {0, 1, . . ., r} and p decomposes into a direct sum relative to these subspaces. We claim that the Uj can be chosen as subspaces satisfying the following relations:
Free modular lattices and their representations
(4.1)
191
Uoj]p (e,) = 0, Uo + S P (e,) = F,
and for any / ^ 0, / E /, (4.2)
ff,p(fy)
= 0,
Uj + 9(ejhj) = p(e,),
where /^ = &y (1) = 2 ef. Step 1. We recall that any element of B+(l) can be written in the form (4.3)
i>«.i =
2 2
i
where a is a subset of / = {1, . . ., r} and a = / - a. We write Vatl = p(va,i) and claim that if in V subspaces Uj, j 6 {0, 1, . . ., r}, are chosen to satisfy (4.1) and (4.2), then for any a CI (4.4)
Va.i^^Uj + Ve.i.
We prove (4.4) first for the case of one-element subsets a = {*}, * 6 {1, • • ., r}, that is, that (4.5)
F m , i = ^, + F 9 ( 1 .
Let p(ei) = Ei and p(hi) = Hi. It follows from (4.3) that r
F{t)f i —Et + S ^ i ^ i and F e>1 ~ 2 ^i^*z- Then we find that r
Ut + VQ,1=.Ut+ 2 ^i^i = Ut + EtHt + S ^i^fBy construction (see (4.2)), Ut + EtHt = Et, consequently, This proves (4.5). In §3 we have proved that i?+(l) is a Boolean algebra, and that v aub l = va l + vb l ^ or a n y subsets a, b C I; in particular, ^o.i — S y{t>, !• ^ follows that every subspace p(vatl) = Va<1 can be represented as a sum F a § 1 = 2 ^{<}, i- Putting F{t}> i = C/* + VQ>1 in this formula, we obtain t£a
This proves (4.4). Step 2. We show that our chosen subspaces Uj are such that F s F 9 f l 0 t/0 © C/i 0 . . . © f/r We write a, = {* + 1, . . .,
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/. M. Gel'fand and V. A. Ponomarev
and W^ = Vahi = p(va). Note that (see (4.3)) Wt •-= yj EtHi + S £/. It follows easily from the relations EiHi C £,. that the subspaces PF*, t £ {0, 1, . . ., r} form a chain Wr c= PF,.., <= . . . c= TF2 cz ^ c= JF0 c= F, where PFr = 2 £ , # , = F 9 ( 1 , PF0 = j] Et = F 7 i l . i=l
i=l
We claim that the Uh j £ {0, 1, . . ., r}, subject to (4.1) and (4.2), are connected with the Wj by the following equations: (4.6)
TF0 + f/0 = V, WQU0 = 0,
and for every t 6 {1, • • ., r) (4.7) ut + Wt = Wt-V (4.8) f / ^ i = 0. Note that (4.6) is the same as (4.1), because Wo = V, Et = 2 p(*i)- We i=l
have proved earlier (see 4.4) that Va>1 = ^] Ut + F 6 t l Consequently, PFt = F a , t = *'
S
i=t+i
i=l
for every a C /.
U
J + ^e.i = S ^ + ^ . Now (4.7) i=*+i
evidently follows from this equation. Note that t
r
W* = S EtHi + S ^ s ^/^« + S £/ = ^ ^ + Ht = //,. From this, i=l
;=*+l
j^t
using (4.2) (t/ r ^ r = 0), we obtain WtUt C HtUt = 0, that is, UtWt = 0. This proves (4.8). It follows easily from (4.6)-(4.8) that r
r
V-=y] Ut + Wr = ^] Ut+ Ve 1 and that this sum is direct. Step 3. We claim that for every i £ (1, . . ., r} (4.9) Et = S^C/ y - + ^ F 0 l l . To prove this we show first that EtUj = 0 for / =^= /, and ^ - ^ = f/z-. For by construction, f/0 21 ^i ^ ° a n d ^ 7 ^ = ^ S Ei = °» ^ + ^*#* = ^iConsequently, for every / ^ 0 we have: 1) U0Ei = 0, 2) UJEi = 0 for / =£ 0, / * i, and 3) ^ - ^ = £/,, Thus, we can rewrite the right-hand side of (4.9) in the following form:
(4.10)
jj EiUj + EtVe.^EiUi + EiVe.^Ut + EtVe.i. j-=0
Let us find EtV6 x. By definition, Fe>i = p(^e i) = P ( h M = fl^ii il
ii
where
Free modular lattices and their representations
Hj—^\Et.
Hence Efl^^j
193
- Et, and therefore
EiVQ,i^EiCn
Hj) = f| EtHj = EtHt.
We have to show that Et = Ut + Vel Et = Ui + £,//,, which is true by construction (4.2). This proves (4.9). Step 4. We combine the results of steps 2 and 3. We have proved that V = ^] Ut + VQtl and that this sum is direct. Further, we have proved that t=0 every subspace Et (Et = p{et)) is representable as a sum 2£j = j\EiUj-\- EiVQtl. Consequently, by Proposition 2.3, p splits into the direct sum p = © py, x © Te ,i, where pj9l=puJ and t 9 f l = p v e.i. Step 5. We claim that pjtl is a multiple of the atomic representation p; lt , that is, p/ f l = p/,i © . . . © Pi.i, with rrij > 0. First we study the representation pOil. It follows from Proposition 2.3 that the subspace Uo is admissible, therefore, p0>i (et) = Uop(et) = U0Et. We have just proved in b) that U0Et = 0 for every /. Thus, the subrepresentation po,i in Uo is such that po.ife) = 0 for every /. If dim Uo = m0 > 0, then po,i is different from zero, and, clearly, splits into the direct sum po.i — ^
x
© . . . © p^
°f atomic representations
Po.i-
Similarly, for the pj,i with / =£ 0, we obtain
0
if
i¥=h
Uj
if
i = /•
If dim Uj = mj > 0, t h e n it is easy t o see t h a t pj.i of a t o m i c r e p r e s e n t a t i o n s pj.i = P J . I © • • • © P ; M -
splits i n t o a direct s u m
PROPOSITION 4.2. Let val be an arbitrary element of B*(\) corresponding to the subset a £ /. Let p be a representation of Dr in V such that the subspace Va x = p(^a.i) *5 different from zero and from V. Then p splits into a direct sum P= (
©
P/.i)©Ta,i,
where a = I - a\ each of the representations p;,i is a multiple of an atomic one: P;,i = P./,i © . . . © PJM with m;- > 0: Also, ral is the restriction of p to Val. The representation Tal is such that if it splits into a direct sum of indecomposable representations xa>1 = 2 TJ, tnen
194
none sum.
/. M. GeVfandand V. A. Ponomarev
of the atomic
representations
pitl
with
i £ a' [} {0} occurs
in
this
PROOF. In Proposition 4.1 we have shown how to choose subspaces Uti t £ {0, 1, . . ., r}} such that p splits into a direct sum P = ( © P;\i) © Te.i relative to Uo, Uu . . . , Ur, and VB , . Here pjtl = p \v. is a multiple of the atomic representation pjtl, and T e = p |Vfl ^ where F e > 1 = p(y e i l ), is the image of the minimal element veA G £ + ( l ) . The subspaces Uf are such that Val = 2 # ; + F 6 l l . 3 a Consequently, ^
P=(
©
P;.i) © ( © PA.I) ©*e.i = (
©
;6a'U{0}
fe£a
iea'LKO}
where x a>1 = (© p fetl ) © x e > 1
plilevi,
is the restriction of p to Ffl x .
We now claim that rfl ! does not contain as direct summands the atomic representations p ; f l , / 6 «' U {0}. We assume the contrary. For example, let x a > 1 =p J -, 1 © xi t i, where / G a\ and let F fl>1 = Wj + F j f l be the corresponding decomposition of Va x into a direct sum of subspaces W}- =^= 0 and V'a x ¥" Va j . Let us find out what the subspace Ta x (hj) is, where hj = 2 et- ^y Proposition 2.3, Ffl j is admissible, therefore *a.i(h3) = F a , l P ( ^ ) = p(i; a .i)p(^) = P(^a.i^).
We recall that vatl=
(] ht
and, by assumption j £ a . Consequently, hjVa>1 = hj (] hi = f] ht = i; a t l . iga'
igo'
Finally, rfl j (/z;) = Ffl x . On the other hand, we have assumed that Ta>1 s* p ;>1 + x^ 4. Therefore, ia,i(hj) = pj.^hj) + T;, I(/I 7 ). Since the representation Pj, x is atomic, we see that p;\ i(et) = 0 if / ¥= /. Therefore, pj.i(^j) = PJM S e* = S Pi.i(g«) = 0Consequently, T
a.i(hj) = 0+ i'at i(hj) g 7 ^ j ^ F a , x . So we have obtained a contradiction. It can be proved similarly that ra l does not contain the atomic representations po.iCOROLLARY 4.1. Every element va r G £ + (l) is perfect. We have already mentioned that every subspace Vatl = p(vail) is admissible. But we can prove the more general assertion that every element va x G B\\) is neutral1) in Dr. COROLLARY 4.2. The Boolean algebra B+{\) consists of 2r elements. PROOF. We choose a representation p which is a direct sum Pi,] © . . . © p r .i of the atomic representations f>f(1for t £ {1, . . ., r}. Then it is easy to see that the representation space V of p is r-dimensional: r
V ~ k?. Here all the subspaces p{et) are one-dimensional, and 2 P(ei) — Vi l > An element a of a modular lattice L is called neutral if a(x + y) = ax + ay for all x, y G L.
Free modular lattices and their representations
195
It is easy to see that the sublattice p(B+(l)) generated by the p(ef) is a Boolean algebra with 2r elements, that is, p(£ +(l)) ^ # ( / ) , where &(I) is the Boolean algebra of subsets of / = {1, . . ., r}. Earlier we have proved that B*(\) is a factor algebra of &(I). Now we conclude that B+(i) s* # ( / ) . In the following sections we show that every element in the cubicle B*(l) is perfect. However, to prove this we need more complicated techniques (the functors ®+ and ®~). 4.2. Representations of the first lower cubicle. The results of this subsection are dual to those of 4.1. The first lower cubicle B~(l) is the sublattice of Dr generated by the r elements ht~ = n et {t ~ 1, . . . , r). In addition, we write h$ = O et. Clearly, HQ C hj for every / =£ 0 and ho = /2zrfy~ for all /, / £ /, / =£ /. Thus, /*o is the minimal element of B~(\). Then any element ufl { G B~(l) can be written in the form vZ, I = ho + S ^r» where a C /. Since /^o = ye i is the minimal element of B~(\), we see that if a =£ 0,then Let p7, i be dual to p fil .Namely, pr, i is the representation in V ^ kl such that po, i(et) = V , pf, i(^) = V for all /, t ^ i, t ¥= 0 and Pi", i(*«) = 0. The representation pr, i is called O~-atomic, and P M = pj", i is called O+-atomic. The dual to Proposition 4.2 is the following assertion. PROPOSITION 4.3. Let p be a representation of Dr in V such that the subspace Vz,i = p(^a,i) is different from zero and V. Then p splits into a direct sum p = p' © xa>1, where ^a.i = p \v- . Here ra x can be represented as a sum xa>1 = 0 pj u where'each pj i w a multiple eU(0}
o/ ^ e atomic representation pjtl. If p' 5p//Y5 /nto J direct sum, then the atomic representations pj, i , / 6 « ^o «or occwr m ^ w 5wm. We have also proved that each element v~ x G 5~(1) is perfect. §5. The functors O+and
In this section we define functors O + and O~ from M{Dr, k) into M{Dr, k)y and use them to study representations of the cubicles B*(l)y I > 1. In essence, these functors were first defined in [5]. A modification of them, under the name "Coxeter functors", was used effectively in [ 1 ]. They also play a decisive role in the proofs of the main results of this paper. However, we still do not understand their connection with the lattice operations sufficiently well.
196
/. M. Gel'fand and V. A. Ponomarev
5.1. Definition of the functors and their fundamental properties. Let p be a representation of Dr in F. We define a space F 1 and a representation p1 in F 1 in the following way:
t" = {(l,, ...,Er)|6i6p(e«), S16« = 0>,
where e,- (/ = 1, . . . , r) are generators of Dr. r
In other words, let R = 0 pfo) and let V: # -> F it 1 r
be the linear trans-
formation defined by V(£i, • • •> Ir) = S £i- Then F 1 = Ker v. We set Gl = {(Slt . . ., £*_!, 0, £,+1, . . ., lr) e /?}, that is, G\ = 0 p(ej). Then Thus, from the representation p £ J2 we have constructed another representation p*£ M. It is not hard to check that the correspondence pH>-pl is functorial. We denote this functor by 3>+. r A representation p"1 is constructed dually. We set Q = 0 (F/p(e,)). We denote by P*: F-> V/p(et) the natural map, and by JU: F -> Q the map defined by JU£ = (0i ? , - - . , ]3r?). Further, we set F " 1 = Coker M = Q/Im MWe define a representation p"1 in F" 1 in the following way. We write Qt - {(0, . . ., 0, pf | , 0, . . ., 0) | I e F}. Let (9: Q -• F" J = Coker /x be the natural map. We set def
where 6Qt is the image of the subspace Qt (6Q( C F" 1 ). It is not hard to check that the correspondence p *-*> p~l is functorial. We denote this functor by (p-(d>-: &-+<%) and we write p"1 = O"p. We describe first some simple properties of the functors O + and O". PROPOSITION 5.1. Let p and p1 == O+p and p"1 = O"p be representations of the lattice Dr in the spaces V, F \ and F" 1 , respectively. r
(I) / / p is such that 2 p(et) = ^. ^^« r
dimF 1 = ^] dimp(ei) — dim F. (II) / / p is such that S.pfo) = F, then dim p1 (et) = 2 P (e7") — ^ i m V.
Free modular lattices and their representations
(III) / / p is such that
197
(1 p(et) = 0, then r
dimF- 1 ^: 2 dim 7/p (a,-) — dimF. (IV) / / p is such that
(] p(et) = 0, then dim p~l(et) = dim F/p(ef ).
PROPOSITION 5.2. Let p be a representation of Dr. , vv/zm? P* w r/ze dwa/ to p (II) / / p = 0 p^ then
p
i^l
(&) 7=1
The proof of these propositions is not difficult and reduces to a direct verification. Next we prove a number of important properties of O + O " . (I) There is a natural way of defining a monomorphism /: <[)~O+p—>- p such that O~O+p is isomorphic to the subrepresentation p |ve { — the restriction of p to F e , x = p(i;e, i)» where vd x is the minimal element of £ + O). (II) There is a natural way of defining an epimorphism p: p - ^ O + O ~ p such that O + O"p is isomorphic to the factor representation p/x7i, where Tj i is the restriction of p to Vjti = p(vjt i), where v] x is the maximal element of B'{\). If p is indecomposable, then so are, as a rule, ®+(p) and <£>~(p) . More accurately, we have the following assertions. (III) If p is indecomposable and O + p =^0, then O+(p) is also indecomposable; in this case the monomorphism i: O-ct) + p->p defined above is an isomorphism. If p is indecomposable and O + p = 0, then p is O+-atomic: p ^ p;\ i for some / 6 {0, 1, . . ., r}. (IV) If p is indecomposable and O~p =7^0, then ®~(p) is also indecomposable, and the epimorphism p: p -> O+O~p is an isomorphism. If p is indecomposable and O"p = 0, then p is O'-atomic: p =^ pj, i for some ; G (0, 1, . . ., r}. We prove (I) and (III) below in the framework of the more general Proposition 5.4. (II) and (IV) are dual to (I) and (III). 5.2. The elementary maps q>j. We denote by cp^ the linear map cpf. V1 ->• V from the representation space V1 of p1 = ®+ p to the representation space V of p that is defined by the formula (5-1)
?r) = h.
We call these maps elementary. It follows at once from the definition of p1 that for any i g {1, . . ., r)
198
/. M. GeVfandand V. A. Ponomarev
(5.2) (5.3)
p*(ef) = Ker
where ht = 2 **• Note that
cpf
does not define a morphism from p1 to p.
The cpi have another property, which could also serve as their definition. First some notation. We note by n the embedding fx: F lc —>/? = © pfe). It follows from the definition V1 = Ker V that the following sequence of vector spaces is exact: V+- R +- F1*— 0. We set G,= {(0, •.. ., 0, g,,0, , . .,0) \tiep(et)}.Then
R = S G, and this sum is i=i
direct. We denote by JI,- the projection Jt,: R -* R with kernel GJ = 2 ^ and image Gt. PROPOSITION 5.3. (I) q)i = VJiiM- ^«<^ (II) p1^*) == Ker cpf = Kerjt|(.i. The proof is elementary. We now describe an important "construction." Later ( § § 6 , 7) we shall prove that this construction "builds up" from V\it = p1(va> t), va, t g B+(l), the subspace Va, Hi = p(va,i+1)1 va l+1^B+{l-\-i), where a is some subset of / = {1, . . ., r). CONSTRUCTION 5.1. Let p be a representation in V, T1 a representation of p1 = O + p, and U1 be the representation space of x 1 . We set U = 2 ^iUx(U
^ V). Then a representation x in U is given by
i(et) - (p^ 1 . PROPOSITION 5.4. Let x1 representation x defined from properties: (I) x ;* O-x1; (II) x1 s O + x; (III) x w fl direct summand Before proving Proposition
Z?e a ^/>ecr summand of p1 = O + p. Then the x1 Z?y r/ze construction 5.1 /20s ^ e following
of p. 5.4, we illustrate it by one of its consequences. r
r
In §3 we have introduced the element ve<1 = f| ht — 2 ethu where r
i=i
i=i
^i == 2 ej- This ve.i is the minimal element of B+(\). We denote by %Q
1
the restriction of p t o F 9 l l = p(^e.i)COROLLARY 5.1. xe, x =^ O-O + p. PROOF OF THE COROLLARY. We have proved in Proposition 4.1 that x 6 ( 1 is a direct summand of p. Therefore,
Free modular lattices and their representations
Since hj = 2
e
t>
w e see
199
that eA- = e{ if / =£ /, and hence
In Proposition 5.4 we set /71 = V1. Then £/ = S
and the
i—1
representation x in ( / i s given by x(et) = cp^F1. By Proposition 5.4, x ^ O-p 1 = ®~® + p. We claim that T = T 8 f l . It follows easily from the definition of the elementary map q>< that q^F 1 = p^,-/^). Therefore,
Moreover, by definition xfo) = q^F 1 = p(e^j). Thus T 8 ( 1 = T ^ d)-O + p 7 and the Corollary is proved. PROOF OF PROPOSITION 5.4. Suppose that p 1 = O + p splits into a direct sum p 1 = x\ 0 Tg. We denote by U \ and C/J the representation spaces of xj and x\. Then V1 = U\ + t/J, (/} C/J = 0, and for every
r
Next we set Rj = S ji^i/J (/ = 1, 2). Note that Rj c /? = © p(^.). i=i
t=i
We list ^some properties of Rx and 7?2a) Firstly, RXR2 = 0. WQ recall (see Proposition 5.3(11)) that p1^.) = Ker jtjfx. Consequently, we can rewrite (5.4) as: (5.4')
Ker ji^pi = U\ Ker jt£jx + U\ Ker jij^i.
By construction, the subspaces U\ and L^J do not intersect. Using this fact and (5.4'), we deduce easily that for any i G / the subspaces nt\iU\ and TCtnUl also do not intersect: (5.5)
(nt\iUl)(niViU\) = 0. r
Note that ni\iUj C Im Jij = Gt. Also, R = 2 ^f > a n ^ this
sum
is direct.
an
From these properties, the definition Rj = V ji/jxZ/J, ^ (5.5), it is not M hard to show that R^2 = 0. 1 b) Next we claim that R^V ) = iiUj (/ = 1, 2). Since /x: V1 -• /? is an embedding, we set /xF 1 = F 1 and fiUj = Uj. By definition of ifiCSi, • • • , £,-, • • • , ^ ) = (0» • • • > 0, f,-, 0, . . . , 0). Consequently, r
2 3xjJJ. f/} ^ U), in other words, £/y C jRy-. Using this relation, the equation i=l
V1 = U\ + U\, and applying Dedekind's axiom, we obtain
200
/. M. Gel'fandand V. A. Ponomarev
Since Ul2C R2, we have RXU\ C RlR2
= 0, hence RiU\
= 0. Consequently,
R{Vl = U\.
(5.6)
The equation R2 V1 - U\ is proved similarly. r
c) We denote by x/ the representation in Uj = 2 9;^) (/ = 1> 2) i=l
by t;(^j) = cpff/J. We now claim that x;- ^ O"x}. r
r
r
We note that vRj = v ( 2 J I ^ C / } ) = 23 V ^ ^ f / J = 2 q>*C/} = £/,. il
il
ii
It follows from this equality that we can define the restriction of y: R -+ V to Rj C R and Uf C F. We denote this restriction by Vj: /?./ ->• f/j. Since V-ff; = Uj, it is an epimorphism. Also, Ker V; = Rj Ker V = RjV1. We have shown (see (5.6)) that RfVl = U}. Consequently, Ker Vj = U). We denote by \ij the embedding Uj C _> /?;-. Clearly, ji; is the restriction of |i to L^ and Rj. Thus, the following sequence of vector spaces is exact:
By definition, the representation T) in Uj is a direct summand of p 1 hence i)(et) = f/Jp 1 ^). According to Proposition 5.2, p 1 ^ ^ = Ker nt\i. Therefore x} (e() = Uj (Ker JI,JA). It clearly follows that ^ £ 7 } ^ C/J/xJfo). By definition, Rj = ^nt\iUji
and as we know nt\iUj C jtf i? = Gz-. Thus the r
sum T] JtjjLit/} is direct, hence i?7- ^ © C/j/xJ-^i). i
i=l
We denote by (jifjji)y: £/y -^ /?;- the restriction of nt\i to £/;? and Rj. By V ^^ = i# Therefore 2 ^/M- = ( S nt) M-
definition of nti
T
i
=
H- ^n particular,
i
it follows that UJ = 2 (jtj^)y. So we have shown "that (I) Rj i S « i ^ } = © U)/^(ei); (II) ^ = 2 i i l i l (Ill) Uj = VjRj ^ ityKer Vj = Rj/Imiij = Coker \ij\ (IV) the representation def
ij in f/y is such that T/(ef) = cp^C/} = SJn^U) = vCrt^t/}) = Vji^tiiU}). By comparing these properties with the definition of ®~, it is not difficult to convince ourselves that T,- ^
/
,
dcf
vi
nt\iU) ^ Vtn^c/j) == x;(e,). Therefore, Rj — ZJ^IV'UJ ^ ® x;(ef). We omit the rest of the proof. (III) We claim that each ty (/ = 1, 2) is a direct summand of p. To prove this, we show that xx 0 x2 = x e>1 , where x 9 i l = p | v
^ and
'e.i
Free modular lattices and their representations
201
VBtl = p(vQtl). Note that in the proof of Corollary 5.1 we have used only Proposition 5.4, (I). Thus, we may now assume that Corollary 5.1 has been proved (that is, t h a t T 9 i l ^ O~O + p). By assumption, V1 = U\ + U\, and this sum is direct. In Corollary 5.1 r
r
we have proved that
53 Vivl
53
^
= Ve. j . Therefore, V e . i = 53 Vi(u\
+
U
D =•
We claim that Ul and U2 do not intersect. In (I) we have shown that Uj = vi?;. Consequently, Ux + U2 = V(#i + R2)There we have also shown that Rx and R2 do not intersect, and that Ker V = V1 = U\ + U\ = VlRt + F 1 i? 2 .Thus, we can write : Ker V = Ri Ker V + R2 Ker V- From this and from RXR2 = 0, it follows easily that (Vi*i)(Vi*2) = 0, that is, Ux U2 = 0. Thus, VQl s Ux 0 £/2- Consequently, T e > 1 = p | VQ i splits into the direct sum iQ}1 = Tj + x 2 , where Xj =
pi = 0
TJ? w
>> 2, x} ^ 0. 77ze« p splits into the direct sum
p = p(i) 0 (® x^), where T;- W constructed from x/ fl5 m 5.1. Moreover, a)
T,
^
O-T};
b) 0 T ; ^ O-O + p ^ x e , i, where x 9 l = p | v +
c) O p(l) = 0
and
p(l) = 0 ^ 0 ^ ,
•
w/z^re p<" 1 w fl« atomic
representation. PROOF. It follows from the proof of Proposition 5.4 that p = p(l) 0 ( 0 TJ), where x, s
J
0 x;- = x e ,i ^ O-(D +p.
By
i=i
Proposition 4 . 1 , p(l) is the direct sum of atomic representations r
P?, i» * € {0, 1, . . ., r } , that is, p = 0 ktpt i- It is not hard to check that <&+pi, i = 0 for every atomic representation p^ lB Therefore, O + p(l) = 0. PROPOSITION 5.5 (I) O " O + p ^ p < = > /or 0// / ; J p ( e .) = v, where F w ^ e representation
space of p.
+
(II) (D O-p ^ p <=> fl pfo) = 0 / o r a// /. PROOF. (I). By Corollary 5.1, O"d) + p ^ p is equivalent t o i e , i = p. By definition, x e t l = p I v 0t ^ where F e > 1 = p(^'e.i) Thus, O-(D+p ^ p <=> F = fl p(^)- Clearly, V = f] p(hj) <=>v = p(hj) = p ( V. et).
202
/. M. GeVfand and V. A. Ponomarev
for all /. (II) This assertion is dual to (I). PROPOSITION 5.6. Let p be an indecomposable representation. (I) / / O+p =^=0 then p1 = ®+p is also indecomposable, and p s* O-p1 = ®~(D+p; (II) / / O+p = 0, then p is ®+-atomic: p ^ p*, i, •* £ {0, 1, . . . , r}; (III) / / ®~p =^=0, then p"1 = ®~p is also indecomposable, and p ^ O+O~p; (IV) / /
and for any / E /,
We define the elementary map
Ker
(5.8)
Next, we set (p(ife, . . ., it) = (P^oCPift+1°- • • ° ^i^ 1 ^ & ^ Z. Thus, cp(4, . . ., it): Vl-+ Vh~K It follows from (5.7) and (5.8) that if if = ij+l, then
• • • » i[
iu . . . , ij with ij£I
= {1, . . ., r } . We define a representation T ° in
U° by setting T°(e;-) = cp}
2
cp(/2, . . ., i z )(7'. Thus, x° is a subrepres-
entation of p°.
PROPOSITION 5.7. Suppose that T1 is a direct summand of
Free modular lattices and their representations
20 3
pl = ((fr+y p°. Then the subrepresentation x° ^ p°, which we construct from T1 by the method described above, has the following properties: (I) T° 5* (O-)V; (II)
x< ^
((D + )'T°;
(III) T ° is a direct summand of the representation p°. REMARK. We shall prove later (in §6) that the Construction 5.2 has another property which is, perhaps, for us the most essential. Let where Vatl = p(uOtl) and va x G B\\). Then x°, which is T ' = p'| V a constructed from xz, is such that T° = p° \Va / + j , where Va, i+i = p{va, l+1) and va / + 1 £ #*(/ + 1). Here va 1 and ufl / + 1 are the elements of 2?+(l) and B*(l + 1) corresponding to the sanie subset a C I. PROOF. We introduce the following notation:
In the space Ux C F x we define a representation x^ by setting e,) = F/l We claim that for all X with 0 < X < / - 1 (5.9)
x^ gg
and that x^ is a direct summand of p* = (O+)^p°. For X = / - 1, this assertion follows from Proposition 5.4, if we replace p by p u i and use the fact that pl = O + p U l . Suppose now that (5.7) has been proved for X = k + 1, . . . , / - 1. We prove it for X = k. By definition ph+l = ® + p \ and we take it as proved that rh+1 is a direct summand of p f t + l . We denote by %h the representation in £/ft = S yhi+ lUh+l
for which xft(ef) = $+1Uh+1.
Then according to
ft
Proposition 5.4 x is a direct summand of ph and %h = ®~x*+l. that xft = x \ Indeed, we take it as proved that Uh+1=
2
cpf+1C/'*+1 =
2
^+2* - • • • U
Ufa =
r y* (pj
6/
==
r ^ j CPJ
ZJ
^P V^ft+2? • • • » ll) U
_
==r
We claim
I- M- Gel'fand and V. A. Ponomarev
204
Thus, xft ^ ®~Tft+1, and fh is a direct summand of p*. Also, by Proposition 5.4, xfe+1 ^ <&+vh. By induction we conclude that these assertions are true for any k > 0. Consequently, x° ^ O-x 1 ^ O - ( O - T 2 ) ^ . . . & (d>-)lxl and, similarly, x1 ^ (
2
_
By T e, z w e denote the
h
representation in Ve j for which
COROLLARY 5.3. %Ql is a direct summand of p°, and T 9 , Z ^ (O-)'(O + ) / p. COROLLARY 5.4. / / pl = (®+)lpP splits into a direct sum pl = 0 xj, M > 2, xj =^0, ^ e « p° is also i l
decomposable:
n
p° =L p(/) 0 ( 0 x;.), w/zer^ x;- w constructed from xj &y ^ e
Construction
3=1
5.2. ^ / - e Ty ^
+ f
(
The proof of this Proposition is, essentially, a combination of the arguments in Proposition 5.6 and Corollaries 5.2 and 5.3. §6. Proof of the theorem on perfect elements 6.1. In this section we prove the main theorem: that all elements of B+(l) and B~(l) are perfect. We assume that the following assertion has been proved: let p be an arbitrary representation of D r , r > 4 , pl = (® + yp, and let V and V1 be the representation spaces of p and p* , respectively. Then (6.1)
P(*,... V ) =
(6.2)
P ( / i i . . . i l o ) =
ii)Vl.
The proofs of these assertions, which are in §7, are the central and most complicated part of this paper. THE REPRESENTATIONS pt+, / AND p7, i-We define representations pl i and P/" i (t 6 {0, 1, . . ., r } , Z = 1, 2, . . .) by means of the atomic representations Pi, i and pr, ithat were constructed in §4. We set, by definition, Pt+,i = PM,
tii
= (Q-)l-ltftu
PF.i = (<J>+)l-1pF.i.
We shall prove later (§8) that pt,i&-Pt,i, where Pt.i are the representations defined in § 1. The functorial definition we have given just now is more convenient in those cases when we need not go deeply into the "inner structure" of Pt, iLet p be a representation of Dr in V. We write dim p = (TI; TO1, . . ., TO1"), where n = dim V and TO* = dim p(et). For
Free modular lattices and their representations
205
example, the atomic representations pj", i and pF, 1 have dim pj t j = (1; 0, . . ., 0) and, for t =£ 0, dim pi i = (1; 0, . . ., 0, 1 , 0 , . . ., 0), where m* = i ; dim po, i = (1; 1, . . ., 1) and, for t =^0 dim pf, i = (1; 1, . . ., 1, 0, 1, . . ., 1), where nt = 0. PROPOSITION 6.1. (I) pjtl ^ (p?, i)*, where * denotes the conjugate representation. (II) The representations pt,i and p7, i of Dr, r > 4, are indecomposable for all * 6 {0, 1, . . ., r} am* / = 1, 2, . . . . PROPOSITION 6.2. We ser dim pjTj = (n,, ,; mj >z , . . ., mj t ,). (I) The sequence nt t satisfies the recurrence relations (6.3)
nt, i = (r — 2)nt, z_x — nt. i_2,
Z > 3,
=
e initial conditions nt x = 1, nt 2 r ~ 2 for t ^ 0 and o l = I « o 2 = r - 1. Hence it is clear, in particular, that, for t =£ 0,nt does not depend on t. (II) TTzere exzs^ intergers mx and m0 t such that n
i
\
^
if
i
mt>l ~\ mi + ( — l)Ui numbers satisfy the relations
if
;=^*
!
m
i
~
m
o, i =
t
0,
rrrn = nt + n M + (-1)', ! rm0, i = n0, t + n0, z_x. «/ stands for nt l with t =£ 0. We prove first Propositions 6.1 (II) and 6.2 by induction on /. Let t =£ 0, so that pf, z is a representation of the first kind. For / = 1 we have already proved that dim pit\ = (U 0, . . ., 0, 1, 0, . . ., 0) and that pt, I is indecomposable. By definition p£ 2 = O~p£i. It is not hard to verify that dim pi 2 = (r — 2; 1, . . ., 1, 0, 1, . . ., 1), where mj 2 = 0. It was shown in Proposition 5.6 that if p is indecomposable and O~p =^=0, then O~p is also indecomposable. Consequently, pt, 2 is indecomposable. We have shown that nt 1 = \ and nt 2 = r - 2 for t ¥= 0. It is also easy to check that (6.4) and (6.5) hold for the numbers m\ x and m\ 2Suppose that Propositions 6.1 (II) and'6.2 have been proved for all k < /, so that the pt+,k, t ^0 are indecomposable, and that the numbers n t k ~ nk (* ^ 0) satisfy the relations * = (r -
2)n fe . x -
nh_2
(k = 3, . . ., / ) .
206
/. M. Gel'fandand V. A. Ponomarev
We now prove these assertions for k = / + 1, where / > 2. a) We show first that pt+,z+i, I > 2, is indecomposable. Note that from (6.6) for r > 4 it follows that (6.7)
nk > 1 for k = 2, 3, . . . , /
We claim that ptj+i = $>~pt,i (t =£0) is different from zero. Assume the contrary, that Q>~pt, i = 0. Since pt, i is indecomposable, pt", i is atomic (see Proposition 5.6): p £ ^ p ~ i . By definition, p7, i is a representation in a one-dimensional space, dim V] x = 1. Since dim F / , = nh we have reached a contradiction to the fact that nx > 1 for I > 2 (see (6.7)). Thus, pt^ + i = ^~Pt,i =^=0. Hence, by Proposition 5.6, pt,z + i and pt,i are indecomposable, and pt, i ^ O+pt+^ + 1 , that is, pj",; ^ O + d)-p^. b) We now prove (6.3)-(6.5) in the case / + 1. In Proposition 5.5 we have proved that if ® + ®-p ^ p, then p(f] ej) = 0 for every /. Hence it follows elementarily (see Proposition 5.1) that r 1
O~p(et) = dim V — dim p(et) and d i m F " = 2 ing this t o p j ^ and pt,i+i — ®~Pt,ii (6.8)
(6.9)
mj,
z+i
dim
®~P(ei) — d i m ^-Apply-
we obtain
= TZf. / — ATZJ, z ,
»*.z+i = 2 wit, i+i — ^ . z -
Since by the inductive hypothesis ntl = nt and mlt raj , = / ? ! / + ( - 1 ) / + 1 , we see that . r nt — mi if m (6.10) *.'+* = \ n z _ T O z + ( _ i ) ^ if
t
= ml if i ^ t and i^^, «.
i=
Thus, we have proved (6.4) in the case / + 1. Substituting (6.10) in (6.9), we find nt l+l = (r - \)nt - rml + (-1) 7 . By the inductive hypothesis we have rml = «/ + «/_i + (~l)1- Consequently, n t i+i = ( r ~ 2)wj - «/_!• This proves (6.3) in the case / + 1. We set, by definition, ml+1 = nx - mx. Then from (6.9), (6.10), and (6.3) just proved for / + 1 it follows that rmi+i = «/ + «/-i + (~1) /+1 . Thus, we have proved (6.5) in the case / + 1. Combining the results of a) and b), we see that we have proved Propositions 6.1 (II) and (6.2) for / =£ 0. The proofs for t = 0 are similar. Now we prove that ptti and p^ t are conjugate. Now p^ i and pjt { are conjugate by definition. Suppose that (pr, i) = (pt, /)* • Then (91 i+i)* = (^+pr, /)* = ( b y Proposition 5.2) ^ O-(pr, z)* ^ ^"pt4; i = pi, i+i6.2. The lattices D19 D2, D3 are known to be finite. Therefore, each of these lattices has only finitely many non-isomorphic indecomposable representations, namely 2, 4 and 9, respectively. Each of these representations
Free modular lattices and their representations
207
can be written either in the form pt, u t 6 {0, 1, « < ,, r } , / £ {1, , . ., r}, or in the form pr, *. For example, the 9 different indecomposable representations of D3 are the following: pt, i, pf+, 2» where 2 6 {0, 1, 2, 3} and pj, 3. Each of these representations can also be written in the form pr, n* Namely, if / =£ 0, then 9t, 1 ^ pr, 2 and pt, 2 = PF, 1 and pj, { ^ po, 3, Po, 2 = Po, 2, po, 3 = po, \ • The "dimensions" dim p^, 1 of these representations are given in the following table: (1; 0, 0, 0), (1; 1, 0, 0), (1; 0, 1, 0), (1; 0, 0, 1), (2; 1, 1, 1), (1; 0, 1, 1), (1; 1, 0, 1), (1; 1, 1, 0). (i; 1, 1, l). Thus, among the indecomposable representations of D3, four are ® -atomic, and four are CD "-atomic, so that the spaces of these eight representations are one-dimensional. Only the representation pj, 2 = Po, 2 is such that dim pj f 2 = (2; 1, 1, 1). +
6 . 3 . With each representation p of D in F there is associated in a natural way an oriented graph T:
V •
•
\
where Ei = p ( ^ ) . The diagram of this graph with unoriented edges is called a Dynkin diagram [ 3 ] :
Then dim p is an integer-valued function on the set r 0 of vertices of the graph (for Dr, the set r 0 consists of r + 1 points). Following the methods of [1], it can be shown that the numbers dim p^ t and dim p^~ l correspond to positive roots of the Dynkin diagram.
6.4. Theorem on the perfectness of the cubicles. As we know already from §3, the /-th upper cubicle B+(l) is generated by the r elements ht(l) = J) eAl),
where ej(l) =
2J
e
* =
S
e , - . , - * . We have also
+
proved there that B (l) is a Boolean algebra. Every element v+(l) E B+(l) can be written in the form:
2 M0 M0 = M0 (n M0), where a is a subset of / = {1, . . ., r} and a - I - a, and where
M0 = 2 MO- ( Note that M 0 +
The minimal e l e m e n t of B (l),
D
fy(0, / ^ 0.)
which we d e n o t e b y vdl,
corresponds to
208
I M. Gel'fand and V. A. Ponomarev
the empty subset a — 0 . Thus,
We now prove a theorem that generalizes Proposition 4.1 to an arbitrary /. THEOREM 6.1. Let p be a representation of Dr in V, and let vel be the minimal element of B+(l). Then p splits into a direct sum
where T 6 , t is the restriction of p to p(vQt j) = VB, i< and each pj, k is a multiple of the indecomposable representation p~jik, that is, ;\ ft
Pi. ft = Pi, k © • • • © pj, ft» rnit
fc
> 0.
The proof is by induction on /c. For A: = 1, the proof was given in Proposition 4.1. We assume that the theorem has been proved for k = I - \ and prove it for k - I. Proposition 4.1 applies to the representation p*-1 = (O + ) Ul p. This means that we can choose subspaces iflr* C Vl~l with the following properties: 1) V1-1 = 2 tfjrl + Fe"}, where VeTi1 = P^M^e.i), and this sum is direct; i=o
'
r
2) p u i splits into a direct sum: pl~l = © p],"} © TJ;}, where p j i = P 1 ' 1 I i-i> u j, i
T
j=o e~i = P1""1 I i-u 3) Pi~i is a multiple of t h e a t o m i c F e, i
representation p£i. The subspaces (7/711 chosen in this way are such that any subspace p ' " 1 ^ , 2) s F1"1, corresponding to yfl j E ^ + (1), can be expressed as a sum
With the help of the construction in 5.2 we build up subspaces Ut / and Vel
such that £ / , . , =
2 *i
(p(ilf . . ., i ^ ) ^ 7 }
and
*l-i
^e.z=
S
P«P(J-l)0(0P,M)0Teii. i=o where pJt t = p | ^ ^ and x 0 ,, = p|^ . Also py, z s ((l)~)Ulpj,~iI. a) We claim that p;-, z is a multiple of the indecomposable representation pj" /. By construction, pj"}^p/ t 1 0 . . . 0 pjf 1. Consequently,
Free modular lattices and their representations
209
b) Now we prove that for any a C / and va l E i?+(/) (6.13)
9(vati)
= ^Uul
+ VBtl.
We denote the subset / — {/} by a.-. Then va.ti = &;-(l) = 2 et.
Hence,
in accordance with (6.11), we can write
PI"1(^(1))= S
U\;}+VlfX
We also write Xj = ^ cp (il9 . . ., it_^) p U l (hj (1)). Let us determine Xj. On the other hand,
But
x,=
2
We assume it as proved that cp(t1, . . ., ii^i)pl~1(et) = p(^i...il_1t) t ¥= //_!, and that q>(il9 . . ., j ; _ i ) p u i ( ^ t) = 0. Consequently, ^ =2
2
p(s...iMt) = p ( S
2
if
.^...<Mt)
So we have proved that (6.14)
22
P(M*))=
It is known (see §3) that va t = n
hAl) for every a =£ /. Therefore,
p(l;at ,) = p ( f| ^-(0) = (1 p(hj(l))- Hence, using (6.14), we find that P(Va,i)=
fl ( 2 jca'
^,^-f-Pe.z).
i£l-{j)
It follows from (6.12) that the sum 2 U^I+VQ,
t
is direct. Consequently,
i
the sublattice of X (V) generated by U}- / and VQ / is distributive. Using these facts, we can show easily that
210
/. M. GeVfand and V. A. Ponomarev
fl ( S ff«.i + Ve.i) = 2ff«.i + F 9 .j. Thus, if a ¥= / , we have proved that (6-13')
9(va.i)-J]U,.l + Ve.l.
It remains to prove (6.13) when a = I. To do this, we have to use the fact that for any /, s G / with / #= 5 we have u7 , = hj(l) + /z s(/). From this and (6.14) it follows easily that p(vI>?)=. 2 Uit j + F e , z. This proves (6.13). In particular, for a = 0 , we obtain p(v e . i) = ^ e . i«
9(1,
S
fi
y definition,
'
M)P(a.i)
Thus, we can rewrite (6.13) in the following way: (6.15)
9(va.i)=
S *i
h-i
It also follows from what we have proved that every subspace ya> l = p(VcLi j) is such that ra, t = p |ya> z is a direct summand of p. For it follows easily from (6.12) and (6.13)'that P = P(Z — 1 ) 0 (
©
P.M) © ta, j .
i£a'U(0}
c) We claim that (6.16)
T O i M = ( e P;,Z)©T6>Z,
, F e .ft= p(^e. &)• BY definition, 0 p;. , 0 T 6> J is the
where x e , k = p | v
e, k r
j
= 0
restriction of p to 2 t/^ z + F e , z. Clearly, S c/;. J + F 9 . «= =
S
S
cp (i,, . . . , i,.,) (y, u];! + vi:l) = ^-i) V - 1 =
q>(£i
S
q> (it, . . -, i « ) S
r
In §5 we have proved that 2 < P i y l = = Z : p(eM = p(^e i)- Applying this result to p U a and p U l ^ O + p 1 - 2 , we find that 2 ^Uiyl"1
= pz~2(^e,i)-
Thus, r
S tfj. z + Ve. i =
i=o
*i
2 *i- a
^P (*i, • • •, iz-2) P'"2 (ve. i).
By induction, we assume it as proved that 2
*i
h-2
T
e. z-i == P l vQ>1
~1
Free modular lattices and their representations
211
7 e t_% = p(ve> ^i). Now (6.16) is proved. By induction, we assume it as proved that p = 0 p;- k © xe, i-t. Comparing this with (6.16), we obtain ;, ft
P = 0 pj,h © x e ^ - 1 j, h
MAIN THEOREM I. In the lattice Dr, r > 4, every element v E B+ = U £+(/) o r u G 5 " = U £"(/) w perfect. This means that for any representation p in any space V over a field k, the subspace p{v) is such that the restriction T = p |P(U) is a direct summand of p. PROOF. In §3 we have proved that B+ = U B\l) is a lattice. In the course of proving Theorem 6.1 we have shown that every element v~ i E B~(l) is perfect. The corresponding result for elements v~j E«5~(/) follows from the duality principle. The following proposition refines the main theorem. PROPOSITION 6.3. Let val E B\l) C Dr', r > 4, and suppose that Va, i = p(^a. z) w different from zero and from V. Then p splits into a direct sum _ P= ( ©
P;.A)©(
©
P;,z)©xa>z,
where xa, i = p | va z» ^«^f ^^c/z representation pj.u and (py, i) w fl multiple of the indecomposable representation p£ ^ <3«
(I) pt,i(x) = Vt,i
for every element x G B+ = U B+(s) such that
x D vt i. In particular, this identity holds for every x E B+(m), m < I. (II) pj", i(y) = 0 for every element y E B+ such that y C ht(l). In particular, this identity holds for every y E B+(n), n > I. PROPOSITION 6.5. Let pjt i be an indecomposable representation of the second kind in F o / . Let f e /_i = 0 ni(l — I) be the minimal element of B\l - 1) and vZt, = S et(l) the maximal element of B\l). (I) Po. ifr) = Vo, i for£Ievery element x E B+ such that x 2 vdl_l. This means that pjt t(x) = VQ, z for every x E B+(m), m < / - 1. (II) po, z(y) = 0 for every element y E B+ such that y C u7 z. r/z/5 m^a«5 //wfr f>o, i(y) = 0 /or ev^rj/ y E i?+(rc), ^ > /. PROPOSITION 6.6. The element fo(l)= S A i o is linearly equivalent to the minimal vdl_l E ^ + (/ - 1), that is, p(/0(0) = p(^e. z-i) /or every representation p. PROOF. In the course of proving Theorem 6.1 we have shown that S
^ M. Gel'fandand
ii-xW1'1
= p(/il...iz_1o).
1-1) =
H
V. A. Ponomarev
Consequently,
P ( / i i . . . i , to) = P(
V--.*i-i
2 *i
2
/*,...*, 4o) -
p(/o(0).
i-i
PROPOSITION 6.7. £ + (/) W £"(/) are Boolean algebras with 2r elements. PROOF. We consider the representation p, the direct sum 0 p£ l t=i
'
of all indecomposable representations pit, i with one and the same /. It follows easily from Proposition 6.3 that all the subspaces p(va, t) for a C / are distinct. Thus, p(B+(l)) as %\I). It clearly follows that£ + (Z) ^ #(7). 6.5. Connection between the upper and lower cubicles in Dr, r > 4. We have proved in §3 that the lattices B+(l) can be ordered in the following way: u* z D v+bj+l for every / and any a, b C I. In particular, the minimal elements v+d z E 2?+(/) form a descending chain U 0 l ^ U0,2 ^ y0,3 ^ • • • • It follows from Proposition 6.7 that all the elements of this chain are distinct. Dual to the v^ j are the elements vjt GB~(l). They form (with respect to /) an ascending chain vj x C vj 2 c vj 3 C . . . . Let p be an arbitrary representation of Dr. As usual, we write Fg, i =p(^e,z) and 77, z = p(^7, /)• In addition, we set F^>0O = O F ^ and Vf^ = U F / / . PROPOSITION 6.8. VJ^ C F ^ /or every representation p of Dr, r > 4. PROOF. The representation space F of p is finite-dimensional. Therefore, there are integers / and m such that Fg, «> = p(^e, *) = Fg, i and "^7,00 = p(vjt m) = F j > m . It follows from the results of this section that p splits into a direct sum p = p(l) 0 x e , z, where T 8 I t = p |y+ i and (O+)'p(Z) = 0. We denote the representation space of p(Z) by U. Since p is decomposable with respect to U and VQJ = Vj^, every x G Dr satisfies p{x) = Up(x) + Fe, oopW. In particular, for the element x = vj m E B~(m), we can write Ff, TO - p(yjf m ) = f/p(yj, m ) + Fg(0Op(i;7, m ) . We claim that (Jp(vjt m) = 0. Hence it follows at once that p(i>i f m)= ^7,m = FJ,ooFj, m , that is, F ^ = F^ m C F0+>oo. Suppose the contrary: that f/p(^7, m ) =7^=0. It follows from the properties of a direct sum that p(l)(x) = Up(x) for the restriction pl^ = p(Z) . Thus, p(Q(*>j,m)= ^p(^7,7w)- If p(0(y7, m ) is non-zero, then, according to the proposition dual to Theorem 6.1, p(l) splits into a direct sum: p(Z) = T(Z) 0 ( 0 pj k), where ( 0 pj h) is the restriction of p(Z) to p(0(?;7,m)- Here "each pj, & is a multiple o f p j ^ . B y assumption, (O + ) z p(Z)=0. On the other hand, (O+)lp(l) = (O+)rx(Z) 0 ( 0 (O+J'pj ft), and we assume U h
that the sum 0 pj
ft
is non-zero. This means that there are / and k such
that pjk = pjh 0 . . . 0 pjki
where m^k
> 0. Then
Free modular lattices and their representations
213
(®+)lPJ,h => @rnj.h(®+)lPlk. By definition (<S>+)lpJtk ^ Plh+i- Since pj, fe is a representation of Dr, r > 4, as we have shown at the beginning of this section, p j 5 ^ 0 for all s > 1. Consequently, (®+)*pj?ft =^0, and hence (O+)lp(Z) = 0. But this contradicts the fact that (O+)fp(Z) =£0. PROPOSITION 6.9. Let p be a representation of Dr, r > 4, />z F, and fef 0 C F/>oo C F0+)OO C F 2? 0. For Z) 4 , the representations of type px were studied in [ 5 ] , where they were called regular. Very little is known about the regular representations of the lattices Dr, r > 5. It is clear (see [ 5 ] , [7]) that the classification of regular indecomposable representations contains as a special case the problem of classifying up to similarity an arbitrary set of linear transformations A
l 9
. . . , A
, n > 2,At:
n
V -> F .
§7. The subspaces p (ea) and the maps cpi
As usual, let p and pl = (® +) l p and V1, respectively, and let (p(il9 (pijo(pi2 o. . . o
p(^i,...i,()
where e
=
be representations of Dr, r > 4, in F . . ., ii): V1 ->• V be the linear map V*1'1 are elementary maps (see §5). prove the formula
9(h» • • «i if) pl (et)i
E Dr is the lattice polynomial defined in § 1 .
i i t
7.1. A lattice in the space J B = © p(e<)* We repeat the construction of the representation p 1 = O + p from a representation p in V. Let r
R = 0 p (a), and let V: R ->• V be the linear map defined by the i=^l
r
formula V(5i, • • ., lr) = S i/i where lt £ p(et).
We denote by Gi the
subspace of R consisting of the vectors (0, . . . , 0,£,, 0, . . . , 0) with It 6 P (ei)>
an
d 1^ GJ ~ 2 £/• Then p 1 = O + p is defined as the representation
in V1 = Ker V for which p 1 ^ ) = V 1 ^ . Let X (R) be the lattice of all linear subspaces of R. We denote by M the sublattice of X (R) generated by the subspaces G 1? . . . , Gr and F 1 . It is easy to see that these subspaces satisfy the following relations: 1°
^
C
7?
2°.
Gi^Gt—
3°.
G{Vl
0 for every /.
= 0 for every i.
214
I-M. Gel'fand and V. A. Ponomarev
From 1° and 2° it is obvious that the sublattice of M generated by Gu . . . , Gr is a Boolean algebra, which we denote by $. Clearly, the maximal (unit) element of $ is R. We define two representations v° and v1 of Dr in R by the formulae ^(efi^Vi
(7.2)
+ Gu
vi(ei) = ViG'i = V*yiGt.
Thus, v°(D r ) E M and v1 (Dr) G M. It turns out that v° and v1 are almost the same as p and p1. More precisely, the following is true. PROPOSITION 7.1. (I) The map V: R-+.V defines a morphism of representations v : v° -> p. Moreover, p(x) — Vv°(x) for every x E Dr. (II) v° splits into the direct sum v° = vj + v°, vv/zere vj ^ Im V = p/,i PJ.I W J/ze restriction of p ro F j , i = p ( 2 ^i) ^
i=l
^fl/ of v° ro F 1 . ^ere vj ^ Ker V ^«^ v i is a multiple of the atomic representation p^,. PROOF. We prove first that v° splits into the direct sum v° = vj + vj. Let U be any subspace complementary to V1 = Ker V (that is, UVl = 0 and U + F 1 = tf). By definition, v°(^) = Gt + V1 D V1. Consequently, v°(et) = v^e^R = v°(e^(C7 + y 1 ) ^ v°(e-)f/ + v^e^^.This means (see Proposition 2.3) that v° is decomposable and that V1 is admissible relative to v° (that is, V^ix) + v°(z/)) = Vlv*{x) + F^v^y) for any r x, y E D ). Since V1 = Ker V is admissible, by Proposition 2.2 the correspondence x>-+ Vv°(.r), x 6 Dr, defines a representation in the space ImV = R/V1. Note that Vv°(ef) = V(^f + F1) = p(ef). Consequently, Vv°(x) = p(x) for every x E Z) r . Thus V defines a morphism of representations: V: v0 -> p. Obviously, Im V = P/,i, where pj.i is the restriction of p to *1
i l
^
Let us verify that Im V ^ vj, where vj is the restriction of v° to U. First of all we note that the linear map \/v: U -> Im V is an isomorphism. ( Vc; denotes the restriction of V to U.) Also, we proved above that (7.3)
v° (a) =
G« + F 1 = v° (Ci) U + v° (e«) F 1 - vJJ (e,) + F 1 .
As we have already mentioned (see 3°),Gt V l = 0; moreover, it follows from UV1 = 0 that v J ^ F 1 = 0. Consequently, (7.4)
dim p(ef) = dim Gt = dim vjjfo).
Thus, Im F = vj. To finish the proof, we note that v}(ef) = F 1 for any i 6 {1, . . ., r}. Consequently, vj is a multiple of the atomic representation po, i (We recall that po, I is the representation in the one-dimensional space W = kl for which po, i(et) = W for all /).
Free modular lattices and their representations
215
We denote by \i the embedding \x: V1 CL> R. PROPOSITION 7.2. (I) The embedding \i: V1 -> R defines a monomorphism p.: p1 -> v1 of representations. Also, v1(x) = jup1^) for any x E Dr. (II) v1 splits into the direct sum v1 = v\ + vj, where v\ is the restriction 1 v | y l vx ^ p1, a«d vj /s 0 multiple of the atomic representation po, iThe proof of this is similar to that of the preceding proposition. 7.2. The connection between the maps (p* and the lattice operations. In this subsection we use as definition of the maps cp* the formula (?i = V^i^t, where nt is the projection onto R with kernel Gj and image Gt. This formula was proved in Proposition 5.2. It is useful for us to have a lattice definition of a projection. LEMMA 7.1. Let R be a finite-dimensional vector space, X a subspace of R, and let n: R -> R the projection with kernel G' and range of values G. Then nX = G(G' + X), where nX is the image of X. COROLLARY 7.2. For every subspace Y1 C V1 and every i £ (1, . . ., r} n^Y1 = GAGi + [lY1). The basic results of this section are formulated below in Theorems 7.1, 7.2, and 7.3. THEOREM 7.1. Let p be a representation of Dr, r > 4, and p1 = O+p. Then 1°. (PiiP'te,...^) = o. 2°.
p(*ji 1 ...t J ) if j =£ rt.
Here, e^ t E Dr are the lattice polynomials defined previously. We shall show that the proof of 2° follows from the next Proposition 7.3. Its proof is very laborious, and is given in §7.3, 7.4. PROPOSITION 7.3. Let p1 = O + p, and let v°, v1 be the representations of Dr, r> 4, in R = 0pfe) defined by v°(et) = Gt + F1, vx(^) = G\V\ i
Then for every ea = e ^
t
and every j =^= ix,
(7.5) V*l+ Gj (G; + vl(ea)) = v°(eJa). PROOF OF THEOREM 7.1. 1°. By definition,
Furthermore, Ker (pt = p1(ei) (see §5.2). Consequently, cp^p1 (e^.,.^) = 0. 2°. Suppose that (7.5) has been proved. For any Y C V\ according to Corollary 7.1, we have cp^F = Vnj\iY = V{Gj(G'j + \xY)). By definition, V1 = Ker V- Consequently, (7.6)
216
I.M. Gel'fand and V. A. Ponomarev
cpipHO = Vv°(e7-a) =
p(eJa).
THEOREM 7.2. Let pl = (O+)'p &e fl representation of Dr. Then for
every sequence
il9 . . . , it, where
ij G {1, • • •» r }» fy ^ *;+ir
0 « ^ every
t ^ it, we have P R O O F . By definition, pk+1 = O + p \ T h e r e f o r e , applying T h e o r e m 7.1 to p* and p f t + 1 , we find t h a t for any a = (/ l 5 . . . , im ) and any / =^= / p * ( ^ . . . . i m ) = q>? + 1 P* + 1 K...i m ), w h e r e q)j Therefore,
+ 1
:
F
ft+1
- > F/l>.
By definition, q>(/3, . . ., / / ) = cp i j o. . , o < p i r
cp(i1, . . ., ^) pl (et) = cpi, o . . . o cpiz p f fe) = cp{l o . . . o cpi^p^" 1 (eiz<) =
(
Pii° • • • ° ^^i-aP1"2 ^z-iM*) = • • • = P K . . J , t ) 7.3. Lattice lemmas. Here we prove t h r e e l e m m a s , which will be used in the p r o o f of Proposition 7 . 3 .
LEMMA 7.2. Let aua2,b, c be four elements of a modular lattice L such that ai C a2. Then (I) (b + a2c) - fll (a2b + c); fll (II)
«2 + &(fli + c) = a2 + (fll + fe)c.
PROOF. (I) Since ax C #2, we have a! = «1<22 and hence, fli (^ + aic) = d\^i{b + <22C)- It follows from Dedekind's axiom that a2(b + a2c)^a2b + a2c==fl2(«2^+c)- Therefore, ax(6 + a2c) = axa2(a2b + c) == a^ajb + c). (II) Since 0j C a2, we have «2 = a\ + a2- Therefore «2 + h{flx + c) = a2 + fl2 + 6(a! + c) = a2 + (a± + ^(aj + c) = = a2 + «! + (ax + fe)c = a2 + (ax + 6)c. LEMMA 7.3. Let a, a , xu . . . , xn be elements of a modular lattice L such that aar = 0 and a + a' = 1, where 0 and 1 are the minimal and maximal elements of L, respectively. Then
PROOF. Let x be an arbitrary element of L. Then a ' + x = \{a + a-) == (a' + a) (a' +x) = a' + a(a' + *). Using this equation, we can write Since a 2 S a (a' + ^ ) , applying Dedekind's axiom we find
Free modular lattices and their representations
217
a (a' + S xt) = a {a' + I a (a' + xt)) = aar + ^a (a' + xt) = %
i
i
= O + ^a(a'
+ Xi)--=^a (a' + **).
LEMMA 7.4. Lef 38 be a Boolean algebra with generators gl9 . . . , gr r
such that 2 gi = 1 and
ftSft
i=l
= O /or ev^r^ /. We sef g* = S ^»
<=j«=i
Then
<^=i
^
7.4. Proof of Proposition 7.3 ( F 1 + ^ ( ^ + v 1 (6 a ))=v°(c / a ))- To begin with we prove a lattice proposition equivalent to the equation l • • -i)
=
0.
PROPOSITION 7.4. Let p1 = O+o a«(i /e/1 v1 Z?e the representation of Dr in R -
0 p(^)
e/e/in^ by vl{et) =
F1 + Gi4 (G-j + v 1 fe l ...i z )) = F 1 /or every a = (iu . . . , /z) G ^(r, /). PROOF. By definition, v1(ei....i.) = v1(e.) S v1(^) = F1G'il 2 v\e 1 Per(a) per(a) 1 Consequently, G^ + v (g^ •) = G\^ and so V1 + ^.(Gi, + v 1 ^ . . . ^ ) = V1 + G i ^ - F 1 + 0 = F 1 . We prove Proposition 7.3 by induction on /, therefore, it will be more convenient for us to prove the following stronger Proposition 7.5, whose first part is equivalent to Proposition 7.3. PROPOSITION 7.5. (I) Let p be an arbitrary representation of Dr, r > 4, let p1 = ® + p, and let v°, v1 be the representations of Dr in r\
R = 0 p(et) defined by the equations vo(et) = Gt + V\ v1^)
= GJF1.
i l
/ o r every (/ 1} . . . , / / ) G ^4(r, /) and every j ^ ilt we have F1 + G ; ( ^ + v ^ . . . ! , ) ) =
(7.7)
vo(ejiv..h).
(II) F o r every a = (z 1? . . . , / / + 1 ) , / > 1, and/ every ^ $ {i1? ?'2} ^ ^ 2 eP ^ " ^ ^ii ta + S ^P) Perxcc) 3er(a> /or every representation i
ar
elements eii
(7.8)
x( e i l S eP) = T ( e i l ( e , + Per(a)
^ linearly equivalent, that is,
S P£r<
w/zere r(a) = r(i 1T . . . , tl+1) = = {p =
fe
. . ., A:,) £ A(r,
I) \ kx $ {il9
fl«^ gfl 2 ep = e a . pena) REMARK. Apparently, in D '
i2},
k2 $
2^g
I. M. GeVfand and V. A. Ponomarev
(7.9)
eu S e^ = eil(et+ psr(a)
2 *()
per(a)
for every t ^{i^ i2). However, we can prove this only for 1 = 2 and / = 3. A proof for / = 2 is given below. Note that (7.9) is a special case of the more general conjecture stated in §3.5 (that ea = e'^y Our proof of Proposition 7.5 is by induction on /. Step 1. We prove (7.7) for a = (i^), that is, we show that if/ =£ iu then V1 + Gj{Gj + v 1 ^ ) ) = v 0 ^ ) . By definition, v 1 ^ ) = F 1 ^ = V1 2 Gt. It follows from / # ix that G'^ D Gj. Hence, applying Lemma 7.2, we can write Gj(G'j + VlG'{i) = GjiG'fi'i, + V1). Also by Lemma 7.4, GjG^ =
S G«.
Using these equations, we find
t=j, ii
Step 2. We prove (7.9) for / = 1, that is, we show that if t ± iu i2, then
By definition T(ilt i2) = {k\k${i1, ^ufe+ e
<+
2 2
^p)=^ii(^+
2
i2}}. Thus,
efe). Since t ¥= / 1? z2, we have
*h = 2 ' eh hence, ^^(^ +
2
e§) = eti{ 2
^h) = en, iv
as required. Step 3. Now we suppose that Proposition 7.5 (I) is proved for all a - (*!, . . . , i\), X < / - 1, and Proposition 7.5 (II) for all a = (il9 . . . , / x + 1 ) , X < / — 1. We show that then Proposition 7.5 (I) is also true for any a = (iu . . . , it). Step 3a. We perform some manipulations with the subspace V1 + Gf(Gj + v H O ) - B y definition, (7.10)
vi^^v*^
We set (7.11)
2 ^)=vi( eil ) 2 v ^ e ^ ^ F ^ 2 v* Psr() perc) per() X=
2 pr( 1 By definition v ^ ) = V G\ C F for every /. Consequently, v x (x) C F 1 1
l
for every element x E Dr. In particular,
2 vl(ep) ^ ^ Perc)
Thus, we can
Free modular lattices and their representations
219
For write vH^a) = V^G'^ 2 ^M = G'uxbrevity, we also set perx)1 (7.12) F + G,(Gi + v 1 ^)) = F(ea). l Thus, F(ea) = F + G;(GJ + Gyr).
def
By assumption, / =£ iu hence G^ = 2J Gt ^GJ. Using Lemma 7.2 as we did in step 1, we obtain (7.13) F(ea) Step 3b. We carry out the proof for r ~> 4 and, by assumption, / =£ il. Then the subset / — {/, ix) = {1, . . ., r} — {/, ij} is not empty, and we can Write / — {/, ix} == {s3, . . ., sr}. We show that F(ea)=Vi + Gj(r^GSk(G'Sk + X) + G8r + X).
(7.14)
In (7.13), replacing X by the equal subspace X + G's X and applying Lemma 7.2, we find that
, (X + G^ {GSi + J 4 G,k + X)). r
Obviously, GL ZD 2 Gsft. Consequently, by Dedekind's axiom, 3
(^3 +
S
^ + I ) = i l Gsk + G;3(GS3 + X).
Thus,
fe=4 fe=4
(«a) = V1 + Gj(X + G;3(GS3 + X) + S 6^5fe). Note that + G'S3(GS3 +X) = (X + G'S3)(GS3 + X) = X + GS3(G'S3 + X). (7.15)
F(ea)
(
Therefore,
^
Applying transformations similar to those we have used to get from (7.13) to (7.15), we obtain (7.14')
F(ea)=Vi + GJ(^G.h(Gh
Step 3c. We show that (7.16)
F(ea) = 7i + G3 ( S GSk (Gs'fe + X) + GSr
r
We assume that Proposition 7.5 (II) has been proved for all X < / - 1, that is, that for every representation x and every a = (i19 . . ., i x+1 ), X < I — 1, and every t ^ il9 i2, we have
220
I- M. GeVfand and V. A. Ponomarev
*(eii 2
ef) = x(eil(et+
S
PSr(a)
*(0-
tJgr(a)
Replacing T by v1 in this equation, we see that for all a=
(,*!, . . . , ij) EA(r,
I)
v1 (e«) = vi (eh) S ^ v* (ep) = v* (^) (vi (et) + % We have written
2 v l W = X. Now we set per()
Then we can write v1(ea) = v1(e\^)X = vl(e'u)X. Since v1(ei ) = VlGft, X C V1 and X C V\ the preceding equation can be rewritten in the following form: vl (ea) = Gi X = Gt X. From this it follows easily that F(ea)d= V1 + Gj(G] + G'UX) - F 1 + Gj(G- + G^X). In the derivation of (7.14), the only property of X we have used is that X C V1. Consequently, we may replace X by X and as a result we obtain (7.17)
Vi + G, ( 2 3 GSfe (G;fc + X) + G^ + X) = = V* + Gy ( S 3 GSfe (G',k + X) + GSr + X ) .
By definition, X C X. Consequently, we also have F (ea) = F + G, ( S G (G; + X) + G + X) <= <=Vi + G} ( j j GSft (C;fc + X) + GSr + X) = T" + G; ( j ( G*h (GSk + X) + G.r + X). In (7.17) above we have shown that the extreme terms of this inequality are equal. Hence so are the first two terms, that is, F (ea) = V* + G, ( £ G$h {G'Sh + X) + GSr + X). ft=3 1
By definition, X - v ^ ) + X = VlG't + X, where / =£ / t , / 2 . We did not impose any restrictions on sr except that sr # /, ix. Now we require that sr ¥= /2• Then we can choose X = V1G'S + X, that is, if sr ^ {/, /1? i 2 }, we obtain
(7.16')
r-1
ri F(ea) = l + Gj ^
k=3
(Czs -J- A ) -\- Ors h
Step 3d. We claim that r
(7.18)
F(e,
+ GAV +
2J k=3
G
sk (G$k "•
k
+ X + V*G;r).
Free modular lattices and their representations
221
Since sr =£ /, we have G'Sr D G;, and so, applying Lemma 7.2 (I) to the right-hand side of (7.16'), we find that F (ea) = 7* + Gj (Vi + G'Sr (GSr + J ? G&k (G'Sk By definition, all the indices s3, . . . , sr are distinct, therefore, r-l
r-l
GI 5 2 Gs 3 2 £* (Gi + x)r
k=3
h
fe=3
h
Usin
h
S
this
relation and Dedekind's
axiom, we obtain F(ea) = F» + G, (7» + ' S GSft (G;k + X) f G; (G,r + X)). h—3
Since V1 D X, in accordance with Lemma 7.2 (II) we have V1 + G'Sr(GSr + X) = F 1 + GSr(G;r + X). Thus, (7.18')
^M =
j
Step 3e. Now we prove that F(ea) = v°(^ ;a ). By definition, Gr and G't determine a partition of the identity in M, that is, GtG't = 0, G* + G\ = / Therefore, applying Lemma 7.3, we find
2 vi(ep))= 2 ^ ( G H - V 1 ^ ) ) . per)
Using this equation and also the fact that / — {/, h} = {s3, . . ., sr}, we can transform (7.18) to the following form: (7.19)
F (ea) = 7 1 + G, (F« + S ft3
S
G (G; + v1 ft
per()
A
2 r By definition, F(a) = {P = (A:x, . . ., ki^) | ^ ^{ii, J 2 }, . • •}• Consequently, two cases for the pair (tfi) - (t,kl9 . . . ,f/_i) are possible: 1) t = kXi or 2) t =£ kx. Earlier, in Proposition 7.4, we have proved that Vl + Gki (G'ki +v1(eki,.mkl_i))
= V1. I f t ± kl9 t h e n b y t h e i n d u c t i v e
1
hypothesis V + Gt(Gi + v^ep)) ="vofap)f where fl3 = (^ fc1? . . . , A:^^. Therefore, we can rewrite (7.19) in the form
F(ea)=V^ + GJ(Gi + ^(ea))^(y^+GJ)(
2
2 ^ M = = vo(^)
2 *¥=i, ' i , hi
Per(a) Let ja = (j, i1} . . . , i z ). T h e n r(/a) = {7 = (A:o, /c1? . . ., & M ) | A:o ^ {/, ^J, ^ ^ {i lf j 2 }, . . .}
It is not hard to check that
(7.7')
(
2
)
222
£ M. Gel'fand and V. A. PonOmarev
So we have proved that if (7.7') F 1 + Gj(Gj + vl(ea)) = v°foa) holds for all a = (zl5 . . . , i\), X < / - 1, and (7.8) T (ei, 2 *P) = T (^ii (** + S *p)) for all a = (z'i, . . . , i\), X < /, then perca) pgr(a) (7.7') is true for any a = (il9 . . . , /,). Step 4. Now we show that if (7.7') holds for any a = (i"i, . . . , i\), X <_/, and (7.8) for any a = (iu . . . , i\), X < /, then (7.8) holds for every a = (/, iu . . . , it) G A(r, I + 1); that is, we prove that for every t ¥= /, /x (7.8')
p(e-)=p(ej
S
«T) = p(e;)(^+ S
V6r()
ey).
Y6F()
To do this we show that (7.20)
Vi + Gi(G'i + vL(ea))=V>(eJ(et+
S eY)). ero')
From this (7.8) follows almost immediately. Just as we have proved (7.14'), so we obtain
F (ea) = F* + G, ( j ] G.fc (G;fe + X) + GSr_t where X = S v 1 ^). We can number the subset / — {/, ix} = {s3, . . ., 5r_1? sr} so that 5^.! ^ i2, that is, 5r_x ^ {/, i l t i 2 }. Then, by arguments similar to those in step 3c, we obtain
(7.21)
F (ea) = V + G,- ( j f GSft ( ^ + X) + 6 ^ + GSf + X + F ' C j .
Using the same techniques as in step 3, we see that (7.21) can be transformed into: (7.22) F{ea) = Vi+GJ(Gi+v1(ea))
= 'tf>{ej(ehi+
2 *,)), vero'a)
where jot. = (/', h, • • • , //)• By construction, 5r can be chosen subject only to the condition /» ii}. Therefore, comparing (7.7') and (7.22), we obtain (7.23)
where tfc{/, ^ J . By Proposition 7.1, for every x in ZX we have Vv°(o:) = p(x). Applying the map V: R ->• F to both sides of (7.23), we obtain P (ej
2 «v) - P ( ^ (^ + S ^v)) , ^ { / , «i}veroa) ver(ja) By construction, p was an arbitrary representation in V. We have now proved (7.8) and with it Proposition 7.5. 7.5. Proof of the formula
Free modular lattices and their representations
fix ...i/O °fDr i s constructed from a sequence a = (ix, following way:
223
. . . , il9 0) in t h e
Pe( where r(a) = (P = (*lf . . ., kt) £ 4(r, 0 | *x £ {»lf * 2 }, k2 Thus, the polynomial of the second kind / ^ / / 0 is defined in terms of the polynomial of the first kind e^, where ]3 E {ku . . . , kj) E ,4(>, /). Note that by the definition of A(r, /), all kt E / = {l, . . ., r). For example, etio = eit S
^ii2o = ^ii 21 0*1*1. where /cj ^ i l s / 2 and A:2 ^ / 2 .
THEOREM 7.3. Let V1 be the representation space of pl = (®+)lpThe proof of this theorem easily reduces to that of the following formulae: (I) cpJF1 = p(/i0), (II)
q>}(p1(/ii...«Mo)) - P (/iii...i M o),
By definition, / ; 0 = ej 2 ^-
^ > 2.
Consequently, (I) can be rewritten in
the following form:
The truth of this equation follows easily from the definition of the elementary map cp). Now (II) is obtained from the following assertion, which is analogous to Proposition 7.3. PROPOSITION 7.6. Let p1 = O+p, and let v°, v1 be the representations of Dr, r > 4, in R = 0 p(e^) defined by the formulae = Gf + F1, vx(^) = GIV1. Then for every fa = f^
//0 ,
l>\,and
F 1 + Gj(G- + v 1 ^)) = v°(fja), where ja = (/, ^'i, • • • , //, 0). The proof of this proposition is, essentially, a repetition of that of Proposition 7.5, with // changed to 0 in a = (il9 . . . , it). A difference occurs only for / = 1, when we have to prove that V1 + G,(Gj + F1) - v°(/j0). Here is a proof:
V* + Gj (G] + F1) = (F1 + Gj) ( 2
(V' + Gt)) =
every
224
^ M. Gel'fand and V. A. Ponomarev
All further steps in the proof of Proposition 7.5 remain valid when ix = 0, that is, when e^ , M / / i s changed t o / ^ . . . ^ o §8. Complete irreducibility of the representations p*, i 8.1. Equivalence of the representations pt, i and pt,i> ^n §6 we have given a functorial definition of the representations pt, i, namely, Ptti = ((I)~)z"1p?, i for / ^ 2. In §1 we have given a constructive definition of the representations pt, i for / > 2, which we repeat here. By Wtfi (t 6 {0, 1, . . ., r } , I = 2, 3, . . .) we denote the vector space over k with the basis r)a, a €z At(r, I) where At(r, I) = {a = fe, . . ., i M , 0 Mfe e / = {1, • • .. r} and ix =£i2i
i2 ^ i 3 , . . ., i z _2 =£ii-x,
ii-
By Z r / we denote the subspace of H^r z spanned by all vectors r ga;k=
S
T
li,...i f t ...i z _ 1 i, w h e r e a = ( i i , . . . , / A , . . .A^u
t),
and the summation is over all a' = (il9 . . . , ik, . . . il_l, r) in which all is with s =A /: are fixed. We set Vt l = Wt x\Zt h and denote by 0 the canonical map ft: H^r / -» F r ; ; we denote the vectors ftr]a by J o . Thus, Vt z is the space spanned by the vectors £ o , o: G ^4r(r, /), for which S iii...i fe ...i z _ 1 t = 0. Now let / be a fixed index / E /. By F;- r t we denote the subspace of Wt j with the basis {t] a }, where a has the form a = (/', *2> • • • » '/-l» ^* We define a representation pf,; in F r / by setting pt,i(ej) = ft^j, t,/To establish the isomorphism pt.i ^ p?, z, J > 2 , it is convenient to introduce auxiliary representations vt x in Wt x. We set vt i(ej) = Fj t [ + Zt j . We now list some of the simplest properties of the vt t. Obviously, the map ft: Wt x -> F r / defines a morphism of representations ft: vt) t -+• pt, i- We introduce the trivial representation T in Z r z by setting %(et) = Ztl for all / G /. It is easy to see that T is a multiple of the atomic representation po, i. It turns out that (8.1)
vttls*
pu®
T.
T o establish this isomorphism, it is sufficient t o choose in Wt x any subspace U such t h a t UZtj = 0 and U + Zt x = Wt i- It can be s h o w n elementarily t h a t vtti = vt, i \v + vttt | Zf z and t h a t vt,i \v ^ pt,i and Vf.^ | Z t i ^ T. T h u s , v/, z differs from pt,i b y t h e trivial r e p r e s e n t a t i o n T. It follows from ( 8 . 1 ) t h a t <3)-vt>l ^ ^ " p f . z ©
Free modular lattices and their representations
(8.2)
225
Q'Vt.i £< ®~Pt,i-
I n a d d i t i o n , w e s e t , b y d e f i n i t i o n , vtti = p t f i = pt, iP R O P O S I T I O N 8 . 1 . For every I > 1
PROOF. We proceed first with an auxiliary construction. a) We denote by Yj the subspace of Wt l spanned by the vectors gjfok = 2 I}™.. -\"-h i«»
wnere
2 < k < I - 1, the index / G / is fixed
and j3 = (72, . . . , z^, . . . , i^\, t). Obviously Yj C Fj>tjZt /. We set r
y = S y ; . Clearly, F C Z f / . W e denote the factor space Wt tY by G and the canonical map PVr z -> Q by 5. It follows from the relation F C Z r z that i>: ^ j -> Ff / splits into the compositum of the epimorphisms
where 0 is the epimorphism with kernel Ker 6 = dZt
/#
From the proof, r
which we give repeat below, we can deduce that Q ^ © p*. i{et). b) We write <2;- = 5F;- r /# By definition, PFt, t = 2 ^ i , *, i a n d t n i s s u m is direct. Clearly, Q = 6 ^ ,
z
= 6 S ^ i , *. z = S ^ i , *, z = S
'
3
have
3=1
'
also defined Ker 6 = Y = S ^;» where y;- C FJ>tl. Hence it is easy to deduce that the sum (? = 2 (?./
is
direct.
c) Let / be a fixed index. We define maps fij>: Wt of the system of equations f
(8 3)
'
0
if
^-^^i^..,^,
/-A
-> Wr z by means
y = i2,
if ,*=«,.
r
We set jut = 2 Pi-
Thus,
3=1 r
fATli2...iz_1t=
^ 2
T
lsi2...iMt =
^a;l,
where a = (/ l5 / 2 , . . . , //_!, 0- Next we set y = 8JJL and 7;- = 5JU;-. Thus, the diagram
commutes.
226
/. M. Gel'fand and V. A. Ponomarev
LEMMA 8.1. The maps y and 7;- defined above have the following properties'. (I) Im 7 = 5 Z r / ; (II) imyj^Qj;'
(III) Ker y, = vtt ,-ifo) = Fjt tt ,_i + Zt, n . A proof of this lemma is given later; first we complete the proof of Proposition 8.1. From (II) and (III) in Lemma 8.1 it follows that Qj g* Wt, z-i/Ker yy = Wt, i-i/v,. i-i(ej). Thus, Qe*®Wtt ^ / v , , ^(ej). It is 5
also clear that the map y: Wt t_x -» Q is such that y = SJA == 6 2 MJ = S ^^J == S 7J- Consequently, O-v<} z - 1 is a representa3
i
j
tion in V = Coker y = Q/lm y. From (I) in Lemma 8.1 we obtain V = Q/lm y = QISZti t = (?/Ker 6 = Vt% t. Therefore ^"v*. z_i(^j) = e(?j = Q(6Fjt t, i) = QFJt t, i = pt, zfe). We have now proved that ®-v t | Z _i s* p/, ^. PROOF OF LEMMA 8.1. From the definition of the maps \XJ and \i it can be verified immediately that f a) Im (8>4)
I c)
HJ
= ^-,
b)
Z,,i = y
Ker ^ = Fjt 1
)
^
tt z _ l f
^
(I) From (8.4) c) we find dZttl = b(Y + = 6(Ker 6 + \iWt, i-i)
= 6\iWtt z-i = yWt, z-i = Im y.
(II) From (8.4) a) we find Im 7; = yjWt, 1-1 = $\XjWt, i-\ = =
SFJt
tt
t =
Qj.
(III) Obviously, x G Ker yj = Ker 6fx;- <=> ji^(a;) 6 Ker 6 <=> jut^o:) 6 (Ker 6)(Im \i3). Using r
the equations Im /i/
=
^} f /> Ker 6 = 2 ^ j
an
^ the relation Y- C F;-1 /5
it is easy to show that (Ker 5)(Im ju;-) = Yj. Thus, we can write (8.5)
z g K e r Y , - ^ ^(x) 6 Fj.
As we have mentioned, Ker /x;- = F;-1 t_x and y;- = \XjZt l_l. Thus, From this and (8.5) it is easy to see that Ker yj = Fjtl_l +Zr/_1.This proves the Lemma. PROPOSITION 8.2. pt,i^pt,i. The proof is by induction on /. By definition, pt, 1 = 9t, i« Suppose that we have proved that pt.k = Pt, k for all k < / - 1. Then we show that p*f 1 ^ pt,i- ^y (8.2), Q>~pt,i-i = O~v^. M , and by Lemma 8.1,
Free modular lattices and their representations
227
0>~vt, i-i = Pt, i> Consequently, ®~p*. ;_t ^ p/,;. By induction, we assume that p,, w ^ p£ z _j. Hence pi, t ^ O~pf , M ^ ®-p£ I M = pt+t z.
8.2. Complete irreducibility of the representations ptt {. In this subsection we explain the basic steps in the proof of the theorem on complete irreducibility. A full proof of the theorem will be published separately. THEOREM 8.1. Let pt, i and (pT,i) be indecomposable representations of Dr, r > 4, in spaces Vtl over a field k of characteristic 0, with dim Vtl > 3. Then pt, i and (pt,i) are completely irreducible, that is p±z(Z)r) ^ P(Q, m), where P(Q, m) is the lattice of linear submanifolds of the projective space of dimension m = dim Vt l - 1 over the field Q of rational numbers. REMARK. The restriction dim Vt l > 3 occurs because the following indecomposable representations are not completely irreducible: 1) all the atomic representations p?", i and pt, I for t £ {0, 1, . . ., r} for any lattice Dr', and 2) the representations p£ 2 and (pjt 2 ), t £ {1, . . ., r}, for D4. For the latter representations, dim Fr 2 = 2. We describe the basic steps in the proof of Theorem 8.1. We denote by L the sublattice of X (Vt, 1) generated by the one-dimensional subspaces kla, a G At(r, 1). Since kla £pj"t / (Dr), it can be proved elementarily that L^pi,z(£r). The proof of the isomorphism L = P(Q, m) when dim Vt t > 3 is based on the following assertions. Let {£i}i£A be a set of non-zero vectors in V. We call the set {h}i£A indecomposable if for every subset B of A (B =7^ 0 , B ~^ A) the intersection of the subspaces VB—-^2}k\i and V-B = 2
&!./ is non-empty:
FB F5- ^ 0 and VB + F^ = V. A set {IJigA in a finite-dimensional space F over a field /c of characteristic 0 is called rational if we can choose a subset B C A of linearly independent vectors {lt}i£B such that for any / € A lj = 2 ailt > where a*£Q. i6B
PROPOSITION 8.3. Le/1 pt, 1 be an indecomposable representation of r D (r > 4) in a space Vt t over a field k of characteristic 0. Then for / > 1, the set of vectors £a, a E A(r, /), is indecomposable and rational. The proof of the indecomposability of the set {la}a£At(r,i) follows easily from the indecomposability of pf,j. The proof of rationality is also elementary. In establishing the isomorphism L = P(Q, m), the central fact is the following theorem, which is of independent interest. THEOREM 8.2. Let {lt}i£A be an indecomposable and rational set
228
^ M. Gel'fand and V. A. Ponomarev
of vectors in a space V over a field k of characteristic 0. / / dim V > 3, the lattice L generated by the one-dimensional subspaces k%t is completely irreducible, that is, L ss P(Q, m), where m = dim Vt x - 1. A proof of this theorem will be published separately. References
[1]
I. N. Bernstein, I. M. Gel'fand and V. A. Ponomarev, Coxeter functors and Gabriel's theorem, Uspekhi Mat. Nauk 28:2 (1973), 19-33. = Russian Math. Surveys 28:2 (1973), 17-32. [2] G. Birkhoff, Lattice Theory, Amer. Math. Soc, New York 1948. Translation: Teoriya struktur, Izdat. Mir, Moscow 1952. [3] N. Bourbaki, Elements de mathematique, XXVI,Groupes et algebres de Lie, Hermann and Co, Paris 1960. MR 24 # A2641. Translation: Gruppy ialgebry Li, Izdat. Mir., Moscow 1972. [4] P. Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 71-103. [5] I. M. Gel'fand and V. A. Ponomarev, Problems of linear algebra and classification of quadruples of subspaces in a finite-dimensional vector space, Coll. Math. Soc. Ianos Bolyai 5, Hilbert space operators, Tihany (Hungary) 1970, 163-237 (in English). (For a brief account, see Dokl. Akad. Nauk SSSR 197 (1971), 762-765. = Soviet Math. Doklady 12 (1971), 535-539.) [6] L. A. Nazarova and A. V. Roiter, Representations of partially ordered sets, in the collection "Investigations in the theory of representations", Izdat. Nauka, Leningrad 1972,5-31. [7] L. A. Nazarova, Representations of quivers of infinite type, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 752-791. = Math. USSR - Izv. 7 (1973), 749-792. Received by the Editors, 10 June 1974 Translated by M. B. Nathanson
Dedicated to P. S. Aleksandrov, who has done so much for the development of general ideas in mathematics
LATTICES, REPRESENTATIONS, AND ALGEBRAS CONNECTED WITH THEM I1 I. M. Gel'fand and V. A. Ponomarev In this article the authors have attempted to follow the style which one of them learned from P. S. Aleksandrov in other problems (the descriptive theory of functions and topology). Let £ be a modular lattice. By a representation of L in an .4-module ./I/, where A is a ring, we mean a morphism from L into the lattice X (A, M) of submodules of M. In this article we study representations of finitely generated free modular lattices Dr. We are principally interested in representations in the lattice £(K, V) of linear subspaces of a space V over a field K (V= Kn). An element a in a modular lattice L is called perfect if a is sent either to O or to V under any indecomposable representation p : L -*• X(K, V). The basic method of studying the lattice Dr is to construct in it two sublattices B+ and B~, each of which consists of perfect elements. Certain indecomposable representations p+t / (respectively, p~tj) are connected with the sublattices B+ (respectively, B~). Almost all these representations (except finitely many of small dimension) possess the important property of complete irreducibility. A representationp:L -*- £(K, V) is called completely irreducible if the lattice p(L) is isomorphic to the lattice of linear subspaces of a projective space over the field Q of rational numbers of dimension n - 1, where n = dim^ V. In this paper we construct a certain special K-algebrdAr and study the representations p ^ : Dr -> XR (A r) ofDr into the lattice of right ideals of Ar. We conjecture that the lattice of right homogeneous ideals of the Q-algebra/4 r describes (up to the relation of linear equivalence) the essential part of Dr.
Contents
§ 1. Basic definitions and statements of results 229 §2. The category ek{L, K) 235 §3. Perfect elements. Elementary properties of the lattices B* and B~ . . 237 §4. Proof that B\\) and B~(\) are perfect. Atomic representations . . . 240 §5. The functors 3>+ and <£>" 243 References 246 § 1. Basic definitions and statement of results
This article is a further development of the authors' paper [ 7 ] , but can be read independently. 1
The second part of this article will be published in these Uspekhi 32:1 (1977), 85-106. 229
230
/. M. Gel'fand and V. A. Ponomarev
1.1. Lattices. A lattice I is a set with two operations: intersection and sum. If a, b E L, then we denote their intersection by ab and their sum1 by a + b. Both these operations are commutative and associative, and, moreover, satisfy the axioms of absorption: a(a + b) - a and a + ab = a. An order relation is defined naturally in a lattice L:aCb*=*a + b= b. It is easy to deduce that aa - a and a + a = a for every a E L. A lattice L is called distributive if for any a, b, c G L (1.1) a(b + c) = ab + ac, (1.2) a + be = (a + b) (a + c). It can be shown that a lattice is distributive if it satisfies at least one of the equations (1.1) or (1.2). A lattice L is called modular if for all a, b, c £ L a C b =• b(a + c) = a + be
EXAMPLE 1. Let A be a ring and let M be a left (right) A -module. Then the set of all submodules of M is a lattice with respect to the operations of intersection (O) and sum (+). We denote this lattice by X(A, M). EXAMPLE 2. In this article we shall most often consider the lattice X(K, V) of linear subspaces of a finite-dimensional vector space V over a field K. When dim V = n (that is, V = Kn), we also denote this lattice by X(V) or X(Kn). If Ux and U2 are subspaces of F, then we denote their sum ana intersection by U1 + U2 and Ux U2. EXAMPLE 3. Let P(F) = Yn(K) be the projective space corresponding to V = Kn+l. Then X{V) is well known to be isomorphic to the lattice of linear submanifolds of the projective space P(F). We call this lattice a projective geometry (PG(F)). Thus, if V = K3, then the elements of the corresponding geometry are the points and lines of the projective plane V2{K). (If a and b are points in Y2(K), then a + b is the line passing through a and b. If A and B are lines, then AB is their point of intersection.) The basic objects of our investigation are the free modular lattices Dr with a finite number of generators (el9 . . . , er). The lattices D1 and D2 are obviously finite. It is not difficult to show that the lattice D3 is also finite (see Birkhoff [3]). The lattices Dr, r > 4, have a very complicated structure. We are by now only close to an understanding of the structure of Z>4 (the factor lattice of Z)4 with respect to linear equivalence, which we define in 1.2). 1.2. Representations. Let I be a modular lattice and X (A, M) the lattice of submodules of a left A -module M. A representation p of L in M is a morphism2 p: L -+ X(A, M). Here, for any x, y G L we have p(x + y) = p(x) + p(y) and p(x>>) = p(x) n p(y), where p(x) and p(y) are submodules of M. The intersection of the elements a and b is often denoted by a n 6 or a A &> and the sum by a u ft or
a\J b. A morphism p:Lx -*L2 of lattices L x and Z,2 is a mapping such that p(xy) = p(*)p(y) and P(* +y) = p(x) + pO) for all x, y e L,.
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Throughout this article we are concerned with only two types of representation. 1) A representation of a lattice L in a finite-dimensional vector space V over a field K (p: L -» X{K, V)). Such a representation associates with elements x, y S L subspaces p(x), piy) C F. 2) A representation of L in the lattice X(A) of left ideals of a ring ,4. We introduce in L an equivalence relation by setting x ~ y if p(jc) = p(y) for every representation p: Z, -+£(K, V) in any space F over any field K. It can be shown by examples (even in D4) that there exist linearly equivalent, but unequal elements. We consider the set whose elements are the classes of linearly equivalent elements o f / / . T h e operations (•) and (+) carry over naturally to this set. We denote by Dr the lattice obtained in this way. The aim of this paper is the study of the lattices Dr. An important technique for the study of Dr is the construction of the sublattice B of perfect elements, to whose definition we now turn. 1.3. Perfect elements in Dr'. An element i; in a modular lattice L is called perfect if for every indecomposable1 representation p: L -+%{K, V) either p(v) = V or p(v) = 0. In the free modular lattice Dr with generators e 1? . . . , er we construct two sublattices B+ and B~, whose elements, as we shall prove later, are all perfect. For every integer / > 1 we construct a sublattice B+(l) consisting of 2r elements. We shall call this sublattice B+(l) the /-th upper cubicle. It is quite simple to construct B+(\). Namely, we set ht(\) - 2 e,-. i*t
Then the upper cubicle B+(l) is the sublattice of Dr generated by the elements hx{\), h2{\),..., hr(\). It is not difficult to prove (see §3) that B+(\) is a Boolean algebra with 2r elements. Thus, B+(\) is isomorphic to the lattice of vertices of an r-dimensional cube with the natural ordering. We proceed to the definition of the cubicles B+(l). The elements of + B (l) are constructed with the help of polynomials e^ iy which are of independent interest. We denote by Air, I) the set whose elements are the sequences a = (ilf i2, . . . , //) of integers 1 < ij < r such that ij ¥= iJ+l for all 1 < / < / - 1. We set, by definition, Air, 1) = / = {!, • • ., r). The elements ea = e^ t are defined by induction on / as follows. If / = 1 and a = (z'i), then ej = et ; if / > 1 and a = (il9 . . . , /z), then ea = et 1 ( 2 eB), where F(o:) C A(r, / - I ) consists of the sequences /*er(r . . . , A:/_1) constructed from a fixed a in the following way: A representation p in a space V is called decomposable if there exist non-zero subspaces Ult U2 in V that are complementary to each other (Ul U2 = 0, Ux ~v U2 = V) and such that p(x) - p(x) Ux + p(x) U2 for every x e L.
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T (a) = {p == (kt, . . . , ki-i) I kt ^ ii9 i2; k2 =£ i2, ia, . . . , k^ *£ U_u it}. For example, if a = (ilt /2)> then T(u) = {?> = (ki)\ki=£iu ^iii2 = eh(
i 2 }, and so
e
2
j)'
Now we define the elements ht(l). We set a£At(r, I)
where At{r, I) is the subset of A(r, I) consisting of all sequences a = (/ l5 . . . , tz_l5 0 in which the last index is fixed and equal to t. Further, we set ht{l)
=
%ej(l). tet
Then we define B+(l) to be the lattice generated by the elements
MO, -..,
W
It is not difficult to prove (see §3) that B+(l) is a Boolean algebra and that elements from different cubicles B+(l) and B+(m) can be ordered in the following way: for every vt E B+(l) and vm £ B+(m) it follows from I < m that Vi D vm. Thus, the set B+ =
u B+ (I)
is itself a lattice. oo A second set B~ = U B~(l) consists of the elements dual to those of /=i
B+. (We say that a lattice polynomial g(el9 . . . , er) is dual to a lattice polynomial /(e 1? . . . , er) if it is obtained from / by interchanging the operations (+) and (H). Thus, for example, ei(e2 + e3 + e4) is dual to ex + (e2eze4). One of the main theorems of this article is the following. THEOREM 1. The elements of the lattices B+ = U B+(l) and B~ = U 5~(/) are perfect. i=i
1.4. Characteristic functions of an indecomposable representation. Let L be an arbitrary modular lattice. Then the set B of perfect elements in L is a sublattice of L (see §3). If p is an arbitrary indecomposable representation of I (p: L -+%(K, V)), then every perfect element has, by definition, the following property: either p(v) = 0 or p(v) - V (V is the representation space of p). Thus, to every indecomposable representation p there corresponds a function xp o n the set B of perfect elements, which is defined as follows: 0 if p(i>) = 0, 1 if p{v) = V.
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We call xp the characteristic function of p. In the lattice Dr we have defined two sublattices of perfect elements B+ and B~. We claim that p(iT) C p(u+) for any representation p{p: Dr -> X (K, V)) (r > 4) and any iT G 5 " and u+ G £ \ We denote by B the sublattice of perfect elements in Dr generated by B+ and B~. From P(V~) C p(i/) it follows that every characteristic function x p defined on B belongs to one of the following three types: 1) Xp(u+) = 0 for some v+ G B\ hence x p (*O = ° for all v~ G B'\ 2) XpOO = 1 for some v~ G B~, hence, x p (V) = 1 for all v+ G B+; 3) x p 0 O = 0 for all v' G 5 " and Xp(^+) = 1 for all v+ G £ + . We denote the last function by x i ; In §7 we prove the following theorem. THEOREM 2. Let p be an indecomposable representation of the lattice Dr (p: Dr -> «£(#, V)) (r > 4). / / the characteristic function \p of p is of the first or second type, then p is defined by its characteristic function uniquely up to isomorphism. We shall find all indecomposable representations corresponding to the various characteristic functions of the first or second type. In the following subsection these representations will be constructed explicitly. As for the indecomposable representations p whose characteristic functions are of type 3 (xp = Xo)> w e know at the moment only that there are infinitely many of them. In the case of D* the classification of all such representations is known [6]. For the lattices Dr (r > 5), the classification of the indecomposable representations with xp = Xo contains as special cases such problems as the determination of a canonical form for several linear operators Al9 . . . , An (At: V ->- V). 1.5. The algebra Ar and the representation pA . Let K be any field. We define the i^-algebra Ar as the associative i^-algebra with unit element e generated by £0> £i> • • • > £/- with the relations (1) (2) (3)
E?=0 Ui=h
(i = l f . . . , r ) , Io2=&), (* = 1, . . . , r ) ,
The standard monomials in Ar are the products ^ . . . ^ %t such that 1 < ij < r, ij * i /+1 for all 1 < / < / - 1, it_x ¥= t, and 0 < t < r. Thus, in a standard monomial £0 c a n occur only in the last place. It is easy to see that any non-zero monomial can be brought to standard form. The standard monomial ^ . . . £/, , £r is also denoted by £a = £Zi A t, where a = ( * ! , . . .Ji_xt). The degree of the monomial £a is the number d(i~a) defined in the following way: d(e) = d(%0) = 0, d(£t) = 1 for every i ¥= 0, d(Za$p) = d(ia) + d%) if ? a ^ ^ 0. The degree of the element 0 is left undefined.
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We denote by V{ (Vl C Ar) the space of homogeneous polynomials of degree /. It is not difficult to show that this introduces a grading in Ar: Ar = Vo © Vx © . . . © Vx © . . . (ViVj C Vi+j), where 1 Vo = KE © £ £ 0 .
In §8 we shall show that the algebras Ar (r > 4) are infinite-dimensional, and that dim Vl > 0 for all / > 0. The algebras A\ A2, A3 are finitedimensional, and their dimensions over K are 3, 5, and 11, respectively. We denote by XR{Ar) the lattice of right ideals of Ar (with respect to the operations of intersection O and sum +). We define the representation pA\ Dr -+ XR(Ar) by setting pA(et) = %tAr, where £tAr is the right ideal generated by £z-, and the et(i = 1, . . . , r) are the generators of Dr. In §8 we shall prove the following interesting theorem, which establishes a connection between the lattice polynomials ea (which were defined in 1.3) and the monomials £a in Ar. THEOREM 3. For every a = (z l5 . . . , il_l, t) E At(r, I) (I > 1) we have pA(ea) = %aAr', where %aAr is the right ideal generated by the monomial =
£/, • • • £/z_ x Sf This result is due to Gel'fand, Lidskii, and Ponomarev. 1.6. The representations ptl. Let A%t be the left ideal generated by the element £ r We introduce the following notation: (* = 0, 1, . . . , r ; Z = 1,2, . . . ) .
£a
According to this definition, Vi 0 = 0 if / # 0 and F o 0 = ^ ? 0 - F° r / = 1, F,- j = K%t if / ¥= 0 and K 0 1 has the dimension r- 1 and is the sum of the one-dimensional subspaces K^o. For I > 2, every subspace Vtl is generated by one-dimensional subspaces K%a%t, where £a£r is a monomial of degree /. We define a representation pt l of Z)r in Vt i as follows. We set where ^^4 is the right ideal generated by £,- (1 < / < r). We define a representation p e of Z)r in i^e (where 8 is the unit element of Ar) by setting p z{ei) = 0 for every et G D r . It is elementary to prove (§8) that pA is isomorphic to the direct sum p A ^ p e 0 p o , o © ( 0 ( 0 p#. i))It turns out that the representations ptj so constructed possess the following remarkable properties. THEOREM 4. (i) The representation pt j(t = 0, 1, . . . , r\ I = 1, 2, . . . ) of two representations ptl and pt>j> such (ii) p 0 0 and pt j(t = 0, 1, . . . , r; /
p00 Dr(r that = 1,
and the representations > 4) are indecomposable. Any (t, 1) =£ (t\ I') are not isomorphic. 2, . . . ) #re the only representa-
tions whose characteristic functions are of the first type (that is, By Ke (respectively K£o) we denote the subspace generated by the element e (respectively, £0).
Lattices, representations, and algebras connected with them
235
X p (V) = 0 for some v+ G B+).
Let V be a linear space over a field K of characteristic 0. A representation p of a modular lattice L in V is called completely irreducible if p(L) C # ( F , K) is isomorphic to the lattice ^(Q n ) where n = dim^ V (n > 3) (that is, p(L) is the projective geometry P ^ - i ( Q ) in > 3) over Q. The following result holds. THEOREM 5. All representations ptl\ Dr ^X(VtiI; K) (r > 4) over a field K of characteristic 0, except finitely many, are completely irreducible. The only representations that are not completely irreducible are the following: a) p 0 0 and pt x (i =£ 0) for any r > 4; b)p / ; ; (i^O)forr = 4. 1.8. The lattice F*. We introduce an equivalence relation R in Dr by setting x = y (mod R) if ptj(x) = ptj(y) for any representation ptl\ Dr -> X{VttU K). We denote the factor lattice Dr/R by F*. We now state an important conjecture about the structure of the lattices F*(r > 5). Let ArQ be the Q-algebra A\ where Q is the field of rational numbers. A right ideal in ^4Q is called homogeneous if it is equal to a finite sum of ideals ftjArQ, ftl G Vtl. The lattice of right homogeneous ideals of AQ is denoted by o//f {ArQ). CONJECTURE. The factor-lattice F* of Dr (r > 5) is isomorphic to the lattice QM(AQ) of right homogeneous ideals of the Q-algebra AQ. We shall soon publish some results obtained jointly with B. V. Lidskii, which bring us close to a proof of this conjecture. §2. The category M (X, K) 2.1. The category ^ (X, K). Let px and p 2 be two representations of a modular lattice L in spaces Vx and V2, respectively. By a morphism ^ : Pi ""* P2 w e mean a linear transformation w: Vx -+ V2 such that u Pi(x) 5 P2(-x:) f° r aH x £ L, where wpi(x) is the image of the subspace PJ(X) under the transformation u. We often denote a morphism w simply by u (the corresponding linear transformation). We denote by Horn (p l 5 p 2 ) the set of all morphisms from px to p 2 . It is not difficult to verify that this determines a category %(L, K) that of finite-dimensional representations over K. In ffl(L, K) the direct sum Pi e P2 of any pair of objects px and p 2 is defined in the natural way. The category M(Dr, K) is the object of our study. It is easy to show that M is additive but not Abelian. Let p E M(L, K) be a representation of a lattice L in a space K. We denote by K* the space dual to V. We define the representation p* G m(L, K) in F* by p*(jc) = (p(x))1 for all x G L (where
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(p(jc))1 C F* is the subspace of functionals that vanish on p(x)). We call p* the representation dual to p. 2.2. Decomposable representations and admissible subspaces. Let p be a representation of a modular lattice L in a linear space V. A subspace U of V is called admissible with respect to a representation p if for all x, y €i L (I) tf(p(s) + p(z/)) = Up(x) + ffp(y). It is not difficult to show that (I) is satisfied if and only if (10 U + p(x)p(y) = (U + p(s))(ff + p(y)). PROPOSITION 2.1. Z,e^ p be a representation of a lattice L in a space V. Let U C V be a subspace of V, U" = V/U be the quotient space, and 6: V -> U" be the canonical map. Then the following conditions are equivalent: 1. U is admissible with respect to p. 2. The correspondence x •-> Up(x) defines a representation in U. 3. The correspondence x *+ 6p(x) defines a representation in U". The proof is elementary (see [3] ).• The representation in an admissible subspace U defined by the correspondence x *-» Up(x) is called the restriction of p to U and is denoted by Pit/PROPOSITION 2.2. A representation p G M(L, K) in a space V is decomposable if and only if there exist non-zero subspaces Uli . . . , Un such that V = Ux © . . . © Un and for every x G L
REMARK l . I f a representation p G M(L, K) is decomposable, with n
p = © Pj and if Ut the subspace corresponding to p,-, then the Ut are admissible. The converse however, is false, that is, if V = © Ut and if each i= l
Ut is admissible, then it does not follow that p splits into the direct sum of their restrictions. QUESTION. Is it true that if U is an admissible subspace, then there is a subspace U' complementary to U (that is, UU' = 0 and U + U' = V) such that £/ and Uf define a splitting of p into a direct sum? PROPOSITION 2.3. ^4 representation p G # (Z)r, ^) ^pto into a direct n
sum p = ® Pi of representations pt if and only if there are non-zero subi=1
n
n
spaces Ux, . . . , Un such that V = © Uj and p{et) = 2 p(et)Uj for every j= l
/= l
/ G {1, . . . , r) , where the ef are the generators of Dr.*
Lattices, representations, and algebras connected with them
237
§3. Perfect elements. Elementary properties of the latticesB+ and i T 3.1. In this section we prove that the set B of perfect elements in a modular lattice I is a sublattice. PROPOSITION 3.1. Let a and b be perfect elements of a lattice L. Then so are a + b and ab. PROOF. If an element a is perfect, then there exists a subspace V complementary to p(a) (that is, p(a)V = 0 and p(a) + V' = V) such that p splits into the direct sum of representations p = pq + p', where Pa = P\p(a) anc * P' = P\v- I n particular, for x = b (3.1)
p(b) = 9(b)p(a) + p(b) V.
Note that p(b)p(a) = p{ba) and, by Proposition 2.1, p(b)V Thus, we can rewrite (3.1) in the following form: (3.2)
= p\b).
p(b) = p(ab) + p'(6).
Now Z? is perfect, consequently, in V there is a subspace V" such that F ' = p'(b) + K" and p = p'b + p " , where p'b = p'|p'(Z>> and p" = p'\v». Thus, p = pq ® p' = pa® (p'b © p") = (pfl e p^) e p". We claim that pa © pj, is the restriction of p to the subspace p(a + &). Indeed, by definition, pa ® p'b - Pf\P(a)+p\by Moreover, using the fact that p(ab) C p{a) and p(Z?) = p(ab) + p'(^) 3 we can write P (a) + 9' (b) = (p (a) + p (ab)) + p7 (&) = p (a) + (p (a&) + p' (6)) Thus, p = pa+b © p". In other words, a + b is perfect. The fact that ab is perfect can be proved similarly." COROLLARY 3.1. The set S of perfect elements of a modular lattice L is a sublattice of L. • PROPOSITION 3.2. Let L be a modular lattice, and p G M(L, K) an arbitrary representation. Then the image p(a) of a perfect element is a neutral element of the lattice p(L) C X{V, K),that is, for any xt, x2 £ L PROOF. The element a is perfect, hence there exists a subspace V such that F ^ p(#) © F ' and p = pa ® p' = P\p(a) ® P\v'- Consequently, for any xt G I we can write p(x() = p{xt)p{a) + pix^V. Using this identity, we obtain p(a)(p(xl) +p(x2)):= p(a)(p(xx)p(a) + + p(xt) V + p(x2)p(a) + p(x2) V) = p{xx)p{a) + p(x2)p(a) + piaXpix,) V + + P(^2)F'). By construction p(#)F' = 0, and a fortiori p(a)(p(xl)Vf + p{x2) V) = 0. Thus, p(tf)(p(*i) + p ( ^ ) ) = p ( ^ i ) + P(a)p(x2).m COROLLARY 3.2. Let S be the sublattice of perfect elements in a lattice L. Then p(S) for any representation p E fi (L, K) is a distributive sublattice of neutral elements of p(L)M CONJECTURE 3.1. Let L be an arbitrary modular lattice. Then an element
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a is perfect if and only if it is neutral. 3.2. Elementary properties of the sublattices B+(l). Now we study some properties of the sublattice B+ in Dr, which, as we shall prove later, consists of perfect elements. The lattice B+ is the union of the sublattices £ + (l), B\2), . . . , B\l), . . ..which are called cubicles. The cubicle B\l) is constructed with the help of special elements ea in the following way. We set
M0 =
2
ea, MO = 2 MO,
<x£At(r, I)
ij=t
where a. = (fl9 . . . , it_lt t) E At(r, I) is a sequence such that /;., t E {1, . . ., 7*}, /;. =£ z /+1 , /,_! T£ r (the definition of e a is on page 69). Now B+(l) is, by definition, the sublattice generated by the elements
MO, ..., ^(0. THEOREM 3.1. B+(l) is a Boolean algebra. This theorem is made more precise in the following proposition. PROPOSITION 3.3. (I) Every element vaJ of B+(l) can be written in the following form:
(i)
vatl= 2 M 0 + 2 MO MO,
where a is an arbitrary subset of I = { 1 , • . ., r). Ifa^I, this can also be written vCtt=
(ii)
fl
h}(l).
(II) Let $(I) be the Boolean algebra of all subsets of I. Then the correspondence a <->• va i defines a morphism vf .^(/)-> B+(l) (that is, %ub),l = va,l + vb,l u %nb),l = va,l n vb,l for any a, b C I). REMARK. In §7 (Corollary 7.2) we shall prove that the mapping vf. J?(/)-> B+(l) is an isomorphism. The proof of Proposition 3.3 is based on the following lattice-theoretical lemma. LEMMA 3.1. Let L be an arbitrary modular lattice, and {e^ . . ., er) a finite set of elements of L. Then the sublattice B generated by the elements hj = 2 ef (j' = 1, . . . , r) is a Boolean algebra. i*i The proof of this lemma reduces to a proof of the formula
2
e
i + 2 ejhj == fl hj,
where <£> C b C /. This formula is easily proved (see [7]) by induction on the number of elements in b. • 3.3. Structure of the lattice B+. We denote by B+ the lattice generated by the sublattices B\l) (I = 1, 2, . . . ). THEOREM 3.2. (I) B+ is the union of the sets B\l).
Lattices, representations, and algebras connected with them
239
(II) The sublattices B+(l) in B+ can be ordered in the following way: let v (l) and v+(m) be any elements of B+(l) and B+(m). If I < m, then v+(l) 2 v\m). REMARK. We shall prove in §7 that, in fact, for I < m strict inequality +
v\l) D v\m) holds. The proof of this Theorem is based on two lemmas. We recall that A(r, I) = {a = (il7 . . ., i.) \ ij 6 / = {1, • • ., r}, where ij =fc / / + 1 for all / < /. LEMMA 3.2. Let p = (i!, . . . , /,) e A(r, I). We set pf = (/j, . . . , i/f /) for j =£ it. Then ep D epf. LEMMA 3.3. Let ht(l) be a generator of B+(l). Then ea C ht{l) for all OL e Air, I + 1). The proof of these Lemmas is elementary (see [7]). Now we prove Theorem 3.2. It follows from Proposition 3.3 that the r +
minimal element in B (l) is vd l - n fy(/), and the maximal element in B\l + 1) is U/f+i = S eAl + 1) = i=i
S
ea. It follows from Lemma 3.3
af=A(r,l+l)
that O ht(l) D ea for every a G A(r, / + 1). Therefore, r
v
el
=
^ ^/(0 — /=1
y~s
^
e
a
= u
/ /+1 • Now if ufl / and uft / + 1 are arbitrary
a(=A(r,l+l)
elements of B+(l) and B+(l + 1), then ^ / 2 ^ / 5 ^ / / + i 5 ^ /+1 J and the theorem is proved. 3.4. The lattice ^~. By definition, the elements of the cubicles B~(l) are dual to those of the cubicles B+(l). For example e^ t = ei% 2 e^ hence, by definition, we set ej .• = e,- + ( n e7). Similarly,
wnrw r, Z)
;><
Each cubicle B~(l) is a Boolean algebra, and the elements v~(l) and v~(m) of distinct cubicles .&"(/) and B~(m) (/ < m) are connected by the relation v~(l) C u"(m). Thus, the lattice B~ generated by the B'(l) is the union 0 £-(/) of the sets B~(l). In particular, just as the maximal and minimal elements vfl and vQl of the cubicles B+(l) form a chain
r
so the maximal elements Vj j = 2 /zz- (/) and minimal elements
240 v
el
I- M. Gel'fand and V. A. Ponomarev =
r
n e
i(O °f
tne
cubicles B (/) are ordered dually:
r _ _ _ _ _ D ti = ve, i s i>i, i s i;e, 2 ^ z>i, 2 £ . . . S i>e, z s Vj
S .• •
Note that the element v^j is dual to i>7/ and u£j is dual to vel. In §7 we shall prove the following proposition. PROPOSITION 3.4. Let v+ and v~ be arbitrary elements of B* and B~ Then p(u~) C p(u+) for every representation p: Dr -> X(K, V)
We also believe that the following is true. CONJECTURE. For every v+ G B+ and v~ G B~
§4. Proof thati? + (l) and ^"(1) are perfect. Atomic representations
By definition, i? +(l) is the sublattice of Dr generated by the elements ^ i ( l ) , h2(l), . . . , hr(\), where hAl) = 2 et. The maximal element in the r
cubicle ^ + (1) is vf j = 2 ^-. We note that u7 j is the maximal element in /= 1
the entire lattice Dr. The cubicle B~{\) consists of the elements dual to the elements of B+(l). (It is generated by the elements hj{\) = O et. In this section we prove that every element of i? +(l) and B~{\) is perfect. 4.1. Atomic representations and the perfectness of i? +(l). We define representations p] t for t G {0, 1, . . . , r}, which we call (+) atomic. By definition, p\ x is the representation in the one-dimensional space V*tl = K for which 1) if t = 0, then p+Oil(e{) = 0 for all 1 = 1, . . . , r; 2) if t ¥* 0, then p ^ ( ^ ) = 0 for i =£ t and p]yl{et) = V*tl. Note that the atomic representations are none other than the representations p 0 0 and pix defined in §1 on page 72. Namely, Po5i - Po,o a n d P u - P u i f * * °THEOREM 4.1. Each element va
x
G J5 +(l) 15 perfect.
The proof of this Theorem is based on the following lemma. LEMMA 4.1. Let p be any representation of Dr in a space V. Then p is isomorphic to the direct sum p = p 0 1 4- py- x + Tj, where Tj = P\P(h.(i)y where p ^ and p~x are multiples of the atomic representations p+Ol and p^j (that is, pQ1 = p+01 + . . . + p*Ofl, where m0l > 0, and, similarly,
PROOF OF LEMMA 4.1. We set p(e z ) = Et and p(hj(l)) = Hj. Thus, the sub spaces Et and H* are such that Hj = 2 Et. We also set
Lattices, representations, and algebras connected with them
241
r
Ho = X Ef. Clearly, Ho has the following property: for every / =£ 0 (4.1)
EJ +
HJ^HO^HJ^H,.
We claim that the element ht = £ e{ of Dr is perfect. We choose sub7 /*/ spaces C/o and Uj in F to satisfy the relations U0H0 = 0, Uo + Ho = F, (4.2) (4.3) ffjfT, = 0, Uj + £ , # , = Ej. We illustrate the subspaces in the following figure:
We shall show that Uf + Hj = Ho. Clearly, EJHJ + Hj = /7 ; . Using this and (4.1), (4.3), we see that Uj+Hf = Uf +EfHf +Hj = Ej +# y =H0. Thus, Uj + i/y = /f0, and this sum is direct. It easily follows from this equation and the definition of Uo that V = UQ e Uj e Z^.. We now show that the representation splits into the direct sum P = Po1 e P/ 1 e Tj, where pr 1 = p | ^ and r ; = p ^ . . For this it suffices to show that for every subspace Et = p{et) (i = 1,. . . , r) (4.4)
p (et) = p («,) f/0 + P (e«) ^ + p (et) Hj. r
Let us prove this. By construction, U0H0 = Uo( 2Ef) = 0. Consequently, /= l
EjU0 = 0 for every i. By construction, Ej = Uj + EJHJ, hence Ej = Ej(Uj + JS 1^) = EjUj + ^yZTy. This proves (4.4) when i = /. If i ¥= /, it is clear that Et C X Ek = H-, that is, £,#,• = £,. Next, it follows from (Uo + t/y)/^- = 0 that (Uo + f/,)^ = 0, and a fortiori UQE; = 6^,. = 0. Thus, Et = E ^ o + EtUf + Eflj. This proves (4.4). It means (see Proposition 2.3) that p = p ^ © p|^. © p|^.. Consequently, y J ° hj(\) is perfect. + Thus, the generators /z^l), . . . , hr{\) of J5 (1) are perfect. By Proposition 3.1, this implies that all elements of B+(l) are perfect, and Theorem 4.1 is proved. To complete the proof of the lemma, it remains for us to establish that p\v and p\(j. are multiples of the atomic representations p*0l and pjx. r
By construction of Uo, U0H0 = Uo 2 Et - 0, and so £/0£z- = 0 for every /. Thus, p\v (et) = p(ej)U0 = EtU0 = 0 for every /. It follows that Po,i = P\u *s isomorphic to the direct sum of the atomic representations
242
Po,i>
/• M. Gel'fand and V. A. Ponomarev that
is
> Po,i - Po,i © . . . © Po,i> where m 0 = dim Uo.
T h e sub space Uj h a s t h e following p r o p e r t i e s : a) UfEf = Uf, a n d b ) UfHf = Uf 2 Et = 0, t h a t is, UjEi = 0 for every i * j . Thus,
h
if
i-h
Consequently, the representation p ; - 1 in £/7- is isomorphic to the direct sum of the atomic representations pjl. Lemma 4.1 and Theorem 4.1 are now proved. COROLLARY 4 . 1 . The morphism of Boolean algebras vx\ 38(1)-* B+(\) defined by the formula a -* va l (where a C / = {1, . . . , r} and va j = 2 et + 2 ejhj(l) G ^ + (1)) w flw isomorphism. i
j l
PROOF. We construct a representation p in F = Kr as follows. Let £i, . . . , £r be a basis for F. We set p(et) = K%t. It is easy to see that p is isomorphic to the direct sum © p] x of atomic representations. It is also /= l
easy to check that p(hA\))
= X K%f, that is, dim p(hA\)) = r~ 1. Hence it /*/ follows easily that p(va x) = X K^. Consequently, for any two distinct
subsets a, b E / the corresponding subspaces p(va j ) and p(vb t) are distinct. This means that p(£ + (l)) = 38 (I) and so B\\) 9* 38 (I).u As we know, any element va 2 G ^"""(l) can be written in the following form: va t = 2 et + 2 ^ ( 1 ) , or, if a ¥= I, va 1 = n / z ( l ) . It follows /so
/G/-fl
'
/e/-fl
from Theorem 4.1 that every element va j is perfect. So we come to a proposition that refines Theorem 4 . 1 . PROPOSITION 4.1. Let p e<%(Dr, K) be any representation. Then p = pa(\) e TQV where ral = P| p(Ufl 1}- Here (i) p _ ( l ) = 0 p; t , and each p,-, is a multiple of the atomic i6(Ia)(J{0}
representation
Jf
J
'
p;- j = pj x e . . . e pf t(mj > 0);
(ii) if Ta j ,yp/z75 z>zto <2 direct sum TQIX— © Ty o / indecomposable representations r;-, r/ze/t «o«e o / ^/ze r;- «re isomorphic to any of the representations p]x for i G (/ - a) U {0}. • We shall use P r o p o s i t i o n 4.1 m o s t often w h e n val = ve>1 is t h e m i n i m a l e l e m e n t of B+(\). T h e space p(vei) is, as it were, t h e s u m of all s u b r e p r e s e n t a t i o n s t h a t are n o t (+)-atomic. N a m e l y , p = (l)
Lattices, representations, and algebras connected with them.
where T61
= p|p(u
)5
243
and if r 0 1 = © ry-, where the r7- are indecomposable,
then none of the r;- are isomorphic to any of the pj x for / E {0, 1, . . ., r}. 4.2. (—)-atomic representations and the cubicle i?"(1). We define representations p~t x as follows: pj t = (p+t j ) * . We call them (—)-atomic. It follows from the definition that each p't x is a representation in the one-dimensional space V~x s £ , and Po 5 l O/) = 0 f o r a n Y *; Pj,\^ed = *Xi i f / ^ ' a n d The lower cubicle B (1) is defined to be the sublattice generated by the r
elements h~AX) 1
_
n ef. Here, v* , = O e.- is the minimal element in B (1) .
i*j
i=\
If any element v~ x G ^"(1) with a ^ 0 (6), can be written in the form v~ x = 2 hj(\). Arguments dual to those used in the preceding subsection ye
show that each element v~ x G B~(\) is perfect. Also, the representation Pa I ~ P|p(u" ) i s a direct summand of p, and p~ j = © mtP~t,\' w n e r e 0. 5. The functors 4>+ and In this section we define functors
yMfo, ....Wi&epfo), 2E* = O}, P1 (*,)={&, . . . . u o . U - y e n where the et are generators of Dr'. In other words: we denote by r
V: © p(et) -* K the linear map defined by the formula V(£l9 . . . , £ , . ) =
2 £,-. We set Ker V = K 1 . Then the following sequence i= 1
of vector spaces is exact: "K
r
V
0 - ^ V1 - > 0 p(e^) —> F, where X: K1 -> © p(e.-) is an embedding. i
i
244
/• M. Gel"fand and V. A. Ponomarev r
We denote by 717 the projection onto the space R = © p(et) (IT: R -+ R) 1=1
with kernel © p(et) and range p(e ; ). We set <^y = V 7r7-X. Then 1*7
pV/) = Ker^y. Thus, from a representation p E 92 = 92{Dr, K) we have constructed another representation p' G M. It is easy to check that this correspondence is functorial. We denote by <£+ the functor p -+ p1. A representation p" 1 is constructed from p in a dual manner. We set r
Q - e (Vlp(et)). We denote by JJL the linear map \x\ V -+ Q defined by 1=1
M£ = (|3i5, • . • , j3rg), where fy: 7 -• V/p(ef) is the canonical map. We set V~l - Coker JJL = Q/Im /x. Thus, the following sequence is exact: V ^
© (F/p^^F-'-^O. 1=1
We set i//y = 07Ty/x, where 7r;-: Q -> 2 is the projection into the space Q = e (F/p(e,)) with kernel © (V/p(et)) and range V/p(eA. Then i=l
i>/
p~1(eJ) = Im i//;-. It is not difficult to see that the correspondence p ^ p~l is functorial. We denote this functor by
pj
t
w a multiple of the atomic representation
p+t
l
f
_ _ _ _
r=e
'
(that is,
P+t,x = P+t,
The following proposition describes the dual properties of <J>". PROPOSITION 5.2. Let p G ^(D r , K). Then the following assertions are equivalent: (i) 3>~p = 0. (ii) p(e}) + ( n p(ez)) = F /or ever^ 7 (where V is the representation space of p). (iii) p = © p?~i where each r=o ' p^j w 6f multiple of the atomic representation p ^ . The proofs of these assertions follow immediately from the definitions." PROPOSITION 5.3. (i) If p ^ © pt, then $ + p = © $+pz- and $
p ^
© <> | (p{). (ii) 77zere exw^ « natural monomorphism
i: &
/= 1
(iii) There exists a natural epimorphism p: p -*
Lattices, representations, and algebras connected with them
245
Let us prove (ii). The map <£y: V1 -• F (where K1 is the representation space of p 1 ) is such that ty($lf . . . , fy, . .. , £r) = £;, where £; G p(ey). r
Here the condition (£ l5 . . . , %r) G V1 is equivalent to 2 £,- = 0, and so 1=1
£• G p(e.) ( 2 p(ez)). Thus, Im
If Si
e
P(Cjhj) = pO;0 ( 2 p ^ ) ) , this means that £y- = 2 £,., where
5, G p(^.). Then « ! , . . . , S/-!, "£/, £ /+1 , • . . , 5r) e K1, hence, S;- G Im^y. Thus, the map
(FVp 1 (et)) = © p(e,-/i/)- This implies that the following diagram is
commutative:
I v
0-^y 1 ^
'
T $ p (ef/^) ^
t* Coker pi' -> 0
Now it is not difficult to check that we can construct a linear map i: Coker JU1 -> V such that the right square of the diagram is commutative. This map / is nothing but the natural isomorphism r
Coker /z1 = 2 p(^//i/). We set $} - dTrjX', where nj is the projection in r
© piejhj) onto the ;-th component. It is easy to see that i=i
Im #} = p(ejhj). We define a representation p in Coker /i1 by setting p ( ^ ) = Inupj. It follows from the definition of 3>~ that p = ^"(p 1 ), where p 1 = ^ + (p), is a representation in F 1 . Consequently, the embedding /: Coker yi -+ V defines a morphism of representations /: ^>~^>+p -* p. The proof of (iii) is dual to the one just presented. r
We n o t e t h a t 2 eihi is t h e m i n i m a l e l e m e n t vex o f t h e cubicle B + ( \ ) . /= l
In §4 we have proved that this element is perfect. Thus, p splits into the direct sum p = p(l) © r 0 1 , where r 0 1 = p| p ( u )? and p(l) is the direct r
sum p(l) = e mtp] x (mt > 0) of the atomic representations p\ x. We t=o state as a separate corollary the properties of 3>""
246
/. M. GeVfand and V. A. Ponomarev
COROLLARY 5.1. (i) There exists a natural isomorphism $+<|>~p 9* p | p ( u . ), where vdl is the minimal element of B+(\).
Here p is r
+
isomorphic to the direct sum p = 4>"4> p © p(l), where p(l) = © mtp\. r=o is a direct sum of atomic representations p] x. (ii) +<£>~p is isomorphic to the factor representation p/p"(l), where p"(l) = P| p(u- ), is the restriction of p to p(vj' x), the image of the maximal eler
ment ofB~(\).Here p s p-(\)®($*$~p) and p"(l)= ®mtpjx (where pjt is t=i
'
the atomic representation).* We state another proposition in a form convenient for a later application, which is a simple combination of the properties of <J>+ and <£". CONSTRUCTION. Let p be a representation in F, let r 1 be a subrepresentation in p 1 = <J>+p> and let U1 be the representation space of r 1 . We set U = 2 ^ t / 1 (1/ C F), where spt: V1 -> F is the standard map ^/(£i> • • • > 5/» • • • » 5r)
=
5i- We define a representation r in C/ by setting
1
r(e z ) = tPjU .
PROPOSITION 5.4. Suppose that p 1 = +p w decomposable: n n p 1 = © TJ-. Then p is also decomposable: p = p(l) © ( © r y), where Tj is obtained from rj by the construction described above. Here n
p
'
(mt > 0)
of the atomic representations p] x, and 4>+pO) = 0." References
[1]
[2]
[3] [4]
[5]
M. Auslander, Representation theory of Artin algebras. I, II, Comm. Algebra 1 (1974), 177-268; 269-310. MR 50 # 2240. - and I. Reiten, III, Comm. Algebra 3 (1975), 239-294. MR 52 #504. I. N. Bernstein, I. M. Gel'fand, and V. A. Ponomarev, Coxeter functors and Gabriel's theorem, Uspekhi Mat. Nauk 28:2 (1973), 19-33. = Russian Math. Surveys 28:2 (1973), 17-32. G. Birkhoff, Lattice Theory, Amer. Math. Soc, New York 1948. Translation: Teoriya struktur, Izdat. Inost. Lit., Moscow 1952. N. Bourbaki, Elements de mathematique, XXVI, Groupes et algebres de lie, Hermann et Cie., Paris 1960. MR 24 # A2641. Translation: Gruppy ialgebry Li. Izdat. Mir, Moscow 1972. P. Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 71-103.
Lattices, representations, and algebras connected with them
[6]
[7]
[8] [9]
247
I. M. Gel'fand and V. A. Ponomarev, Problems of linear algebra and classification of quadruples of subspaces in a finite-dimensional vector space, Coll. Math. Soc. Ianos Bolyai 5, Hilbert space operators, Tihany (Hungary) 1970,163—237 (in English). (For a brief account, see Dokl. Akad. Nauk SSSR 197 (1971), 762-765. = Soviet Math. Dokl. 12 (1971), 535-539.) I. M. Gel'fand and V. A. Ponomarev, Free modular lattices and their representations, Uspekhi Mat. Nauk 29:6 (1974), 3 - 5 8 . = Russian Math. Surveys 29:6 (1974), 1-56. V. Dlab and C. M. Ringel, Representations of graphs and algebras, Carleton Math. Lect. Notes No. 8 (1974). L. A. Nazarova and A. V. Roiter, Representations of partially ordered sets, in the coll. "Investigations in the theory of representations", Izdat. Nauka, Leningrad 1972, 5-31. Received by the Editors, 9 April 1976
Translated by M. B. Nathanson
LATTICES, REPRESENTATIONS, AND ALGEBRAS CONNECTED WITH THEM II1
I. M. Gel'fand and V. A. Ponomarev Contents
§6. The representations p] t and p ^ 249 §7. The perfectness of the lattices i?+ andi?~. Characteristic functions . . 252 §8. The algebras Ar and the representations p r / 258 §9. Complete irreducibility of the representations p+tl and p~tl 264 References . ' . . . ' 271 §6. The representations p\A and pf~,
We define the representations pj"/ (/ = 1, 2, . . . ) as follows: p*tl is the atomic representation (see §4), and for / > 1 we set inductively
The representations p ^ are, by definition, dual to the p]/5 that is,
Thus, the p ^ are the (-)-atomic representations, and it follows from properties of the conjugation functor (see Proposition 5.3 (iv)) that
In §8 we shall show that the p*tl are essentially the same as the psl, whose definition in terms of the algebra A was given in § 1 . More accurately, we shall show that pQtl_x = p©,/ and pjt = p j z if/ =£ 0. The functorial definition of p+tj is more convenient when we are interested in such categorical properties as decomposability and when there is no need to investigate the intrinsic structure of the representation. DEFINITION 6.1. The dimension of a representation p G M(Dr, K) in the 1 The first part of this article was published in these Uspekhi 31:5 (1976), 71-88 = Russian Math. Surveys 31:5 (1976), 67-85.
249
250
/. M. Gel'fand and V. A. Ponomarev
space V is the sequence of integers: dim p = {n\ m l 5 . . . , m r ), where n = dim V and mt = dim p(ef). For example, the atomic representations p+0 l and p* ^ r ^ o n a v e dim p+0 ! = ( l ; 0 , . . . , 0) and dim p*>1 = ( 1 ; 0, . . . , 0,' 1, 0, . . . 0), where the 1 stands in the (t + l)st place. PROPOSITION 6.1. Let p G J2(Dr, K) be an indecomposable representation. (I) Then there are the following possibilities for <£+p: a) 3>+p = 0 <*» p s p* z /or some f G { 0, 1, . . . r } , b) $ + p =£ 0 «-* (3>~Vp s p); /zere, f/ze representation
n{=z^]mi — n,
m)= ^mi — n.
(II) There are the following possibilities for <J>~p: a) ($-/> = 0) <-=> (p s p - j /or some f G {0, 1, . . . , r}), b) (4>~p =^ 0) «==> (4>+
PROOF. (I) a) and b), except for the assertions about the dimensions, clearly follow from Proposition 5.4. From the same proposition we find that 3>~
'
/
for all /, and so p{eihi{\)) - p(e^)V = p(et). Therefore, we can rewrite the diagram (5.1) in the following way:
2 0p (eiht)
> Coker p,'
> 0.
i
From this we find that dim V1 = nl = 1 dimp(^)-dim V= i
f i= 1
The formula m) = 2 mt - n is also easily proved. Part (II) of Proposition 6.1 follows by duality. THEOREM 6.1. Let p be an indecomposable representation of the lattice Dr', where the number r of generators of Dr is at least 4. Then there are the
Lattices, representations, and algebras connected with them. II
251
following three mutually incompatible possibilities: (I) ( ( $ + ) M p =£ 0 and (<$>y p = 0) *=> (there is a t G {0, 1, . . , r } i p S p ^ ) ,
(Ill) V /fw ((<*>+)'p * 0 <mrf (4>T P # 0). PROOF. Let p be an arbitrary indecomposable representation. There are the following possibilities: 1) there is an / > 0 such that (&)l~l p ¥=0, (4>+)zp = 0; 2) there is an m > 0 such that ( r f ^ p ^ O , (3>~)mp = 0; 3) ( 4 0 ' p ^ 0 and (SO'p =£ 0 for every / > 0. We consider these cases separately. 1) (&)lp =£ 0, (<&+)7P = 0. We write (&)*~lp = pl~l. It follows from Proposition 6.1 that p 7 " 1 , as well as p, is indecomposable, and that p s ( $ - ) ' - i p ' - i =(4>-) / - 1 (^ >+ ) / ~ 1 p. Since ^ p 7 " 1 = 0, it follows from Proposition 6.1 that pl~l = p*tl. Consequently, p = ($~)l~lp*tl = p]j(<J>~)mp ='0. Arguments similar to those used in 1) 2)(^')m~1p^0, m l show that ($>-) - p^p-s>l and p *± (&)m ~l p~s x = p " m . For the proof of the theorem it remains to show that if p E ^ (//, K) and r > 4, then 1) and 2) are mutually exclusive. Let p = p+tl. We claim that ($~)m p]t =£ 0 for every m. By definition, (<|)")mp*/ = p* / + m .Thus, we must prove that p\k =£0 for every k > 0. By Proposition 6.1, dim p\k ~ (jitk^m\k^ • • • ' m r J t ) c a n ^ e computed recursively from the formulae r n
t, h — ( r " ~ l ) ^ , f e - i — Z J ^ J ,
fe-Ti
^ ? , k = nt, fc-i — ^ J , / i - i -
il
It is not difficult to deduce from them that for r > 4 the terms of the sequence {ntl} (/ = 1, 2, . . . ) can be found from the recurrence relation nt, i = (r — 2)nt> i-i — ntt z_2,
and the initial conditions nt w
o,i
=
!» ^0,2
=
r
"
x
Z>3,
= 1, nt 2 = r —2 for t =£ 0, and
!•
For r > 4 the terms of {ftf>/} increase monotonically with /, and so all the p+tJ are different from zero. Thus, if ($ + ) / - 1 p ^ 0 and (4>+)zp = 0, then (4>-) m p^0 for every m, that is, 1) and 2) are mutually exclusive. REMARK. The lattices D1, D2, Z)3 are finite and each has only finitely many indecomposable representations (up to isomorphism). The numbers of these representations of/) 1 , D2, and D3 are 2, 4, and 9, respectively. If r > 3 and p is an indecomposable representation of Dr, then there are positive / and m such that (^>+)/p = 0 and (<£~)mp = 0. Therefore, each indecomposable representation of Dr, r < 3, can be described both in the. form p+tl and pf~m . For example, in D3 the following isomorphisms hold: Po,i — Po,3> Po,2 ~Po,2»Po,3 — Po,i- ^ n e dimensions of these representations are, respectively', (1; 0,0, 0),'(2; 1, l', 1), and (1; 1, 1, 1).
252
/. M. Gel'fand and V. A. Ponomarev
§7. The perfectness of the latticesB + and B~. Characteristic functions
7.1. Proof of the theorem on the perfectness of the sublattices B+ and B~. This proof is based on the following proposition. (As usual, we write p 1 = $>+p> where V1 and V are the representation spaces of p 1 and p ; pj\ V1 -* V is the standard map pj(^ , . . . , £ / , . . . , £ r ) = £7-; and et t are the lattice polynomials in Dr(r > 4) (see § 1 , p. 70).) PROPOSITION 7.1. For every p G M(Dr, K), r > 4,
O, p(eHv.Ai),
when
j^iim
We omit the proof of this proposition. It is the central and most complicated part of [7]. We recall that the generators ht(l) of the sublattice B+(l) are defined in the following way: et(l)=
2
e{
ht(l) =
x
^1ei(l),
where et(l) is the sum of all possible et f t in which the last index is t. COROLLARY 7.1. For every p G @(Dr, K), r > 4, p(MZ+l))= S W'ihtV)). j=i
The proof obviously follows from Proposition 7.1 and the definition of ht{l)M FUNDAMENTAL THEOREM 7.1. All the elements of the lattices B+ and B~ are perfect. Before proving this theorem, we prove the following proposition. PROPOSITION 7 . 2 . (I). Every element hj(l)eB+(l) (j = 1, . . . , r) is perfect, that is, for every representation p G £R{Dr, K) in V
where Tjj = p |p (/z.(/)) u pu = p \Ut and U is a space complementary to p(hf(l))in V. (II). The representation pu satisfies the relation Pu = ( © Pt.h) © P o , Z © P ; , I, 0
where each representation pt k and ps i is a multiple of the indecomposable representation p] k and p+s lt that is, pt k = p] k ® . . . © p] k> where mt k > 0. m
t,k
(III). IfTj i splits into a direct sum Tj t = © Tf of indecomposable representai= 1
Lattices, representations, and algebras connected with them. II
253
tions rt, then none of the Ti are isomorphic to any of the representations The proof is by induction on /. For / = 1 the corresponding assertion was proved earlier in Lemma 4.1. Now we assume that the proposition has been proved for the elements hj(l - 1). Then the representation p 1 = 4>+p can be written as where p* = p 1 i^i, U1 is a complement to pl(hj(l-
1)) in V1, and
We apply the construction of §5 to p 1 . By Proposition 5.4, p is isomorphic to a direct sum p = p (1) © pu © r7- /, where p (1) = © mt p] x (mt > 0) and f=0
the representations pu and r;- z are constructed from p* and rjA_x as in §5. Then r
Tj i is the restriction of p to the subspace 2 ^ ( p 1 (fy(/ — 1))). From Corollary 7.1 /= r
it is clear that this subspace is nothing but p(/z;(/)). Thus, r;-1 is a direct summand of p. This proves part (I) of Proposition 7.2 (that is, we have shown that the element hjj is perfect). Before proving parts (II) and (III) of Proposition 7.2, we now prove Theorem 7.1. PROOF OF THEOREM 7.1. Since the elementshx(l), . . . ,hr(l) are generators of the lattice B+(l), and since each of them is perfect, the entire sublattice B*(l) is perfect (that is, consists of perfect elements). We have proved in §3 that B* is the union of the setsi?+(/). Therefore, B+ is also perfect. The perfectness of B~ follows by duality. COMPLETION OF THE PROOF OF PROPOSITION 7.2 (II). We may assume by induction that we have proved that pi = © P*. * © Po, i-i © Pi, i-i. where each of the ptk and ps l_l is a multiple of the indecomposable representation p+t k and p+sj_\. By Proposition 5.4, pu = ~ p^ , where pu is the representation constructed from plu. Consequently,
0
By assumption, p
t k
= p + t k © . . . © p \ k, m m
rk
t,k
^~P+tk = P+t)k+\ by definition. Consequently,
> 0. Moreover,
254
/. M. Gel'fand and V. A. Ponomarev
Pf,h
t
pt
k
where
t,k and each such r e p r e s e n t a t Kk
= $~pt k_l,
posable r e p r e s e n t a t i o n p] k . Finally, bearing in m i n d t h a t p ( l ) = © f=0
we can write p
p()©(
e
pu)epo,iepM©^i
© p*.* e p o . i e p ^
!<*<*
0
This proves part (II) of Proposition 7.2. Part (III) is proved similarly (here we have to use the isomorphisms r;-1 = ^~rjl_l 2LndTJl_l = $ + r^ /). As we know, any element of B+(l) can be written in the form va / = 2 et(l) + 2 e^DhM) or, if a =t I, va t = n /2-(/), where a is a zGfl
/e/-a
'
j&l-a
subset of / = { 1, . . . , r}. The following proposition makes more precise the properties of a perfect element. PROPOSITION 7.3. Ler p G M {D\ K) by an arbitrary representation. Then p £* (
where
ra x - p | p ( u
representation
p+tk
0
p,,ft)© (
^ ^/7G? eac/z pt and p]
h
k
0
P., i) © T a, Z,
and ps t is a multiple
Here, ifral
of the
splits as Tal - © rt into i= 1
indecomposable a direct
sum
of indecomposable representations Tt, then none of the Ti are isomorphic to any of the representations p] k (t = 0, 1, . . . , r; k = 1, . . . , / - 1) or p+sl(se ( / - a ) U {0}). We omit the simple proof of this proposition." COROLLARY 7.2. (i). The correspondence a\-*-\)ai defines an isomorphism of the Boolean algebras vt: % (I) ->• B+(l). (ii) Let va i and vb m be arbitrary elements of B+(l) and B+(m). If I < m, then'val D vbm. P R O O F . We consider t h e r e p r e s e n t a t i o n p = p*t © p\ , © . . . © p+rj. By Proposition 7 . 3 , if a and b C / and a =£ b, t h e n the subspaces p(va z) and p(vb z) are distinct. H e n c e , if a ¥= b, t h e n val # vbl. r
Let ve i_i = n hj(l) be the minimal element in B+(l~ 1), and i-1
r
Vj t = 2 et(l) the maximal element in B+(l). It is not difficult to check that the /= l
subspaces p(vd j_i)
and p(U//) can be described as follows:
Lattices, representations, and algebras connected with them. II
p (vQ.
M
+
) =©^
f=O
,I
H
P (VI, 0 = 0
255
Vt, u
where V]j is the representation space of p+tl. Thus, p(vel_l) 3 p(u//), and so y 0 /_i is strictly larger than VJJ (vdl_l D VJJ). Now if va j _ 1 GB+(l~ 1) and vb / EB+(l) are arbitrary then u a,/_i 3 u 0,/-i ^ y /,/ 3 vbj. Consequently, vaj_l D vb /.• 7.2. The connection between the representations of B+ and B~. We have proved that the elements ve j (where vQ l is the minimal element of B+{1)) form a decreasing chain: vdl D vd2 D . . . D u 0 z D . . .. Dual to the elements vd i of Dr are the maximal elements vfj of the lower cubicles B~(l) r
(v7,=
2 /*."(/), where/*."(/) = O ^7~(/)). These elements also form a chain: '—
y
/",i
Cu
1
'-A
7,2
'
C - . c v f j C . . .
Let p E ^ (Z)r, Z). We write Vel = p(ve j) and Vfj = p(vjj), and we set V* = H F . , and FJ = n F / , . /=i
'
/=i
'
PROPOSITION 7.4. F ; C Vl for every representation pe M{Dr, K),r>4. PROPOSITION 7.5. Every representation p G & (Z7, K),r> 4Js isomorphic to a direct sum p = p~®p@p+, where p~' = P\y^> ( P ~ 0 P X ) = P | F + - Here (^>+)/p+ = 0 for some I > 0, (4>")/p" = 0 for some / > 0, and for every l> 0
The proof of these propositions follows easily from Theorem 6.1 and Proposition 7.3.• _ Note that in Proposition 7.5 p+ = 0 pj" /, where each p\ x is a multiple of the indecomposable representation p+t j = (&mt l p]lt mt / ^ 0, similarly, p " s 0 p ~/5 where each pf; is a multiple of the indecomposable representation t, i
p^j. A classification of the p x is known at present only for the lattice D4 [6]. The classification of the representations px ofDr, r> 5, seems at present a hopeless problem. 7.3. Indecomposable representations and characteristic functions. Let B be a sublattice of perfect elements in a modular lattice L. With each indecomposable representation p: L -> ^ ( Z , F) we associate a function xp on the set B in the following way: 0 1
if p(v) = 0, jf p (y) — y,
where v G B and F is the representation space of p. This x p is called the
256
/. M. Gel'fand and V. A. Ponomarev
characteristic function. The set with two elements {0, 1} can be regarded as a lattice (a Boolean algebra), which we denote by 2. The characteristic function xp then becomes a lattice morphism xp : B ->• 2. It is clear from the definition of the characteristic function that if two elements vx and v2 of B are linearly equivalent, then Xp(^i) = Xp(v2). Thus, xp is completely determined once its corresponding map xp : B -+ 2 is specified. We do not know the entire lattice of perfect elements in Dr (r > 4). We know only its two sublattices B+ and B~ C Dr. We denote by B the lattice generated by B+ and B~, and by B the factor lattice of B by the relation of linear equivalence. We know that p(iT) C p(v+) for any representation p & 91 {Dr, K) (r> 4) and any v+ E 5 + and v~ GB~. Consequently, 5 = £ + U £~ (that is, if uGJ?, then either^ G B+ or v G 5"), and if u+ G B+ and iT G 5", then v~ C D+ in i?. We shall study the characteristic functions xp- B ^ 2 (B C Dr). It follows from the ordering of B that there are three kinds of morphisms X: B -* 2: Either (1) there is an element v+ G i?+ such that x(u+) ~ 0- Then XOO = 0 for every v~ G 5"; or (2) there is a u ' G F such that x(V~) = 1Then x(y+) = 1 f° r every v+ G B+] or (3) x(v+) ~ 1 f° r every v+ G 5 + and X(t>~) = 0 for every iT G B~. We denote the last function by Xo- Characteristic functions of the first kind are denoted by x+> a n d those of the second kind denoted by x~THEOREM 7.2. Let p be an indecomposable representation of Dr, and suppose that its characteristic function xp is of the first or second kind. Then p is determined by its characteristic function uniquely up to isomorphism. Moreover, if xp is a function of the first kind, then p ss p+j for some t G {0, 1, . . . , r } and I G { 1, 2, . . . } ; // xp is a f u n c t i o n of t h e s e c o n d k i n d , t h e n p = p ~ m f o r s o m e s £ { 0 , 1 , . . . , r } and m G { l , 2, . . . } . Before proving this theorem, we define morphisms xj" / : B ~* 2 and Xr~/: B -> 2. F o r every / = 1, 2, . . . we define 1 if 0 if
m
where v*m is an arbitrary element of B*(m). Z, or if
m=i
and yj, = fes(m), where
where hs(m) (5 = 0, 1, . . . r) are generators of the cubicle B*(m). The characteristic functions xf";: ^~* 2 are defined similarly, namely,
Lattices, representations, and algebras connected with them. II
257
1 if 0, I (Vm) = j %b,l(Vm)=\
1 if I < m 0 if m < /
Q
if
or if m = Z and /*7 (w) s z;^, or if m — l and v~n^hr&{m), where
s^t.
LEMMA 7.1. Let x: B -> 2 be an arbitrary morphism. Then there are the following three possibilities: 1) x is a morphism of the first kind { that is, 3v+ G B+\x{v+) = 0). Then X = Xt,i for some t G {0, 1, . . . , r) and I = {l, 2, . . . } ; 2) x is a morphism of the second kind {that is, 3v~ G B~\ x(v~) = 0- Then = X Xs,m f°r some s G { 0 , 1, . . . , m} and I = { 1, 2, . . . } ; 3) X = Xo> that is, x(v~) = 0 for every v~ G B~ and x(v+) = 1 f°r every + v G B\ The proof follows easily from the description of the lattices B+ and B~.* PROOF OF THEOREM 7.2. We claim that each morphism xtj- B -• 2 has one and only one indecomposable representation p such that xp = Xt,i anc ^ p 9* p+t j . We first consider the case t =£ 0. The equality Xp = Xr / indicates that p(/zf(/)) = 0. It follows from this and from Proposition 7.4 that
where each ps k and p s t is a multiple of the indecomposable representation p+sk and p*j. But p is indecomposable, by hypothesis, and so p = p+sk, where for the pair (s, k) either k < I and s is arbitrary, or (5, k) = (0, /), or
(s, k) = (t, I). The equality x p = x]ti also indicates that p(ht{l)) = V for any i ¥= t. It follows from p{ht{l)) = V, from Proposition 7.2, and from the indecomposability of p that p cannot be isomorphic to any p] k with k < / or 0, fc) = (0, /) or 0, fc) = (z, /). This reasoning applies to all / (z ^ 0Therefore, the only possibility is that p = p+t t. This proves that (xp = xj,/) =* (p = Pr+,/)Together with Lemma 7.1 this means that (x p = Xr/)<===>> (P — P?,/)- The proof for the case f = 0 proceeds similarly. The proof for characteristic functions of the second kind (xp = Xr"/) proceeds dually." Little can be said about indecomposable representations p for which Xp = Xo (that is, x p (v+) = 1 and x p 0 O = 0 for all v+ G B+ and v~ G 5"). They are precisely the representations that are not annihilated by any of the functions (^)+)/ or (^~) z . Their classification is known only for the lattice Z)4 [6]. There are infinitely many such indecomposable representations p G J?(D4, K) and each of them has not only integer invariants, but also a continuous invariant X G K (similar to an eigenvalue of a linear transformation).
258
/. M. Gel'fand and V. A. Ponomarev
One might suppose that the equality xp = Xp' = Xo> where p and p' are non-isomorphic indecomposable representations, is a consequence of the fact that we do not know the entire lattice B of perfect elements in D 4 . However, we believe that this is not the case. We offer the following conjecture. CONJECTURE. The lattice B of perfect elements in Dr 0 > 4) is the union of B+ and B~. §8. The algebras Ar and the representations pt j 8.1. The algebras Ar. Let K be a field. We define the ^-algebra Ar as the associative ^-algebra with unit e and with the generators £ 0 , £ l 5 . . . , £r satisfying the following relations: r)f
We write ^4 instead of Ar when this cannot lead to misunderstandings. By a standard monomial in Ar we mean a product £,- . . . £• J r , where 1
Z—1
for every 1 < / < / - 1 +i and We denote this standard monomial by £z- , . . £z- t = S a , where ex = (/ x , . . . , / ^ j , 0 is a sequence of indices. Clearly, ev every non-zero
monomial in Ar can be put in standard form. The degree of £a is the integer d(£a) defined as follows: dtto) = d(e) = 0; d&t) = 1 for every i ^ 0; and d(SJfi) = d(ga) + tf(^) if £a£/3 ^ 0- ^ n e degree of zero is undefined. We denote by Vl (Vt CAr) the space of homogeneous polynomials of degree /. For example, V1 is the space that contains the monomials £i> • • • > £r> £i£o> • • • > Mo of degree 1, and dim Vx = 2r - 1. It is easy to see that in this way ^ becomes a graded algebra: Ar = Vo © V1 © . . . © F/© . . .; KjFy C F / + / , where F o =Ke ®K%0. Direct calculation shows that the algebras^ 1 , A2, and A3 are finitedimensional over K of dimensions 3, 5, and 11, respectively. Let us show, for example, that A2 = Vo © Vx, and that dim^ A2 = 5. Note that ^ ^ = 5 ? + * i * 2 = ( 2 5 / ) t 2 = ( 2 ^ ) ( ^ 0 ^ ) = ( 2 i= l
Similarly, £ 2 £i
=
/= l
^o)fe=0.
/= l
05 and so V2 = 0. Thus, Vx = 0 for every I > 2. In every
Lattices, representations, and algebras connected with them. II
259
algebra Ar we have dim Vo - 2 and dim Vx = 2r- 1. Therefore, dim A2 = dim Vo + dim Vx = 5. In the same way it can be shown that A3 = Vo © Vx © V2, and that dim Vo = 2, dim Vx = 5, dim V3 = 4. As regards the algebras Ar, r > 4, they are all infinite-dimensional, as we shall show later, and the numbers dl = dim Vt form a sequence strictly increasing with /. 8.2. The representations pA and pt t. We define a representation pA\ Dr -+ XR{Ar))of Dr in <£*C<4r) of right ideals of A = Ar in the following way. We set
where %{A is the right ideal generated by £ /; and ^- is a generator of Dr (i = 1, . . . , r). If x and j ; are lattice polynomials in Dr', then, by definition, p^ (x O;;) = p^ (x) O p^ (y) and pA(x +y) = pA (x) + pA(y), where pA(x) and p^Cv) are right ideals of A. Let ;4£r be the left ideal generated by £ r For every / > 0 and ^ = 0, 1, . . . , r, we set (8.1)
Vtil = (Att)[]Vh
It is immediately clear from this formula that Vt 0 = 0 for z ^ 0 and VOtO = K^o. It is also easy to see that all the subspaces Vt l (i ^ 0) are one-dimensional: Vt x = K%f. Any space Vt /} where (t, /) = (0, 1) or / > 1, is the sum 2 ^ a ? f of all subspaces K%a%t such that d ( f a ^ ) = /. We define a linear representation pt f. Dr -* % (K, Vt> i) in Vt t by setting (8.2)
Pt.i(ej)
= Vttln(tjA)
(/=1, ...,r),
where J;-^4 is the right ideal generated by £;-. We also define the representation p e in the one-dimensional space Ke so that p e (^) = 0 for all / = 1, . . . , r. EXAMPLE 1. pUl\ Dr -* X{K, Vi^lO ¥= 1) is the representation in the space Vitl = K%t such that pitl{et) = K%t and P/fl(ey-) = 0 if / =£ /. EXAMPLE 2. p 0 j : Dr -• ^ (/iT, Fo, I) is the representation in F o>1 that is r
generated by the vectors ^^0,
. . . , £ £0 ( 2 £/£o
=
0)- Clearly,
/=0
dim V01 = r- 1. By definition, for this representation pOtl(ej) = K$j%0. PROPOSITION 8.1. For every algebra Ar (r > 4) (8.3)
PA =
Pe © Po, o © ( © ( 0 Pt, 1=1
i))>
t=0
The proof reduces to a verification of two elementary assertions. 1) For every / > 1 the space Vl is isomorphic to a direct sum:
260
/. M. Gel'fand and V. A. Ponomarev
Vt s* Vo, i © Vu l © . . . 0 Vr, ,. 2) Clearly, every ideal %tA = p(e,-) is homogeneous with respect to the r
grading A = Vo © Vx © . . . , that is, £z-.4 = © (F, n £z-,4). To complete i= l
the proof of the proposition it remains to check that every subspace Vt O ^ splits into a direct sum (vt n itA) ~ ©
Pit z
(eo = © (Ff, z n 5i^) - © {Ait n ^ n 5^).
•
REMARK. (8.3) also holds for the algebras A1, A2, and A3. True, in these cases the sum on the right contains only finitely many terms. For e x a m p l e , A3
= p
© p
© ( © ( © pt / ) ) . 1=1 t=o
8.3. The connection between the representations ptj and p\j. By definition, the atomic representation p] l in the one-dimensional space V\x = K is such that: a) p+0A{et) = 0 for every / = 1, . . . , r\ b) pl.iej) = Vjtl and pj^ie-) = 0 if i ± j . Thus, Po. i^Po.o and Pi, i = Pi,i (i¥=0). The representations p* z, / > 1, were constructed from the atomic represdef
entations by means of the functors $ , namely, pt every ^ = 0, 1, . . . , r. PROPOSITION 8.2. For every I > 1, Po, z-i = po, i, pt,i = P+tj,
l
=($)
pt
x
for
f=l,...,r.
For / = 1, as we have already mentioned, the proposition is evident. For / > 1 the proof is by induction. We assume that the isomorphism Pj l = pjl (or Poj-l — Po/) n a s already been proved and we claim that similar isomorphisms hold for I ¥= 1. To see this, obviously, it is enough to prove that (8.4) (8.5)
Let us prove (8.5), say. We denote by £;-L the linear map of A into itself defined by the formula £/L : x -• fyx for every x G ^4. The map 0) a c t s on the standard monomial %a G Kf z_x by the formula °
if
/ = ii,
Hence ^/jL Kf>/_j_ C Vtl for / # 0; moreover, it is easy to see that
Lattices, representations, and algebras connected with them. II
261
r
Let fx denote the map Vo /_ x -> © p 0 /(e7-) defined by the formula /= l
lix^-fax,
. . . , lrx). r
r
Let 0 denote the map 0: e p 0 z(g.) -+ F o z such that 0(JC 15 . . . , xr) = 2 xb /=i
'
'
i=i
where xt E p 0 /(^). Thus, we have a sequence (8-6)
Fo, ;_i A
J> p0, i (ej) \
Vo> t
for which dfx = 0. It is clear from the definition of <£~ that to establish the isomorphism &~poi_1 = pol we have to prove the following: a ) Po,Md — ^o,/-i 'Poj-iiejYi b) the sequence (8.6) is exact at the middle term, that is, Ker 6 = Im /x; c) Vo t = Coker JU (however, if Ker 0 = Im /x, then Voj = Coker ju, because 0 is an epimorphism). We omit the simple proof of these assertions." COROLLARY 8.1. The algebras Ar, r > 4, are infinite-dimensional, and the numbers nx = dim Vl form an increasing sequence n0
(jer(a)
y4(r, / - l)and
We denote by At(r, /), / E {0, 1, . . . , r}, the set whose elements are sequences j3 = (/j, . . . , / j _ l f /) such that ij G { 1, . . . , r}, z';- ¥= ij+i, //_! =£ /. For every j3 E ^40(r, /), / > 1, we define e
t = eii..-il_io ^
J
= ei
4
S 761X3)
^Y» Y
where F(j8) C A(r, I - 1) and F(j3) is constructed from the sequence 0 = (/l5 . . . , //_!, 0) in the following way: r (P) == {Y= (^i» • • •' ^-i) I ki *£ ^ii ^
262
/. M. Gel'fand and V. A. Ponomarev
For example, eiio=eil
2**i
eiii 2 o = e i i ( 21
f^ii
*ftifcS).
fei^in'2 ft 2=^ 12
The next theorem establishes a remarkable connection between the monomials ea and the representation pA: Dr -> # f l (-4) (p^ (e,) = %jA). THEOREM 8.1. Let pA be the representation of Dr defined above in the lattice of right ideals of A. Then for every a = (i1, . . ., ij_lt t) G At(r, /): PA ( e a) = %aA> where %aA is the right ideal generated by the element This theorem is based on the following proposition, which is proved in [7]. Note that this proposition refines Proposition 7.1. PROPOSITION 8.3. Let p be an arbitrary representation of Dr (r > 4), and p 1 = 3>+p- Let
= P (*j, o), c
(piiP1 (e»4.. .i^t) = 0,
1
PiP K...i / _1 O=Pfei 1...iz _ 1t),
if i¥=i.
•
We have proved that the representations ptj satisfy &+ptj = Pf,/_i, and that the map J/Z : Vtl_l -• Fr>/ is the one we have denoted by «p;-: F 1 -> F. Consequently, Proposition 8.3 can be restated for s G {0, 1, . . . , r) in the following way. LEMMA 8 . 1 . For every
m >
1 and every a = (il9 . . . , il_l,
Ps, m(*iio)=SiiV 8f
t) ^At{r,
/)
m-i,
Ps, m^tiia.-.i^^) =SiiPs, m-1 ( e i2 . . .tM <) i
where £z- p ( ^ ) w f/ze /mage of the subspace p(e^ ) C Vs m_l under f^ ^ .• PROOF OF THEOREM 8.1. Since p^ splits into a direct sum of representations p5 m , we can write (8.7)
PA('«)= 2
S P..m(«a).
Let us find the subspaces ps m{e0L)-ps (et t t) for various s and m. For this we have to analyze the cases t =£ 0 and /• = 0. We first consider t=£0. I. a) Let /• ¥= 0. We determine ps m {et A t) when m
Lattices, representations, and algebras connected with them. II (8.8)
pSt
m
{eit... i^t)
= ^ t P«, m-1 (*i2. .. i^t)
263
~
= 111 (Ii2ps, m-2 (^3. . .*/_!*)) =
By definition ei
t
Consequently, ps l(ei '
t
Eet (I — m + I) C hp(I — m + 1) for any p ^ t. t)^Ps
t TYl '"
I— 1
i(hv(l-m '
+ 1)). As we know, the ele-
"
ment hp(l - m + 1) is perfect, and Ps \ — P*s \ if s =£ 0, and p 0 t = p+0 2 . Therefore, if / > m, then psA(hv(l-m + 1)) = 0, and a fortiori Pf.e ( % ...//_! r) = °- T h u s ' i f m < l> t h e n Ps.m <<ei1 .../ /M r) = °b) We now determine ps m(ei t t) for m = /. By analogy to the chain of equalities (8.8), we find
If s = 0, then, by definition, POti(et) = K£t£0 = £t(K%0), and if s = t, then ps i(et) = ATJf = £rA" and p^ i(e r ) = 0 for s ¥= t. Thus, we can write
{
lii...il_itK%o
if
5=0,
0
if
s^t.
c) For m > / we obtain v"'y/
P«» m( g i 1 ...i / _ 1 «) = = Si 1 . • . i ^ P a , m-Z+l(^*)«
By definition, for n > 1 the subspaces ps n(et) Ps n^et^
=
%t^s n — \' Consequently, ps
m((?f
/
satisfy t)
=
%j
f-
?^s
m-/ #
^
can therefore rewrite (8.9) as follows: Ps, mfax)= Ps, m (eit. . .tz_t<) ==&!•. . i ^ j ^ . m-l = \aVs, m-l d) Now we insert all the relevant expressions for ps m(ea) This gives
(fn > Z). in (8.7).
= 2 Ps, I {ea) + 2 P«, l+l (e*) + • • • + 2 Ps, Z+n (
2 5=0
,
2
.iB 3=0
2 2
n=l s=0
e
264
/. M. Gel'fand and V. A. Ponomarev
The proof that ea = ei{. . . eij_i 0 proceeds similarly, with the help of the formula p
o m
( e i 0 ) = ^ Vo>m_1 .m
§9. Complete irreducibility of the representations p+t t and pj
t
In this section all vector spaces are finite-dimensional over a field K of characteristic zero. 9.1. Systems of vectors. Let R C V be a finite set of elements of a finitedimensional space V. Then i? is called a system of vectors in V if all elements a E i? are non-zero and i? generates V. A subset i?' Ci? is called a subsystem of R if R' generates V. EXAMPLE 1. A root system (for the definition, see [4]) is a system of vectors. EXAMPLE 2. Let Vt! C Ar be the representation space of pt /. Then the set of £a = £z- ,- f (monomials of degree /) is a system of vectors. A subset B C R is called a Z?as/s o/ i? if B is a basis of V. A system 7? is called indecomposable if for every subset Rx C R the intersection of the subspaces Vx = X Ka and V2 = 2 Koc is nonempty. (Ka denotes the one-dimensional subspace generated by a.) We introduce several concepts, which allow us to give a convenient criterion for the indecomposability of a system R. Let B = {al9 . . . ,an} be a fixed basis of R. We associate with each vector /3 E R a subset (chamber) C^ C B in the following way: and
^=7^0).
Two vectors o^- and ay in B are called simply-connected if they belong to the same chamber (that is, if there exists a 0 £ R such that P = biai + bjOLj + 2 £*;%, where &,- =£ 0 and £.- =£ 0). Two basis vectors at and ak are called connected, in symbols c^- ~ ak, if there is a sequence of vectors af = ot°\ a^l\ . . . , a (m) = ak such that every pair ct^, oSJ+l^ is simply-connected. If we take every vector a E B to be connected to itself, then it is not difficult to check that connectedness is an equivalence relation. A basis B is called connected if every pair of vectors at, oy GB is connected. PROPOSITION 9.1. The following assertions are equivalent: (i) Every basis B of R is connected. (ii) The system R is indecomposable. We claim that a basis is not connected if and only if the system is decomposable. 1) Let B be a disconnected basis of R and Bl C B be a non-trivial connected component. Then for every vector 0 E R with chamber CB
Lattices, representations, and algebras connected with them. II
either C, CBt or C, C^-B^.
265
We set Vx = X KocmdV1=
2
aE5,
a^B-Bl
Ka. It
follows easily from the definition of a chamber that j3 E 2 Aa. Thus, for every vector j3 E i?, either j3 E Fx or j3 E F 2 . This means that R is decomposable. 2) If R is decomposable, then there exist two non-empty complementary subs e t s ^ andi? 2 (7?j DR2 =0,Rt + R2 = R) such that the subspaces F- = 2 A$ (z = 1, 2) are disjoint. Let B be any basis of R. Clearly, each basis fieR
vector belongs to one of the subspaces Vx or F 2 . Thus, the set B splits into two complementary subsets Bx and B2, where a E Bj <=» a: E F;-. Here i?;- is a basis of F^O' = 1,2). Let /3 E/? be an arbitrary vector. Then the condition of decomposability implies that either jft E Fi orjSE F 2 . Now j3 G Fz- clearly implies that Cp C ^.. Consequently, B is not connected." An indecomposable system R is called minimal if for every ]3 E 7? the subsystem i?' = /? -{]3}is decomposable. PROPOSITION 9.2. Let R be a minimal indecomposable system and B = {ax, . . . , an } a basis of R. Then: a) the number m of elements in R - B satisfies 1 < m < n; b) for each (3 E R - B the chamber Cp C B contains at least two elements; c) the set R - B can be numbered in such a way that R - B = { ft, . . . , Pm} and for each j > 1 the chamber C; = Cfi. /-I
/-I
i= 1
z= 1
satisfies 0 C C1 H ( 2 C) C £ Q (where C denotes strict inclusion) and the it
subsets 2 Q /orm a strictly increasing chain: i 11 i=
^...crS We omit the simple proof of this proposition." Let R be a system in V = K71. The lattice generated by the onedimensional subspace Ka, a E R, is denoted by >//(i?). A system /? in V is called completely irreducible if c#(i?) ^ <£(Qn), (where ^(Q n ) is the lattice of linear subspaces of Q"). A system R is called rational if there is a basis B = { ax, . . . , an } C /? such that each vector j3 G 7? can be written as a sum j3 = 2 A,-o:z- with rational coefficients %i. /= l
It is easy to show that if R is rational, then every vector /3 E /? for any basis 5 = { a J } C i ? can be written as a sum ]3 = 2 X/a:z- with Xj G Q. l
266
/. M. Gel'fand and V. A. Ponomarev
THEOREM 9.1. Let R be an indecomposable rational system in V = Kn over a field K of characteristic 0. / / dim V = n > 3, then R is completely irreducible, that is o£ (R)s*X (Qn). The proof requires several Lemmas. MAIN LEMMA 9.1. Let V = Kn, where K is a field of characteristic 0, n
and let R = {<x0, c^ ,. . . , an } be a system in V such that 2 OL = 0. / / n > 3, then R is completely irreducible, that is, o/H (R) =^ X (Qn)). We do not give the rather tedious proof of this lemma. We only remark that it is very similar to the standard procedure for introducing coordinates in a projective space. REMARK It is easily seen that any n vectors {a t } of the system R in Lemma 9.1. form a basis of V. LEMMA 9.2. Let R be a system as in Lemma 9.1 {that is, R C Kn, R = {a0, . . . , o^}, 2 at = 0), and let o/f (R) be the lattice i=0
generated by the subspaces Kat. Let xQ, xi9 . . . , xn be non-zero elements {that is of Q/IL (R) such that any n of them are linearly independent V i i\i*jxi ^ xk ~ 0)- Then x0, xlt . . . , xn are generators of S {R). k * i,j
PROOF. It follows easily from the lemma that thexz- are one-dimensional subspaces of V. Therefore, by applying the fundamental lemma, we see that each n
element xt can be represented in the form xt = Kfb where ft - 2 X^o:,- and Since dim V = n and all ft =£ 0, clearly, the n + 1 vectors fo> fi> • - • >fn
a r e unear
n
l y dependent, consequently,/ 0 = S jjfj with yt E Q.
By assumption, any n of the vectors f0, fl , . . . , / „ are linearly independent. Starting from this it is not hard to show that all the coefficients yt in the n
expansion f0 = X ytft are non-zero. /=i
We set /o = - / 0 and f( = yifi if i ^ 0. Then the system R' ~ { /o» fu •••>/«} satisfies the conditions of Lemma 9.1, hence, S (Rr) ^ X (Qn). Since Kf\ = x{ and xt G Jl (R), it follows that Jl (Rf)<=<>$ (R).
It follows from the isomorphism S (Rr) ^ X (Qn) that S (Rf) contains n
every one-dimensional space of the form Ky, where y - 2 [ijl with n
i= l
li: G Q. By definition, f'( = yt 2 X/.-a.-. Hence it is easy to show that the /=i
Lattices, representations, and algebras connected with them. II
267
OLj can be expressed rationally in terms of the fj. This means that Koij G G#(i?') and so nM{R') so^(fl). We have shown earlier that dl{R')<^J({R). Therefore, S{R') = oM(R). m LEMMA 9.3. Let R = {a 0 , ai, • • • > an + i} be a system of n + 2 vectors m
n+l
in V = Kn such that 2 OL = 0, 2 OL = 0, where 1 < k < m < n. If i=o
j=k
'
n > 3, ^^« /? is completely irreducible. PROOF. By assumption, the vectors a7- generate V. The conditions of the lemma then imply that B = { a l 5 . . . , c^} is a basis of R. We consider separately the cases m = 1, k = n, and 1 < m, k < n. n+l
1) Let m - \. This means that a 0 4- c^ = 0 and 2 a, = 0. Since /=i
7
Ab: 0 = Koii, t h e s u b s p a c e s ^a z - ( / = 1, 2 , . . . , « + 1) a r e g e n e r a t o r s o f o/^(jR),Thus, this case r e d u c e s t o t h e m a i n l e m m a . n
2) Let k = n. This means that 2 <x- = 0 and a
4- a:
+1
= 0 . This is
clearly another application of the main lemma. 3) Let 1 < m and k < n. We claim that <M (R) contains the element Kf, n
m
/=i
i=i
n
where / = 2 a.-. We write Vo = 2 KOL, VX - 2 Ka, (that is, /=* m
dim F o = m, dim F! =n~ k + 1). The condition 2 o^. = 0 implies that the »=o subspace y0 = ^ a 0 is defined on the basis { ax, . . . , a.m } of Vo by the system of equations
Here it is clear that each individual equation xt = xi+1(l < / < m) defines in VQ an (m - l)-dimensional subspace, which we denote by Wt. We write yj for the line Kotj. It is easy to check that Wj = (y0 + • • • +J>/_i +^i + ^ / + i +J^/+2 + - • • + J ; m M 1
Here each individual equation Xj = x/+1(A: < / < n) is that of the subspace
268
I- M. Gel'fand and V. A. Ponomarev
We define elements Lt in Jl (R) as follows:
f (*/o+i/*-i+£* + */
...+J/n)
for
•. + yn+i)
for
It is not difficult to verify that in the coordinate system {a x , . . . , <xn} the hyperplane Lt(\ < / < n) is defined by the equation xf = xi+l. Consequently, the system of equations
defines a one-dimensional subspace Kf, where / has the coordinates w-l
(1, 1, . . . , 1) in the basis {o^ ,. . . , an ). It is also clear that Kf = n Lt. Since i= 1
the Lt are elements of S (R) we see that Kf E cd(R). We denote the system {oclf . . . , an, - / } by R. Obviously, this system satisfies the conditions of the main lemma 9.1, hence Jt (R) s* X (Qn). It follows from Kf e Jl (R) that GM{R)^QM (R). Owing to the isomorphism S (R) ^ X (Qn) the one-dimensional subspaces m
n
y0 = Ka0 = K ( 2 a,-) and yn + l = Kan + 1 = K( 2 a ) belong to Consequently, oM {R) <=oM(R). We have proved above that e//(S)e c//(i?). Therefore &ft(R) = <JZ (R) s «^(Qn). • PROOF OF THEOREM 9.1. We choose in i? any minimal indecomposable subsystem Rf and claim that R' is completely irreducible, that is S(R') - £(Q n )). Let B = { « ! , . . . , an } be a basis of /?'. We denote by m the number of elements in R' - B. It follows from Proposition 9.2 that 1 < m < «. We recall that with each element /3 £ /?' we associate a chamber C^ (a subset of B) by the following rule: (ai G Cp) ^=^ (|3 = \-az- + 2 X;-a;., with ;>/ Xz- # 0). It follows from Proposition 9.2 that if 0 G / * ' - £ , then the number of elements d(j3) in Cp is at least 2 (d(0) > 2). We break the proof into several steps and use induction on the number m of elements of R' - B. Step 1. Let m = 1, that is, R - B = {j3}. Since Rf is indecomposable it n
evidently follows that Cfi = B, that is, |3 = 2 X,-^, where all Xz- ^ 0. We z=l
set a[) = - 0 and ot- = X.- o:.- for / > 1. Since 2 a- = 0, the system
Lattices, representations, and algebras connected with them. II
269
^1 = (ao> • • • > a'n } cleai*ly satisfies the conditions of the main Lemma 9.1, and so Jl (R!) = Jf(Ri) s* X (Qn). Step 2. Let Rf - B = {ft, j32}. It follows from Proposition 9.2 that the chambers Cx and C2 of ft and ft satisfy Q U C2 = 5, Cj n C 2 ^ 0 , C Hence it is easy to see that the basis B can be numbered in the following *,
n
way: B = {OL1, . . . 9an}, where ft = 2 X^. and ft = 2 /i/°9> with 1
/
1 < s0 < sx < n, and all of the coefficients Xt and JU-; are non-zero. (Note that sx = d(Cx) and n -s0 4- 1 = d(C2), where d{Ct) is the number of elements in Cz. Here, sx ~s0 4- 1 = d(Cx O C2).) We consider separately the cases d(Cx O C2) = 1 and d ( d n C 2 ) > l . Step 2a). Let d(Cx n C 2 ) = 1. This means in other words that sx =s0 =s, that is, ft = 2 X^-a,- and ft = 2 j^-ay, where 1 <s
_1
Vs
Vs
^
«+i
C6 = -^- a;-, if s < / < w; aw + 1 = — ft. Then 2 ^ = 0 and 2 a' = 0, and / = 0
j
= s
{a\, . . . , ocn } is a basis of F. Let i?x* denote the system (oco, . . . , cx^ + 1 ). Clearly, ^ x satisfies the conditions of Lemma 9.3, and so o/fl (Ri) = X (Qn). From the construction it is clear that QM(R')=^ Qff (R{) thus, G ///(ir)^#(Q n ). Step 2b). Let d(Cx n C 2 ) > 1. In other words, we can write 13J = 2 XjCX;, ]32 = 2 jHyO^, where 1 < s0 < sx < n. We use the notation Vx = 2 ATc^ and F 2 = 2 AToy. From 1 < s0 < sx < n it follows that 1-1
/= s0
dim Fj = sx > 3 and dim F 2 = n - s0 + 1 > 3. Let ^1 = {^i5 • • • > a s ? i^i} and /? 2 = {c* v . . . , an9 ft). Thus, 7?^- is a system in Fz- (/ = 1, 2). The same arguments as in step 1 show that Rx and R2 are completely irreducible, that is, o# (i?0 ^ ^ (QSI),
&tf (R2) ^ X (Q n" so+1 ).
Since J/(Rt) ^ X (QS1), the lattice Jl (Ri) contains every one-dimensional subspace of the form Kx, where x = 2 7 ^ with a,- G Q. In particular, if 1=1 a0
= — 2 OLh t h e n Ka0
G
G///(/?!).
It follows from Lemma 9.2 that the subspaces Ka0, Kax,. . . , Kas can
270
/. M. GeVfand and V. A. Ponomarev
be chosen as new generators of o#(i?i) that is, GM{Ri) = j(i (Ka0, Kax,. . . , Kas ), where &tft (Ka0, . . ., Kas ) denotes the lattice generated by the one-dimensional subspaces Ka0, .. ., Kas . The same statements can be made about the lattice (J((R2), generated by the system R2 = {OLS , . . . , an, j3 } in V2. Namely, o/ft (R2) contains def
K a n + l , w h e r e an + 1 =
"
-
2 ay a n d J l ( R 2 ) = Jf2 (Kas ,. . . ,K<xn, K a n + l ) .
We write
Kax,
. . . , Kan,
Kocn + 1). New generators for dl (R!)
*i
n+ l
can be chosen so that 2 az- = 0 and 2 a.- = 0. Consequently, the *-o /=s0 conditions of Lemma 9.1 are satisfied and &ft (Rr) = ^ (Qn)Step 3. We now turn to the general case R' — B ={13!, . . . , 0m + 1 } . From Proposition 9.2 it follows that we can renumber B so that in the new numbering the vectors j3y- G Rf — B can be written in the following form:
S2
Pm+1—
a
S i^m+inLm
^m
J= s m + 1
where C;- is the chamber of the element fy, and Lk =
{l, 2, . . . , sk) =
A:
= 2 Cy. From Proposition 9.2 it also follows that 1 < Si < s2 < . . . < sm < sm + l = n. Since m > 1, we have sm > 3, and so by induction we may assume that the proposition is proved for the def
system Rm = { otx,. .. , as , ft, .. ., j3m } in Vm =
s
m
2 Kat. This means
that the lattice oflm = &ff {Kax, . . . , Kas , ^j!?!, . . . , Kf$m ) is isomorphic to #(Q s m ). Consequently, there is an elemenT>> = Ka0 in oMm where def
&0 =
s
m
- 2 a,-. Here, the elements Ka0, Kax, . .. , A^a,
are generators of
7=1
c//m. Hence we conclude that Q4( (R) = <#(KOLU . . . , Kan, Kpl9 . . . , A/3OT+1) = , ATQ:O, ^T]3W + 1 ) . Consequently, the arguments of step 2 apply
Lattices, representations, and algebras connected with them. II
271
') and so
Step 4. Now we return to the case of an arbitrary (non-minimal) system R. Let R' = {<*!, . . . , a rt , &, . . . , ft } be a minimal subsystem. It is clear that a basis B = { a{, . . . , <*„ } of Rf is also a basis of # . Let R - R' = { 7 1 , . . . , 7fr}. Since R is rational, each vector 7^ can be represented in the form yf = 2 A^-, where X.;. G Q. Since ^(iT) £g #(Q"), we see that A ^ e oM(R'). Consequently, <M(R) = (#(#') o* #(Q n ). • THEOREM 9.2. All but a finite number of the representations ptl\ Dr -* X(K, Vtti) over a field K of characteristic 0 are completely irreducible. Only the following representations are not completely irreducible: a) p 0 0 and pt x(i ^ 0) for any r > 4, b) p/f'2(i 9^0)'/or r = 4. PROOF. The representations p 0 0 and pz- x are not completely irreducible, since dim Vo 0 = dim F M = 1. It is also easy to find that dim Vi2 = 2 for r = 4 and z ^ 0, so that the systems pf 2 (/ ^= 0, r = 4) are not completely irreducible. For all other representations pt / we can show that dim Vt / > 3. To prove the complete irreducibility of the other representations pt /, we have to verify that: 1) the system of vectors Rt l = = {^}, aEAt (r, /), is rational, and that 2) the system Rt / is indecomposable. The rationality of Rtj follows easily from the fact that the complete system of equations satisfied by the vectors £a consists of 2 ti , „/. / t ~ ^' where the summation is over all vectors £a in s
! "" j— 1
i + 1 ' " l— 1
which the indices ik (k ¥= /) are fixed. The indecomposability of Rt t clearly follows from that of the representations pt /. References
[1]
[2]
[3] [4]
[5]
M. Auslander, Representation theory of Artin algebras. I, II. Comm. Algebra 1 (1974), 177-268; 269-310. MR 50 #2240. - and I. Reiten, III, Comm. Algebra 3 (1975), 239-294. MR 52 # 504. I. N. Bernstein, I. M. Gel'fand, and V. A. Ponomarev, Coxeter functors and Gabriel's theorem, Uspekhi Mat. Nauk 28:2 (1973), 19-33. = Russian Math. Surveys 28:2 (1973), 17-32. G. Birkhoff, Lattice Theory, Amer. Math. Soc, New York 1948. Translation: Teoriya struktur, Izdat. Inost. Lit., Moscow 1952. N. Bourbaki, Elements de mathe'matique, XXVI, Groupes et algebres de Lie, Hermann et Cie., Paris 1960. MR 24 # A2641. Translation: Gruppy i algebry Li, Izdat. Mir, Moscow 1972. P. Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 71-103.
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[6]
[7]
[8] [9]
/. M. GeVfand and V. A. Ponomarev
I. M. Gel'fand and V. A. Ponomarev, Problems of linear algebra and classification of quadruples of subspaces in afinite-dimensionalvector space, Coll. Math. Soc. Ianos Bolyai 5, Hilbert space operators, Tihany (Hungary) 1970, 163-237 (in English). (For a brief account, see Dokl. Akad. Nauk SSSR 197 (1971), 762-765. = Soviet Math. Dokl. 12 (1971), 535-539. I. M. Gel'fand and V. A. Ponomarev, Free modular lattices and their representations, Uspekhi Mat. Nauk 29:6 (1974), 3-58. = Russian Math. Surveys 29:6 (1974), 1-56. V. Dlab and C. M. Ringel, Representations of graphs and algebras, Carleton Math. Lect. Notes No. 8 (1974). L. A. Nazarova and A. V. Roiter, Representations of partially ordered sets, in the coll. "Investigations in the theory of representations", Izdat. Nauka, Leningrad 1972, 5-31. Received by the Editors 9 April 1976
Translated by M. B. Nathanson
