CEJM 2 (2003) 141 156
On stabilizability of e v o l u t i o n s y s t e m s of partial differential e q u a t i o n s on IR~ × [0, +oc) by t i m e - d e l a y e d f e e d b a c k controls L.V. Fardigola*
1 Depar'trnent of Mathematical Analysis, Khar'kiv National Univer'sity, 4, Liber'ty sqr'., 61077 Khar'kiv, Ukraine 2 Mathematical Division, Institute .[or"Low Ternper'atur'e Physics ~4 Engineering of the National Academy of Sciences of the Ukraine, 47, Lenin Ave., 61103 Khar'kiv, Ukraine
Received 2 October 2002; revised 17 J a n u a r y 2003 Abstract: In this work we obtain sufficient conditions for stabilizability by time-delayed feedback controls for the system
0,~(x, t) ot
A (Dx) ,~(x, t) - B (Dx) u(x, t),
~ ~ R 'r~, t > h,
where D~ (-iO/Oxl,...,-iO/Ox.r~), A(u) and B(u) are polynomial matrices (rn x rn), P(D~)w(., t - h) is a control, h > 0. det B(~r) ~ 0 on R '~, w is an unknown function, u(., t) Here P is an infinite differentiable matrix (rn x rn), and the norm of each of its derivatives does not exceed F(1 + 1~12)~ for some F , 7 E R depending on the order of this derivative. Necessary conditions for stabilizability of this system are also obtained. In particular, we study the stabilizability problem for the systems corresponding to the telegraph equation, the wave equation, the heat equation, the SchrSdinger equation and another model equation. To obtain these results we use the Fourier transform method, the Lojasiewicz inequality and the Tarski Seidenberg theorem and its corollaries. To choose an appropriate P and stabilize this system, we also prove some estimates of the real parts of the zeros of the quasipolynomial det { IA - A(~r) + B(~r)P(~r)e -ha } . @ Central European Science Journals. All rights reserved.
Keywords: stabilizability, feedback control, delay, partial differential equation, Fourier transform MSC (2000): 93D15, 35B37, 35A22
* E-maih
[email protected]
142
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Introduction
One of the most generally accepted ways to study control systems with distributed parameters is their i n t e r ) r e t a t i o n in the form dw dt
- Aw + Bu,
t > 0,
(1)
where w : (0, +oo) ?-{ is an unknown function; ~ : (0, +oo) are Banach spaces; A is an infinitesimal operator in 7-(; B : H
, H is a control; V-t, H > 7-( is a linear b o u n d e d
operator (see, e.g., [2], [3], [10], [12], [13] [16], [18], [19]). An important advantage of this approach is a possibility to employ the ideas and techniques of semigroup operator theory. At the same time it should be noticed that the results on operator sernigroups t h a t are most substantial and important for applications deal with the case when the semigroup generator A has a discrete spectrum and may be treated in terms of its eigenvalues and its eigenelernents. These assumptions correspond to differential equations in domains bounded with respect to space variables but they are not true for domains unbounded with respect to space variables. In the present paper, we consider the following system
0w (x, t)
Ot
-A(D~)w(x,t)-B(D~)u(x,t),
. • R '~, t > h ,
(2)
where D~ = (-iO/Oxl,...,-iO/Ox.~), A(cr) and B(cr) are polynomial matrices (rn x rn), det B(cr) ~ 0 on IR'~, w : IR'~ × (h, + c o ) > C "~ is an unknown function, u : R 'r~ x (h, + c o )
> C "~ is a c o n t r o l , h > 0.
In Section 6 we investigate stabilizability of systems of the form (2) t h a t correspond to the telegraph equation, the wave equation, the heat equation, the SchrSdinger equation and another model equation. We use the following Sobolev spaces
c~ = {g • cq(Rr~) I Ilgll~ < + ~ } , < -- {g I [v~ • [0, h]g(.,~) • c~]
A[Voz
•
NS~(I~I _< q ~ D~g • C(R ~ x [0, h]))] A
Liiiglll8< ÷~]},
IIIgll18 = sup{llg(.,~)ll81 ~ • [0, h]}, cZ = {g I [v~ • [0, +oo)g(.,~) • c~] A [v~ • NS~(I~I _< ~ ~ Dgg • C(R ~ x [0, +oo)))] f[sup{llg(',t)ll~ It • [0, +oo)} < +oo]} where q • No = 1N n {0}, 7 • Z, (~ = ((~s,...,(~,~) • No is a multi-index and as + . . .
c4 - ~ =
I~1
=
+ ~,~, I" I is the Euclidean norm or R ~. w e also use the spaces C £ = N Of, U c~, c~_~ = qENo
U c~, ~ ~yER
=
N c~-~, s = qENo
N c%, and for P • ~ , denote qENo
by P(D.) the following operator P ( D . ) f = ~ - l ( p ~ f ) , f • S~ where ~ is the Fourier transform operator (such an operator P is called pseudodifferential).
L.V. Fardigola / Central European Journal of Mathematics 2 (2003) 141 156
143
In a d d i t i o n , we a s s u m e t h r o u g h o u t t h e p a p e r t h a t ~ _> 0 a n d h > 0 are fixed. Definition
1.1. S y s t e m (2) is said to be stabilizable in C4- ~ if t h e r e exists a m a t r i x
(rn x rn) P E k4 such t h a t for each p E N0 t h e r e exists q E N0 w i t h t h e p r o p e r t y t h a t for every s o l u t i o n of this s y s t e m w i t h t h e control
u(x,t) = P ( D . ) w ( x , t - h)
(3)
u n d e r t h e initial c o n d i t i o n w E eq we have w E C~r a n d
II (.,t)ll
,0
ast
,
(4)
Such a m a t r i x P is called a stabilizing m a t r i x for s y s t e m (2), a n d such a control u is called a stabilizing control for this s y s t e m . In t h e stabilizability p r o b l e m u n d e r c o n s i d e r a t i o n , a delay a p p e a r s in t h e control because, in fact, a w control c a n n o t be realized i n s t a n t l y ( w i t h o u t a delay). To investigate s y s t e m (2), we use t h e Fourier t r a n s f o r m m e t h o d t h a t was p r o p o s e d by I . C . P e t r o w s k y [14] to s t u d y t h e well-posedness p r o p e r t y of t h e C a u c h y p r o b l e m for e v o l u t i o n s y s t e m s o n a layer R'r~x [0, T]. L a t e r this m e t h o d was generalized by I . M . C e l f a n d a n d C . E . S h i l o v [7]. If we a p p l y t h e Fourier t r a n s f o r m to a control of t h e form (3) we o b t a i n (:Y'u(., t))(~) =
P(~)(:Yw(.,t-h))(~).
Hence for a fixed ~0 E R 'r~ we have a linear o p e r a t o r
d e t e r m i n e d by t h e m a t r i x P(cr0). T h a t is w h y we can c o n s t r u c t a stabilizing m a t r i x P(cr), t r e a t i n g er E R '~ as a p a r a m e t e r .
However, if we w a n t to i n t e r p r e t
$-l(P(er)$w(.,t)) in
s o m e way t h e n we have to a s s u m e t h a t P b e l o n g s to s o m e class of functions. It is n a t u r a l to a s s u m e t h a t P is such a f u n c t i o n t h a t
:Y<(P(~):Yw(.,t)) = P(Dx)w(.,t),
is a p s e u d o d i f f e r e n t i a l o p e r a t o r . It is well k n o w n [7] t h a t in t h e case B(cr) -
i.e.
P(Dx)
0 all solutions of s y s t e m (2) t e n d to
0 as t ---* + c o iff Vcr E R '~ sup {~A I d e t ( I A - A(cr)) = 0} < 0 where I is t h e i d e n t i t y matrix.
It can be s h o w n t h a t in t h e case B(cr) y! 0 t h e answer to t h e q u e s t i o n of
w h e t h e r all solutions of s y s t e m (2) w i t h control (3) t e n d to 0 as t ---* + c o , d e p e n d s on t h e zero dispositions of t h e q u a s i p o l y n o m i a l det { I A - A(cr)+ B(cr)P(cr)e -ha} w h e r e P
E Z4 [1].
If all solutions of t h e s y s t e m do t e n d to 0 as t --+ + c o , t h e n for each
zero k0(cr) of this q u a s i p o l y n o m i a l we have ~k0(cr) < 0, cr E R ( S t a t e m e n t 5.2).
The
a s y m p t o t i c b e h a v i o u r of q u a s i p o l y n o m i a l roots is well k n o w n (see, e.g. [1]). Moreover, for an a r b i t r a r y q u a s i p o l y n o m i a l t h e r e are necessary a n d sufficient c o n d i t i o n s for n e g a t i v i t y of real p a r t s of its roots [15]. U n f o r t u n a t e l y , these c o n d i t i o n s have such a c o m p l i c a t e d form t h a t it is i m p o s s i b l e to a p p l y t h e m to a q u a s i p o l y n o m i a l d e p e n d i n g on a p a r a m e t e r . So, using these c o n d i t i o n s it is n o t possible to choose a p a r a m e t e r ( P ) in order for t h e real p a r t s of its roots to be negative. T h a t is w h y we have to investigate d e p e n d e n c e of t h e zero dispositions of t h e q u a s i p o l y n o m i a l { + / J e -h~ on /3 E C in Section 4. It enables us to choose a n a p p r o p r i a t e P a n d stabilize s y s t e m (2). However, A, B a n d P also d e p e n d o n a p a r a m e t e r (or E R'~).
T h a t is w h y in Section 3, using t h e Tarski
S e i d e n b e r g t h e o r e m , its corollaries a n d t h e gojasiewicz inequality, we have to o b t a i n
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some estimates for semi-algebraic functions on semi-algebraic sets and apply t h e m to the quasipolynomial det { I A - A(n) + B(n)P(n)e -h)'}. Put An(n) = s u p { ~ k I det{Za - A(n) + B ( n ) P ( n ) ~ -"~} = 0}, W ( F , 7 ) = {n • R'~ld[n,N{detB}] < 1P(1 + In12)n} where F > 0, 7 • R, N{H} is the set of the real zeros of a polynomial H and d(n, M ) is the distance between a point n and a s e t M C R '~ (if N { d e t B} =(~ t h e n W ( F , 3 ' ) = ( ~ for all P > 0 and 3' • R). In Section 5 we prove the following Theorem
1.2. Assume that A and B satisfy the conditions Vn • W ( < , ~ l )
a0(n) < 0,
(5)
I det B(n)l 2 _> ~2(1 + Inl~)~ 2,
(6)
vn • R ~ 1 - hA0(n) _> ms(1 ÷ Inl~) ~3
(7)
vn • R'~\w(@I, ~ )
where ¢)s,¢)2 > 0, ¢)3 • (0,1], ~ s , ~ 2 , ~ 3 • Q and ~3 -< 0. Assume also that R > 0, (I)2(I)3R > e, r • Q, ~2 + ~3 + r _> 0 and B' is the adjoint matrix for B (if rn = 1 t h e n we set B ' = 1). T h e n system (2) is stabilizable in C4-°° and
P(n) -
1 e
det B(n)R(1 ÷ Inlb r B'(n)~ ~ u ) I det B(n)l~Rh(1 ÷ Inl~) r ÷ 1
(s)
is a stabilizing matrix for this system.
In Section 3 we prove that conditions (5), (6) are necessary for (9) and (7) is necessary for (10). Theorem
1.3. If system (2) satisfies the following two conditions Vn • R'"
[det B ( n ) = 0 W • R r~
:- A0(n) < 0],
1 i 0 ( n ) < )7
(9) (10)
then this system is stabilizable in C~ °°. Moreover, condition (9) is necessary for stabilizability of (2) in C4-°°, and in the case m = 1, condition (10) is also necessary for stabilizabilty of this system in C~ °°. In addition, in R e m a r k 5.4 we show t h a t for each rn E N there exist polynomial matrices A ( m x m) such t h a t condition (10) is necessary for the stabilizability of system (2) in C4-~. Finally, in R e m a r k 5.5 we show t h a t for the stabilizing control ~t corresponding to the matrix P constructed in Theorem 1.2 we have ~t E C~8 and II~t(.,t)ll _< #(t)IIIwlll q for some s E No where # E C[0, + c o ) , #(t) , 0 as t , +co. Note that, as a rule, a control of the form (3) with a di.ffer'er~tialoperator P(D,) destabilizes (2) and it can be unbounded. Note also that the problem of stabilizability by feedback control without delays (h = 0) was investigated in [5, 6] for equations and systems of the form (2).
L.V. Fardigola / Central European Journal of Mathematics 2 (2003) 141 156
2
The
sketch
of our
145
study
Consider system (2) with a control of the form (3):
- A (D.) ~ ( . , ~) - B ( D . ) P (D.) ~ ( . , ~ -
Ot
h),
. • E ~, t > h,
(11)
under the initial condition
~(.,t)
= ~o(.,t),
(12)
• • R r~, t • Eo, hi.
wherew•C~r,w °• eq,p,q•N0. Applying (formally) the Fourier transform (with respect to z) to problem (11), (12), we obtain
t>h,
- A(o->(o-, t ) - B(o-)P(o->(o-, t - h), at ~(o-,t) = ~°(o-,t), t • [0, hi,
(13)
(14)
where ~ ( . , t ) = Z ~ ( . , t ) , ~0 = Z~0. Assuming that V(cr, t) = e-tA(~)v(cr, t), V°(cr, t) = e-tA(~)vO(cr, t) we reduce problem (13), (14) to the following problem
av(o-,t) - B ( ~ ) P ( ~ ) v ( ~ , t - h), at v(o-,t) = v°(o-,t), t • [0, h].
t > h,
(15) (16)
Now we put
P(G) ~ /3(G)B-I(G)C hA(°-)
(17)
where /3 is a scalar function, /3/det B • ~V[. Substituting P in (15) we obtain
av(~,t)
de
+/3(~)v(~,t-
h) = 0,
(18)
t > h.
Set k(/3, t)
= 0 if t < 0, k(/3,0) = 1, k(/3,t) = Ji~)e~t/({ +/Je-h~)d{ if t > 0 where c is greater then the suprernurn of the real parts of the roots of the quasipolynornial + / J e -h~ (k(/3, t) does not depend on c). Here and henceforth throughout the paper we let J)~)f(A) dA = V.P. f+o~ f(c + i#)d#, V.P. means the principal value of the integral. According to [1, Theorem 6.2, 6,4] and Lernrna 4.4 we conclude that
v(~,t)-
k(9(~),t-h)v°(~,h)-9(~)
~ohk(9(o-),t-~--h)V°(o-,~-)a~ -,
is a unique solution of (18), (16), cr E R 'r~, and V(cr, Setting K(cr, t) = etA(~)k(/3(cr),t) we obtain that
_> h, (19)
.) E C[O, +co), V(cr, .) E US(h, +co).
h
~(~,t) - K ( ~ , t -
h ) ~ ° ( ~ , h ) - 9(~)
0•0
K(~,t-
7- h>°(~,~)&,
_> h,
(2o)
146
L.V. Fardigola / Central European Journal of Mathematics 2 (2003) 141 156
is a solution of (13), (14), (7 E IR'~, and v((7,.) E C[0, + o c ) , v((7,.) E Cl(h, +(x~). One can see t h a t to study the solution (20) of problem (13), (14) we should investigate some properties of the roots of the quasipolynomial ~ ( / 3 , ~ ) = ~ + ~e -h~ where /3 depends on (7. It is not very easy. But if we consider an arbitrary matrix P E 3V[ t h e n we should investigate properties of the roots of det H(%, (7) where H(%, (7) = I% - A((7) + B((7)P((7)e -ha. It is a more complicated problem. Note t h a t if P has the form (17) t h e n H(%,(7) = ~ ( / ~ ( ( 7 ) , I ~ - A((7)) and detH(%,(7) = 0 e=~ [~{0 ~ C ~ 0 ~ C ~(/~((7),{0) = 0 A det(I%0 - A((7)) = 0 A ~ = ~0 + {0]. Thus in this case the zeros of H are d e t e r m i n e d by the zeros of ~ and the s p e c t r u m of A. Finally, applying the inverse Fourier transform with respect to (7 to the solution v of problem (13), (14) we obtain a solution of problem (11), (12) and study its properties. Thus to investigate the stabilizability problem under consideration we (1) analyze conditions (5) (7) of T h e o r e m 1.2 and conditions (9) (10) of T h e o r e m 1.3 (Section 3); (2) study the zero dispositions ofl}((/~, ~) and properties of k(/~,t) and K((7, t) (Section 4); (3) build and study a stabilizing matrix P of the form (17) (Section 5); (4) apply obtained results to the telegraph equation, the wave equation, the heat equation, the SchrSdinger equation and another model equation (Section 6).
3
An analysis of the conditions of Theorems
1.2 a n d 1.3
In this section we prove that conditions (5), (6) are necessary for (9) and (7) is necessary for (10). L e m m a 3.1. Suppose t h a t condition (9) holds for a polynomial matrix (rn x rn) A. T h e n there exist (I)1, F > 0, ~ 1 , 7 E Q such t h a t (5) is true. Moreover, there exist F > 0, 7 E Q such that
A0((7) < - r ( 1 + 1(71~)~,
(7 ~ W ( ~ ,
~),
(21)
P r o o f . We can represent the set W((I)I,~I) in the form W ( d p l , ~ l ) = {(7 E IR'r~ 13rl E R'r~Kdet B(~) = 0 A I~ -- (71 < r ( 1 + 1(712)~1]}. Let ~(r) = inf{d > 0 13(7 E R'~3~I E R'~[d = 1(7 - ~]1 a X0((T) _> 0 a det B(~]) -- 0 a 1(71 -- ~]}. From (90 it follows t h a t L,(r) > 0 (r _> 0). It is easy to see that for every r0 > 0 there exists C(r0) > 0 such t h a t L,(r) _> C(r0), r E [0, r0]. Using the Tarski Seidenberg t h e o r e m [17] and its corollaries [8, A p p e n d i x A] we obtain t h a t u(r) = + o c as r ) + o c or u(r) = Nr2~1(1 + o(1)) as r ) + o c where N > 0, ~1 E Q. Therefore L,(r) _> 2(I)1 (1 + r2) vl, r _> 0, where (I)~ > 0, ~ E Q. Hence (5) holds. Applying the Tarski Seidenberg t h e o r e m [17] and its corollaries [8, A p p e n d i x A]
to ~(r) - supTA0((7) I (7 ~ w ( ~ , ~ ) a 1(71 -- ~} we conclude that ~(~) _< r ( l + ~ F , r _> 0, where F > 0, ~ E Q. We use the same reasoning for obtaining this estimate as we did for obtaining the analogous estimate for L,(r). Thus (21) is true as was to be proved. Lemma
3.2. Let (I)l,P > 0, ~1,~ E Q be constants such t h a t (21) holds. T h e n there
L.V. Fardigola / Central European Journal of Mathematics 2 (2003) 141 156
147
exist (I)2 > 0 a n d ~2 • Q such t h a t (6) is true. P r o o f . From [9, L e m m a 2], we get
I det B( )I _> ~ (1 +
1 12)
(diet, N { d e t B}]) 9 ,
cr • R '~,
(22)
where 13 > 0, (~ • Q , / 3 • Q, m o r e o v e r , / 3 > 0 if N { d e t B} ¢ (~ and /3 = 0 otherwise. Hence (6) holds. The l e m m a is proved. Lemma true.
a . a . Let (10) hold. T h e n there exist (ha • (0, 1] and ~3 • Q such t h a t (7) is
P r o o f . Taking into account (10) a n d applying the Tarski Seidenberg t h e o r e m [17] a n d its corollaries [8, A p p e n d i x A] to #(r) = inf{1 - hA1 I 3A2 • R 3or • R ' ~ [ d e t ( A ( c r ) (A1 + iA2)I) = 0 A Icrl = r]} we conclude t h a t (7) holds. We use the same reasoning for obtaining this e s t i m a t e as we did for obtaining the analogous e s t i m a t e (5). The l e m m a is proved.
4
Properties of the resolving function
K(t, ~)
In this section we study the complex roots of the q u a s i p o l y n o m i a l fie(/3, ~) and e s t i m a t e the functions k(/3,t) a n d K(cr, t). D e n o t e A(/3) = sup { ~ Lemma
[ ffC(/3,~) = 0}.
4.1. For all/3 • C we have A(/3) _> - 1 l b . Moreover, A(/3) = - 1 / h iff/3 = 1/(eh).
P r o o f . Let h = 1. Let us prove t h a t for all /3 • C \ { 1 / e } there exists { • C such t h a t :}f(/3,{) = 0 and R{ > - 1 . Suppose /3 = be~ where b _> 0, ~ • I - r e , r c). Assume F = F1 [..J F2 [..JF3 [..J F4 where F1 = { - 1 t•
I-1,N]},
Ca =
i(t-
~) I t • I-re, re]}, F2 = {t + i ( - r c + ~ ) l =
I-1,N]},
N > 0. Assume also t h a t D is the d o m a i n b o u n d e d by the curve F. Let N > 0 be sufficiently large. W h e n { goes a r o u n d the curve F, the a r g u m e n t i n c r e m e n t of J£(/3, {) is equal to 2re if 0 _< be < 1 and it is equal to 4re otherwise. F r o m the a r g u m e n t principle we conclude t h a t there exists at least one zero of ~(/3, ~) in D. If fl = 1/e t h e n ~ = - 1 is a zero of J£(/3, {). Thus for h = 1 we have A(/3) _> - 1 . Moreover, A(/3) = - 1 iff/3 = 1/e. Let 0 < h ¢ 1. Set ~ = ~/h. T h e n A(/3) 2 - 1 / h . Moreover, A(/3) = - 1 / h iff /3 = 1/(eh). The l e m m a is proved. Lemma
4.2. If 0 _3 _< 1/(eh) t h e n A(/3) _< 0 a n d A(/3) _< ( - 1 + v/e(1 - e/3h))/(e/3h2).
P r o o f . Set { = ( x + i y ) / h , x ¢ R, y E R. We have gf(/3,{) = 0 i f f x + / J h e - * c o s y = 0 and y - ~3he-x sin y = 0. It is easy see t h a t if the second equality is satisfied t h e n y = 0 or x _< ln(/Jh) _< - 1 _< hA(/3) (see L e m m a 4.1). Let y = 0. T h e n the first equality is valid iff x + / J h e - * = 0. Since 0 _3 _< 1/(eh) t h e n this equality holds at least for one z. Let
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L.V. Fardigola / Central European Journal of Mathematics 2 (2003) 141 156
x0 be the m a x i m u m o f x such t h a t x + / ~ h e -~ = 0. It is easy to see t h a t - 1 _< x0 _< 0 and hA(/~) -- x0. Because of e -~ _> x 2-[- ( 2 - e ) x - [ - 1 , x • [ - 1 , 0 ] , we obtain t h a t if
~h(x 2 + (2 - e)x + 1) _> - x a n d x _> - 1 t h e n x _> x0. It follows t h a t hA(/~) -- x0 _< xl where xl is the greatest root of the e q u a t i o n ~h(x 2 + (2 - e)x + 1) = - x . It is easy to see t h a t xl _< ( - 1 + V/e(1 - e~h))/(e~h). The l e m m a is proved. Lemma
4.3. Let 0 _3 _< 1 / ( e h ) , c(/~) • R, ,k(/~) < c(/~) < 1lb. T h e n
08 ~
< - I~(~)1 ~÷1 I ~ ( ~ ) - ~(~)1 ~(~÷1)'
t>o '
~>0, -
(23)
where M~ > 0, w > 0. P r o o f . Since ]~(~1 + i~2,/~)] 2 is a real analytic function of (~1,~2,/~) on R a t h e n from [11, Section 17] we conclude t h a t ]~(~1 + i~2,/~)] _> f~(d[(~l,~2,/~),N]) ~, I~11 _< l / h ,
1~21 _< 2/h, 0 <_ ~ <_ l/(~h) where f~,w > 0, N = {([1, [2, /~) • R 3 I X(~l + i~2,~) -- 0}. Therefore 2
1
I#1 ~ ~, 0 ~ ~ ~ ~ .
~ ( ~ ( ~ ) + ~#, ~) _> ~ I~(~) - x(~)l" ,
(240
We have k(~,t) = Ji~(~)) et~/~ d~ - ~ Ji~(~)) e(t-h)~/(~J~(~'~))d~' t > 0. Taking into account the inequality 1c(/3) + ip[ <_ 21~(c(/~) + i#,fl)[, I#1 _> 2 / h , and (24) we get
Ik(9,t)l _< e ~(9/' { 2 ~ + ~
2
e(~)2 + #2 +
-2/h ~le(~)l le(~) - ~(~)1 ~
Thus (23) holds for I = 0. Since (0/0/3) ~ ( 1 / ~ ( / 3 , ~ ) ) = (--1)~l!e-~h~/(~(~,~)) ~+1 t h e n
Olfl
< - Kle~(~)t 21+1
(~(9)2 + #2)(~+1//2 +
d#
-2/~ ~ 1~(9)- a(9)l ~
, t > 0,
where Kl > 0, I > 0. It follows from here t h a t (23) is true for I > 0. The l e m m a is proved. Lemma (5)
4.4. Let ~1, ~2 > 0, ~3 • (0, 1], 7)1, 7)2, 7)3 • Q, 7)3 _< 0 be constants such t h a t
(7) hold.
1 ]det B(cr)]2Rh (1 + 1~12)r where R > 0, ~ 2 ~ 3 R > e, Let ~(cr) = eh ]det B(~)12Rh (1 + 1~12)r + 1
r • Q, 7)2 + 7)3 + r _> 0, and let K(cr, t) = etA(~)k(/3(cr), t). T h e n for each multi-index (~
IID~K(~,~)II ~ ~,~, (1 Jr 1~12)~'~' ~-~<1+,~,/' where L > 0, I • Q, ~1~1 > 0, ~Cl~I • N. P r o o f . Let us prove t h a t for each multi-index >
(~,~) • R '~ ~ [0, + ~ ) ,
(25)
L.V. Fardigola / Central European Journal of Mathematics 2 (2003) 141 156
149
where 2k/~l~l> 0, nl~ I E R. At first we consider (or, t) E (R'r~\W((I)I,~I)) x [0,+oc). W i t h regard to L e m m a s 3.1 3.3 we get V / 1 - e£(cr)h < (1 + 1~12)~/(O2Rh). Taking into account L e m m a s 4.2, 3.3 and setting c(/3(cr)) = ( - 1 + eV/1 - e/3(cr)h)/(e/3(cr)h 2) we obtain that hA(/3(cr)) < According to L e m m a 4.2 we have Ic(/~(cr))- a(~(~))l _> ( ~ - v~)~/1- e/~(cr)h/h >_ ~(1 + I~lb-b÷'r/~ where b -- deg(det B), K > 0. Now assume that (or, t) ~ W ( ( I ) ~ , ~ ) x [0,+oc). Set c(cr) = L' (1 +l~lb ~" where L ' = min{1/h,r/2}, l'= m i n { 0 , 7 } . Hence A0(cr) + c(cr) < -12(1 + I~1~/~/2. According to L e m m a 4.2 we have
I~(~(~)) - a(~(~))l _> r (1 ÷ I~1~)~/2.
All this implies that for each multi-index p we have I D ' ~ ( ~ ) I _< 7~,,, (1 + I~lb r'~', cr ~ R '~, where 7~1~1, rl~ I ~ R. W i t h regard to L e m m a 4.3 we have t h a t for each multiindex ff an estimate of the form (26) is valid. Using [7, Chapter 1, §6] we obtain I I D g ( ~ ) l l _< ~ , , , ( 1 ÷ [~[)"~'~'e~a°(~), ~ ~ N ~, where Adl~ I > 0, m M e R. Therefore (25) is true. The l e m m a is proved.
we
Taking into account the estimate for get
D~e tA(~) obtained in the proof of this l e m m a
L e m m a 4.5. Let (I)1, (I)2 > 0, (I)3 E (0, 1], ~1~2, ~3 E Q, ~3 -< 0 be constants such that (5) (8) hold. T h e n for P defined by (8) we have P E ~V[.
5
C o n d i t i o n s for stabilizability
S t a t e m e n t 5.1. Assume t h a t for system (2) conditions (9) and (10) hold. T h e n there exists a matrix ( m x m) P E ~ such that for any p E N0 there exist q E N0 and a continuous function p(t) on [0, ÷ o c ) , p(t) > 0 as t > ÷ o c , such t h a t for each solution w of system (2) with control (3) under the initial condition w E e q we have w E C~ and
vt > h
II~(,t)l15 s ~(t)II1~111~.
(27)
P r o o f . It follows from L e m m a s 3.1 3.3 t h a t conditions (5) (7) hold. Let P be a matrix of the form (8). T h e n B(cr)P(cr) - /3(cr)I for /3 defined in L e m m a 4.4. From L e m m a s 4.4, 4.5 we have that P E ~ and estimate (25) is true. Let g' be the dual space for g. Let p E N0 be fixed. Assume that q _> p + ~+,r~+l + 7, w0 e e~ where ~+,r~+l is the constant from estimate (25). For system (12) consider a problem under initial condition (12). W i t h regard to L e m m a 4.4 we conclude that v defined by (20) is a solution of (13), (14) in g' and v(cr,.) E C [ 0 , + o c ) , v(cr,.) E Cl(h, ÷oc). Hence w(.,t) = ~ - l v ( . , t ) is a solution of (11), (12) in g'. Now we prove t h a t w E C~ and (27) is true. Henceforth t h r o u g h o u t the proof we assume t h a t x E R '~, cr E R '~, t _> h. Let e(x) be an infinite differentiable function on IR'~, let s u p p e C { x E R ' ~ I I x I _< 1}, and let E e ( x - k ) - 1. Denote w~(x,t)° IEZ ~
150
L.V. Fardigola / Central European Journal of Mathematics 2 (2003) 141 156
~(.>o(.
+ k,t), %(.,t) o = ~Yw°(.,t). W i t h regard to (20) we have
vk(cr, t ) = K ( c r , t-h)v°(cr, h)-B(cr)P(cr)
/0
K(cr, t - r - h ) v ° ( c r ,
r) dr(g'), t > h (28)
is a solution of (13), (14) w i t h v ° = v ° therefore w k ( x , t ) - (~Y-lvk(.,t))(x) is a solution of (11), (12) in $' with w0 __ wk0 where k • Z '~. Obviously, IIwO(.,T)II q <_ MIIwO(.,T)IIq (1 + Ikl) ~, T • [0, h], where M > 0 does not d e p e n d on r • [0, h] and k • g 'r~. T h e n we have craD~ (cr~v°(cr, t)) _< c II1~°111~(1 + Ikl)~ where C > 0, 191 + I~1 -< q, I~1 -- ~ + 7 + 1. W i t h regard to (28) a n d L e m m a 4.4 this gives D~ (cr~vk(cr, t)) <_ C'lllw°lll q (1 + Icrl)-'~-le-tC(l+l~l)'(l + Ikl) ~ where C' > 0,
I~1 -- deg B +ec~+.~+s +rt + 1,191 _< p. Applying the inverse Fourier t r a n s f o r m with respect to or, we get D~wk(x,t) <_ C*u(t)IIIw°lll~ (1 + I,I)-('~÷~÷~)(1 + IklF where C* > 0, u(t) = (1q-t) 1/I if I < 0 a n d u(t) -- e x p { - t L } otherwise. Since ( l + l k l ) _< (1 + 1 , + k l ) ( l +1,1) t h e n D 2 w k ( x , t ) _< C*u(t)IIIw°lll~ ( l + l . + k l F ( l + l . I ) -'~-1. Hence w ( . , t ) - 2 k ~ . w k ( . - - k , t ) , w • (?~r and (27) is true. It remains to show t h a t the solution w is unique in C~r. It is sufficient to prove t h a t for system (11) the initial p r o b l e m under the condition w(x,t) = 0, x • R 'r~, t • [0, hi, has only the trivial solution w in C~r. Let w be a solution of this problem a n d let w • C~r. According to the initial condition we have w (x, t) - 0 on IR'~ × [0, hi. Suppose w (x, t) - 0 on R '~ x [(k - 1)h, kh] and prove t h a t w(x,t) = 0 on R '~ x [kh,(k + 1)hi, (k • N). It is easy to see t h a t each solution of system (11) on R '~ x [kh, (k + 1)hi under the initial condition w(x,t) = 0, x • R 'r~, t • [(k - 1)h, kh] is a solution of the Cauchy problem
0w (z, t) Ot
-A(Dx)w(x,t),
w(x,0)=0,
x • R '~, t • [ k h , ( k + l ) h ] .
(29)
W i t h regard to [7, §4] t h a t gives w(x,t) = 0 on R '~ x [kh, (k + 1)hi. Thus w(x,t) = 0 on R '~ x [0, + c o ) . The s t a t e m e n t is proved. It follows front the proof of S t a t e m e n t 5.1 t h a t conditions (5) (7) are sufficient for stabilizability of (2) (moreover, in this case P defined by (8) is a stabilizing control). Thus T h e o r e m 1.2 is proved. Statement 5.2. If for system (2) there exists or0 • IR'~ such t h a t Ap(cr0) _> 0 for all matrices (m x m) P • 3V[ t h e n this system is not stabilizable in C4- ~ Proof.
Let P
E 3V[ and Ap(cr0) _> 0 for some cr0 E R 'r~. Let det{%0I - A(cr0) +
B(cro)P(cro)e -ha°} = 0 and A0(cr0) = ~A0 _> 0 for A0 ¢ C. Also let v0, I~01 -- 1, be a vector such t h a t ( k 0 I - A(cr0) + B(Cro)P(cro)e-ha°)Vo = 0. Consider system (2) with the control u ( . , t ) = P ( D x ) w ( . , t h), i.e. the system of the form (11), under the initial condition w(x,t) = exp{tA0 + i(x, Cro)}Vo, x • R '~, t • [0, hi, where (.,.) is the scalar p r o d u c t corresponding to the E u c l i d e a n n o r m in R '~. It is easy to see t h a t
w(x,t) - exp{tA0 + i(x, ~0)}v0, x • R '~, t • [0, + c o l , is a solution of this problem. Since
I w ( z , t ) l - e x p { t ~ a 0 } and ~a0 _> 0 t h e n
lira
t----+--co
IIw(.,t)ll ° > 0, i.e., condition (4) is not
satisfied. Therefore system (2) is not stabilizable in C4- ~ . The s t a t e m e n t is proved.
L.V. Fardigola / Central European Journal of Mathematics 2 (2003) 141 156
151
From this statement and Lernrna 4.1 we obtain C o r o l l a r y 5.3. If the system (2) is stabilizable in C~- ~ t h e n (9) holds. Moreover if, in addition, rn = 1 then (10) also holds. W i t h regard to Statement 5.1 and Corollary 5.3 we conclude that Theorem 1.3 is true. R e m a r k 5.4. For all rn E N, it is possible to construct a polynomial matrix A (rn x rn) such that for some or0 E R '~ it has the form aI where a >_ 1/h. Let P E 3V[ and let #j, j = 1,m, be the eigenvalues of the matrix B(cr0)P(cr0). T h e n d e t { / ~ 0 I - A(cr0) + TD+ B(cr0)P(cr0)} = H / = I ( ~ - a + #/e-h~). W i t h regard to L e m m a 4.1 we obtain that Av(cr0) > 0. Taking into account Statement 5.2 we conclude that condition (10) is necessary for stabilizability of the system (2) with this matrix A in C~-°+. R e m a r k 5.5. Let conditions (9), (10) be valid for system (2). The stabilizing matrix P E 3V[ that has been found in the proof of Statement 5.1 has the form (8). It can be shown t h a t if u(x,t) = P ( D , ) w ( x , t h) t h e n u(x,t) = W ( x , h , t ) where W ( x , r , t ) is a solution of the Cauchy problem
OW(z'T't)-A(Dx)W(z,T,t),
z E R '~, ~ ~ [0, hi,
[Idet B (D~)I 2 2~h (1 + ID~12)'r + 1] B' (D~)W(x, 0, t) =detB(Dx)R(l+lDxl2)'rw(z,t-h), z E R '~,
(30)
(31)
and t _> h is a parameter. From [4, Corollary 2] we obtain that for each s E N0 there exist p E N0 and C > 0 such t h a t for w(.,t) E C~ (t >_ h) the function W(z,T,t) is the unique solution of the problem c I I w ( . , t - h)ll~, ~ ~ [0, h], t _> have t h a t for each s E N0 there #(t) > 0 as t > + o c , such
s (30), (31) for t >_ h, W(.,T,t) E C 7s and IIW(',~, ~ )11~ _< h. Taking into account L e m m a 4.5 and Statement 5.1 we exist q E N0 and a continuous function #(t) on [0, + o c ) , that for each solution w of system (2) with control (3)
under the initial condition w E e q we have ~t E Cv8 and
6 6.1
II~(,t)ll; _< #(t)II1~111~, t _> 0.
Applications The telegraph equation and the wave equation
Consider the system
( cOwl
- ~
+ b~ (D~) ~ + b~ (D~) ~ z E R 'r~, t > h ,
( 0t
- 2 k - ~ , ~ + z X ~ + b ~ (D~) ~ + b~ (D~) ~ OZ
(32)
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L.V. Fardigola / Central European Journal of Mathematics 2 (2003) 141 156
where B
{bij }i,j=l 2 is a p o l y n o m i a l matrix, k • R. This system corresponds to the telegraph equation. It is easy to see t h a t =
I~1 _> Ikl
-Io-I~ 2k
k + @k~-I~1 ~, I~1 ~ Ikl
Let us consider three cases: i) k < 0, ii) k = 0, iii) k > 0. i) Let k < 0. T h e n A0(0r) < 0 if 0r ¢ 0 and A0(0) -- 0. Taking into account T h e o r e m 1.3 we o b t a i n Statement
6.1. Let k < 0. T h e n system (32) is stabilizable in C4- ~ iff det B(0) ¢ 0.
Let us find a stabilizing control of the form (8) for this system, given t h a t det B(0) ¢ 0. P u t ~2 = d(0, N { d e t B } ) / 2 , N" = {0r E R '~ II det B(~)I _< ~ } , ,(~) -- sup{~ ~ a; I I~1-r}, r _> 0. Using the same reasoning as for obtaining e s t i m a t e (5) in Section 3 we conclude t h a t , _< ( P s ( I + r 2 ) ~*, r _> 0, where (Ps > 0, ~ • Q. Hence W((Ps, ~ ) n N" n N { d e t B}. Therefore estimates (5), (6) hold w i t h these (Ps, (P2, ~ a n d ~2 = 0. Obviously, e s t i m a t e (7) is true for ~3 = 1, ~3 = 0. P u t R = 2 e / ~ 2 , r = 0. Erom T h e o r e m 1.2 we conclude that
P(~)-
2detB(~) B,(~) [ ~ Idet g(~)122& + ~2
s i n ( h v / , c r , 2 - k 2) (
(32) in the case k < 0. ii) Let k = 0. Then system (32) corresponds to
/
-k
1
is a stabilizing m a t r i x for
th~ ~
~v~teo~.
We have A0(~) = 0,
0r E R a. F r o m T h e o r e m 1.3 we o b t a i n Statement
6.2. Let k = 0. T h e n system (32) is stabilizable in C~- - o o iff V0r E R 'r~
det B(0r) ¢ 0.
(33)
Let us find a stabilizing control of the form (8) for this system, given t h a t (33) holds. W i t h regard to (22) we conclude t h a t there exist ~2 > 0, ~2 E Q such t h a t [det B(cr)[ 2 _> (P2(1 + [cr[2)~2, cr E R 'r~. Hence estimates (5), (6) hold w i t h these (P2, ~2 and a r b i t r a r y ~1 > 0, ~1 E Q. Clearly, e s t i m a t e (7) is true for ~3 = 1, ~3 = 0. P u t R = 2 e / ~ 2 , r = 0. From T h e o r e m 1.2 we conclude t h a t
P(~)-
2detg(~) g'(~) ( cos(hl~l) (sin(hi-I))/1"1 "~ I Idet g(~)122& ÷ ~2 ~,-I~1 sin (hl~l) cos (hl~l) )
is a stabilizing m a t r i x for (32) in the case k = 0.
L.V. Fardigola / Central European Journal of Mathematics 2 (2003) 141 156
153
iii) Let k > 0. T h e n k _< A0(cr) _< 2k, cr E R'% A p p l y i n g T h e o r e m 1.3 a n d R e m a r k 5.4 we o b t a i n Statement
6.3. Let k > 0. If h < 1 / ( 2 k ) a n d (33) holds t h e n s y s t e m (32) is stabilizable
in C4- ~ . If (33) is not t r u e t h e n this s y s t e m is not stabilizable in C4- ~ . Let us find a stabilizing control of t h e f o r m (8) for this s y s t e m , given t h a t h < 1 / ( 2 k ) a n d (33) holds. W i t h r e g a r d to (22) we c o n c l u d e t h a t t h e r e exist dp2 > 0, ~2 E Q such
that I det B(~)I 2 > ¢)2(1 + 1~12)~2, ~ ~ R ~. Hence e s t i m a t e s (5), (6) hold w i t h these do2, ~2 a n d a r b i t r a r y dol > 0, ~1 E Q. Obviously, e s t i m a t e (7) is t r u e for dp3 = 1 -
2kh,
~3 = 0. P u t R = 2 e / ( e 2 ( 1 - 2kh)), r = 0. F r o m T h e o r e m 1.2 we c o n c l u d e t h a t
P(~)-
2det
B(cr)
g,(cr)[ekhsin(hv/lcrl2-k2) ( -k 1 I
I det g(~)122eh + ~2(1 - 2kh)
+1o_12
it _1o_12/)
is a stabilizing m a t r i x for (32) in t h e case k > 0.
6.2
The
Consider
heat equation
the heat equation Ow -Ot
A w + B ( D . ) u,
. ~ R r', t > h,
where B is a p o l y n o m i a l . It is easy to see t h a t A(cr) = A0(cr) =
(34)
-I~12 on
R r'. A c c o r d i n g
to T h e o r e m 1.3 we o b t a i n Statement
6.4. E q u a t i o n (34) is stabilizable in C4- ~ iff B ( 0 ) ¢ 0.
Let us find a stabilizing control of t h e f o r m (8) for this s y s t e m , given t h a t B ( 0 ) • 0. P u t do2 =
d(O,N{detB})/2.
Using t h e s a m e r e a s o n i n g as in e x a m p l e 6.1 (k > 0) we
c o n c l u d e t h a t e s t i m a t e s (5)
(7) hold w i t h s o m e dpl, ~1, this dp2, ~2 = 0, dp3 = 1, ~3 = 0. 2B(cr)e-hl~l 2 P u t R = 2e/alP2, r = 0. F r o m T h e o r e m 1.2 we c o n c l u d e t h a t P(cr) = is
IB(~)122eh + +2
a stabilizing m a t r i x for (34).
6.3
The
Consider
SchrSdinger
equation
the SchrSdinger equation Ow
Ot
= i A ~ + B (D~)~,
• ~ R ~, ~ > h,
(3~)
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L.V. Fardigola / Central European Journal of Mathematics 2 (2003) 141 156
where B is a polynomial. It is easy to see t h a t A ( a ) = - i l a l 2, A0(a) = 0 on IR'~. According to T h e o r e m 1.3 we o b t a i n Statement
6.5. E q u a t i o n (35) is stabilizable in C4- ~ iff g a • IR'~ B ( a ) ¢ 0.
Let us find a stabilizing control of the form (8) for this system, given t h a t B ( a ) a • R '~. W i t h regard to (22) we conclude t h a t there exists dP2 > 0, ~2 • Q such IB(a)l 2 _> dP2(1 + la12) ~ , a • R '~. Hence estimates (5), (6) hold with these dp2, ~2 arbitrary (I)x > 0, ~x • Q. Evidently, e s t i m a t e (7) is true for (I)3 = 1, ~3 = 0. R = 2e/alP2, r = 0. From T h e o r e m 1.2 we conclude t h a t P ( a ) =
# 0, that and Put
2B(a)e-ihl~l 2 is a IB(a)122eh + +2
stabilizing m a t r i x for (35).
6.4
A Model e q u a t i o n
Consider the e q u a t i o n
Ow
O2w
O2w
a-7 - a.-7 + o.-7 Clearly, A ( a ) -
10(a) -
O2w
O2u
2-57xo.2 + ~ 1-
(<-
o.xo.--------7~+ '~'
a2) 2, B ( a ) -
* • R2' ~ > h
(361
det B ( a ) - < a 2 + 1. Hence (10)
holds for h < 1. We have 1 0 ( a ) - - ( a ~ - a~ + 1 ) / a ~ _< - 1 < 0, a • X { d e t B } , therefore
(9) is valid. Taking into account T h e o r e m 1.3 we get t h a t system (36) is stabilizable in
C4-~. Now we construct a stabilizing m a t r i x of the form (8) for this system.
We have
a 0 ( a ) _< - 1 / 2 , a • W ( + x , ~ x ) , where +x = 1/4, ~x = - 1 / 2 , F = 1/2, 7 = 0 . Let us obtain an e s t i m a t e of the form (22) for B ( a ) = axa2 + 1. At first assume t h a t la21 _> I
1/4.
T h e n lax + l/a21 _> d[a,N{detB}].
Hence laxa2 +
112 Z la2121ax + 1/a21 Z la212(d[a,N{detB}]) 2. Since la21 Z (laxl + la21)/2 Z lal/2 Z (1 + la12) x/2/10, it follows that I 1__ (1 + lal 2) (d [a,X{det B}]) 2 . -
100
(37)
Obviously, if la21 _< lax l a n d lal _> 1/4 then the e s t i m a t e (37)is also true. If lal _< 1/4 then d[a,N{b}] _< 2 therefore (1 +lal2)(d[a,N{detB}]) 2 _< 4 and laxa2 + 11 _> 1 - l a x a 2 1 _> 15/16. All this implies t h a t (37) is true for all a • R 2. Therefore [axa2 + 112 _> 1/1600, a • R2\W(dpx,~x). Hence dP2 = 1/1600, ~2 = 0. On the other hand, IA0(a)l _< 1, a • R 2 Therefore ¢~3 = 1 - h, ~3 = 0. P u t r = 0, R = (1601e)/(1 - h). From T h e o r e m 1.2 we conclude t h a t P ( a ) -
1601(axa2 + 1) eh(x-(~*-~)~) is a stabilizing m a t r i x 1601eh(axa2 + 1) 2 + (1 - It)
for system (36).
Acknowledgments The a u t h o r was partially s u p p o r t e d by INTAS, G r a n t #99-00089.
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155
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[17] A. Seidenberg: \A new decision method for elementary algebra", Ann. Math., Vol. 2, (1954), pp. 365{374. [18] J.M. Sloss, I.S. Sadek, J.C. Bruch, S. Aldali: \Stabilization of structurally damped systems by time-delayed feedback control", Dyn. Stab. Syst., Vol. 7, (1992), pp. 173{178. [19] C.C. Travis and G.F. Webb: \Existence and stability for partial functional di®erential equations", Trans. Am. Math, Soc., Vol. 200, (1974), pp. 395{418.
CEJM 2 (2003) 157{168
Rigidity and ° exibility of virtual polytopes G. Panina¤ Institute for Informatics and Automation V.O. 14 line 39 St.Petersburg, 199178, Russia
Received 15 November 2002; revised 15 January 2003 Abstract: All 3-dimensional convex polytopes are known to be rigid. Still their Minkowski di¬erences (virtual polytopes) can be ®exible with any nite freedom degree. We derive some su¯ cient rigidity conditions for virtual polytopes and present some examples of ®exible ones. For example, Bricard’s rst and second ®exible octahedra can be supplied by the structure of a virtual polytope. ® c Central European Science Journals. All rights reserved. Keywords: Virtual polytope, rigidity, ° exion, Bricard’s octahedron MSC (2000): 52C25
1
Introduction
¤
Roughly speaking, a virtual polytope (introduced originally in [5]) is the Minkowski di®erence of two convex compact polytopes. This can be made precise in di®erent ways ([14], [5], [7], see also [15] for a particular case of virtual polytopes); one can even get down from Picard groups of toric varieties keeping in mind the correspondence between polytopes and vector bundles [4], but all reasonable ways lead to one and the same notion. Virtual polytopes form an abelian group with respect to the Minkowski addition «. In this paper we use a representation of the group of virtual polytopes as a subgroup of the group of units (i.e. the group of all invertible elements) of the ring of polytopal functions. It is possible to give mutually consistent de¯nitions of faces, edges, volume, mixed volume, support function, support oriented planes, and fan for virtual polytopes. All algebraic properties of the above notions for convex polytopes remain preserved when passing to virtual polytopes ([5],[12], [13]). For example, the volume V (¶ K « · L) E-mail: [email protected]
158
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is a polynomial in ¶ and · not only for convex K and L, but also if K and L are virtual polytopes. On the other hand, some classical properties are not preserved: volume of a virtual polytope can be negative, the fan can have non-convex cells, the classical inequalities need additional conditions [9], the Minkowski uniqueness theorem fails [13]. We shall discuss the classical rigidity problems applied to 3-dimensional virtual polytopes. A virtual polytope K is said to be °exible if there exists a nontrivial (not generated by a movement) continuous family of virtual polytopes Kt; t 2 [0; 1]; K = K0 such that all Kt have the same combinatorics and corresponding faces of Kt and K di®er by a movement (see Section2 for the de¯nition of movements for virtual polytopes). Such a family is called a °ex of K . A non°exible virtual polytope is called rigid. The rigidity property is not preserved when passing to virtual polytopes: there exist °exible ones. Some of Bricard’s °exible octahedrons can be viewed as virtual polytopes (see Section6). Besides, there exist simple examples of °exible virtual polytopes with any freedom degree. Their fans are disconnected (see Section4). We derive a su±cient condition of rigidity which re¯nes the rigidity theorem for herissons [15]. Theorem 1.1. A virtual polytope with a convex fan is rigid. I thank V. Alexandrov and I.Sabitov for useful and pleasant conversations.
2
Preliminaries
In this section we give a brief sketch of all necessary constructions. For details see [5], [12], [13]. We limit our discussion to 3-dimensional polytopes. Denote by P the set of all compact convex polytopes in the Euclidean space E3 with a ¯xed origin O (degenerate polytopes are also included). It is a semigroup with respect to the Minkowski addition «. Denote by P ¤ the Grothendieck group of P. The element of P ¤ that is the inverse of K is denoted by K ¡1 . A function F : R3 ! Z is polytopal if it admits a representation of the form F =
X
a i IK i ;
(1)
i
where ai 2 Z, Ki 2 P, and IKi is the indicator function of the polytope Ki : IKi (x) = 1 if x 2 Ki ; 0 otherwise:
(2)
The set of all polytopal functions is denoted by M. It is endowed with two ring operations. The role of addition is played by the pointwise addition, denoted by +. The multiplication is generated by « and is denoted by the same symbol. The unit element of the ring M is obviously the function E = IfOg .
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159
Identifying convex compact polytopes with their indicator functions, we get an inclusion º : P » M. Keeping this identi¯cation in mind, we write for convenience K instead of IK . The following theorem shows that all elements of the semigroup º (P) are invertible in M. Theorem 2.1. (On Minkowski inversion) [5] For any compact convex polytope K , we have (¡ 1)dim K IRelint(sK) « K = E;
(3)
where s is the central symmetry mapping (with respect to the origin O) , Relint(sK ) is the relative interior of the polytope sK (i.e., the interior taken in the a±ne hull of K ). Hence the inclusion P » M induces an inclusion P¤ » M: Remark 2.2. The function (¡ 1)dim K IRelint(sK) is a polytopal function because it can be represented as a linear combination with integer coe±cients of indicator functions of the polytope K and its faces of all dimensions. De¯nition 2.3. The image of the latter inclusion is called the group of virtual polytopes. For convenience we denote it by the same letter P¤ . The sets P ¤ and M do not coincide. For example, any polytopal function of type F = IK1 + IK2 with K1 ; K2 2 P is not a virtual polytope. Necessary and su±cient conditions for a polytopal function to be a virtual polytope are discussed in [12] and [13]. The dimension of a virtual polytope K is the dimension of the a±ne hull of its support. The group G of movements of E 3 acts on the ring M: for u 2 G ; F 2 M, put (uF )(x) = F (u¡1 (x)). Thus it makes sense to speak of movements of virtual polytopes. For a convex polytope K , we obviously have: Iu(K) = u(IK ). De¯nition 2.4. Let K be a virtual polytope. Then there exist convex polytopes L and M such that K = L « M ¡1 . The support function hK of the virtual polytope K is de¯ned to be the di®erence of support functions of L and M : hK = hL ¡
hM :
(4)
This de¯nition is consistent with other de¯nitions of support function [5], [12]. Since we have de¯ned the support function, we have the notion of support oriented planes of a virtual polytope as well. P De¯nition 2.5. [13] Let K 2 M; K = i ai K i with Ki 2 P. Let li (¹ ) be the support hyperplane to Ki with the outer normal vector ¹ . The polytope Ki» = K i \ li (¹ ) is
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called the face of the polytope Ki with the normal vector ¹ , while the polytopal function P K » = i ai Ki» is called the face of the polytopal function K with the normal vector ¹ .
A face of a virtual polytope is a virtual polytope as well. The 0-dimensional, 1dimensional and 2-dimensional faces are called vertices, edges and facets respectively. Virtual faces behave with respect to the Minkowski addition like convex ones: Theorem 2.6. [12] In the above notation, K1» « K2» = (K 1 « K2 )» :
(5)
De¯nition 2.7. The n-volume of an n-dimensional virtual polytope K is de¯ned as the integral with respect to the Lebesque measure: V (K ) =
Z
K (x)dx:
(6)
En
Thus it makes sense to speak of areas of faces and lengths of edges for virtual polytopes. Remark 2.8. A 1-dimensional virtual polytope (virtual segment) K is de¯ned uniquely up to a translation by its length l = l(K ). If l ¶ 0, then K is the segment of length l. Otherwise it is the virtual segment inverse (see Theorem2.1) to the segment of length ¡ l. Therefore for any pair of points A 6= B , there exist 2 virtual segments with vertices A and B. Given two segments K1 and K2 in E 1 , we easily have l(K1 « K2 ) = l(K 1 ) + l(K2 ):
(7)
De¯nition 2.9. A fan § is a ¯nite collection of compact spherical polytopes on the unit sphere S 2 (possibly nonconvex and disconnected ones) such that ° U; V 2 § ) U \ V 2 §; ° [§ = S 2 ; ° U 6= V 2 § ) RelintU \ RelintV = ;: The fan of a virtual polytope is de¯ned below analogously to the classical de¯nition of the outer normal fan. For a virtual polytope K 2 P, its fan §K is the collection of spherically polytopal sets f§K (¸ )g, where ¸ ranges over the set of faces of K , and §K (¸ ) = clf¹ jK » = ¸ g(cl denotes the closure:)
(8)
These polytopal sets are called cells of the fan. Similarly to the convex case, the support function of K is linear on each cell of §K . And similarly to convex polytopes, the fan of a virtual polytope K can be de¯ned as the minimal fan for which hK is linear
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161
on each cell. The 0 -dimensional cells are called the vertices of the fan. The set of all vertices of a fan is denoted by §0 . It equals the set of normal vectors of all facets of K . The collection of all 1-dimensional cells of § is denoted by §1 and is called the skeleton of §. The support of the skeleton supp §1 is the union of all 1-dimensional cells of §. We say that K ¯ts a fan § if § is a re¯nement of §K . It means that hK is linear on each cell of §. For a unit vector ¹ and a real number h, denote by e(¹ ; h) the plane whose equation is (x; ¹ ) = h. Note that a convex polytope K 2 P is uniquely de¯ned by the set §0 = f¹ i g and the values hK (¹ i ) = hi . Indeed, the collection fe(¹ i ; hi )g is the collection of a±ne hulls of its facets. However, this assertion fails on virtual polytopes. In this case we have more freedom: we are free to choose any fan with vertices in f¹ i g. The only thing we have to worry about is the consistency condition which is motivated by the following remark. Remark 2.10. Let K be a virtual polytope. For any cell ¬ of §K with vertices f¹ 1 ; : : : ; ¹ k g, the planes e(¹ 1 ; h1 ); :::; e(¹ k ; hk ) have a common point. This point is the vertex of K corresponding to the cell ¬ . Consistency condition. We say that a fan § and a function h : §0 ! R satisfy the consistency condition if for any cell ¬ of § with vertices f¹ 1 ; : : : ; ¹ k g, the hyperplanes e(¹ 1 ; h1 ); :::; e(¹ k ; hk ) have a common point. The following theorem demonstrates that virtual polytopes are not uniquely restored by the set f¹ i g and values hi . It also shows that in the virtual case the cells may be non-convex, disconnected or of a complicated topological form. Theorem 2.11. [13] Let a fan § and a function h : §0 ! R satisfy the consistency condition. Then there exists a unique virtual polytope K such that §K ¯ts § and hK (¹ i ) = hi .
3
Nets
A net N is a pair (§; l) where § is a fan and l is a function l : §1 ! R. We say that N is proper if for any vertex ¹ of § , we have
L2§1;
X
l(L)L» = 0;
(9)
L is adjacent to »
where L» is the unit vector tangent to S 2 and parallel to L at the point ¹ (see Fig.1). The net of a virtual polytope K is the pair N (K ) = (§(K ); l); where l(L) equals the length of the edge whose spherical image is L. Obviously, the net of a convex polytope is proper.
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Fig. 1
De¯nition 3.1. Given two nets N1 = (§1 ; l1 ) and N2 = (§2 ; l2 ), we de¯ne the sum N1 + N2 = (§; l) in a natural way: ° Start by the fan §0 consisting of all sets of type Ui \ Vj where Ui 2 §1 and Vj 2 §2 . De¯ne the function l0 : (§0 )1 ! R as X X l 0 (L) := l1 (L) + l2 (L): (10) LµL12§11
LµL22§12
° Now erase those edges L of the fan §0 for which l0 (L) = 0 and unite those cells which are adjacent to the erased edges. Thus we get a fan § and a net N= (§; l), where l is the restriction of l0 on the §1 . This net is called the sum of N1 and N2 . Since each virtual polytope is the Minkowski di®erence of two convex ones, it follows from linearity that the net of a virtual polytope is proper as well. Proposition 3.2. 1. For any proper net N , there exists a unique (up to a translation) polytope K = K (N ) such that N = N (K ). 2. For any proper net N = (§; l) with l ¶ 0, there exists a unique (up to a translation) convex polytope K = K (N ) such that N = N (K ). Proof. The assertion 2 has already been proved in [11]. To prove 1, it su±ces to note that N is representable as a di®erence of two positive nets. Let K1 and K2 be the convex polytopes corresponding to these nets. Remark 2.8 and Theorem 2.6 imply that N (K1 « (K2 )¡1 ) = N , which concludes the proof. Thus we have an isomorphism between the two groups: Ã ¡ ! The group of virtual polytopes The group of proper nets, + factorized by translation, «
De¯nition 3.3. A net N = (§; l) and a fan § are called convex if all cells of § are convex. A net N = (§; l) and a fan § are called disconnected if supp §1 is disconnected.
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163
The net of a convex polytope is always convex, but the converse is not true. A fan § is said to be °exible if there exists a nontrivial (not generated by a rotation of the sphere) continuous family of fans §t ; t 2 [0; 1]; § = §0 such that all §t have the same combinatorics and the fans §t locally coincide up to a rotation of the sphere, i.e., for any vertex ¹ of §, there exist neighborhoods U and Ut of ¹ and ¹ t respectively such that the fans § and §t restricted to these neighbourhoods di®er by a rotation. Proposition 3.4. Let K be a virtual polytope. Its fan §K is °exible if and only if K is °exible. Proof. Any °ex of K generates a °ex of §K . Conversely, given a °ex §t of §K , we have a continuous family of virtual polytopes corresponding to the family of nets (§t ; l) where the function l is inherited from the net of K .
4
Flexible polytopes with disconnected nets.
We will show that there exist proper nets with any prescribed number d of connected components. Such nets are obviously °exible (and the corresponding virtual polytopes are °exible) since we can move the componens independently. Their freedom degree is at least 2(d ¡ 1). Consider a tetrahedron T with face normals ¹ 1 ; ¹ 2 ; ¹ 3 and ¹ 0 such that ¹ 1 ; ¹ 2 ; ¹ 3 lie near to ¡ ¹ 0 . Consider now the virtual polytope T 0 = T « (T »0 )¡1 . The support of §1 (T 0 ) lies near ¹ 0 (see Fig.2). It is easy to ¯nd a collection of tetrahedrons fTi gdi=1 such that the supports of §1 (T i0 ) are disjoint.
Fig. 2 The net of T 0
The virtual polytope T10 « ::: « Td0 is °exible with the freedom degree 2(d ¡
1).
164
5
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Rigidity of virtual polytopes
Theorem 5.1. A virtual polytope with convex fan is rigid. [13] Proof. First recall the Cauchy Lemma which was used by A.D. Alexandrov when proving the rigidity of convex polytopes and which will be used by us when following Alexandrov’s proof. Lemma 5.2. Let G be an embedded connected graph on the sphere S2 such that no 2-dimensional cell is bounded by only two edges. Suppose that a sign + or ¡ is assigned to some edges of G such that for any vertex A of G, one of the two statements is true: 1. The sign changes at least 4 times when going around A. 2. No edge adjacent to A has the sign. Then the set of signed edges is empty. Let K be a virtual polytope with a convex fan which is assumed to be °exible. Let Kt be its °ex. For some edge of the graph §1K , assign the sign as follows. If the corresponding dihedral angle of Kt increases during the °ex, the sign is +. If the angle decreases, the sign is ¡ . If the value of the angle doesn’t change, we assign no sign. Consider a vertex A of K , the corresponding cell ¬ of the fan and the complex C(K; A) of facets of K adjacent to the vertex A. There exists a the convex cone CR (K; A) with the vertex A and with the same normals of facets as C (K; A). Each °ex of C R (K; A) generates a °ex of C(K; A) and vice versa. Furthermore, a dihedral angle of C(K; A) increases during a °ex if and only if the corresponding dihedral angle of C R (K; A) increases during the generated °ex. Therefore either the sign changes at least 4 times when going around ¬ or no edge of ¬ has a sign. Consider the graph G = (§1K )¤ dual to §1K and endowed with the sign inherited from §1K . The graph G satis¯es the condition of Cauchy Lemma. A particular case of this theorem (the rigidity of herissons) is proved in [15]. A herisson (as considered in [15]) is a virtual polytope K with a convex fan and with convex-like facets, that is, cl supp K » is convex ( cl denotes the closure.) But there exist many non convex-like virtual polytopes (see [13] for their diversity).
6
Flexible virtual polytopes with connected fans. Bricard octahedra.
De¯nition 6.1. [16] Consider 4 points A; B; C; D 2 E 3 such that AB = CD and AD = BC. The closed broken line ABCDA has a symmetry axis l (it is the line containing the middle points of AC and BD). Consider two more points Q and Q0 which are symmetric with respect to l. Then the closed simplicial complex generated by the collection of
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165
triangles fABQ; BC Q; C DQ; ADQ; ABQ0 ; BCQ0 ; CDQ0 ; ADQ0 g is called the Bricard’s ¯rst °exible octahedron. De¯nition 6.2. [16] Consider 4 points A; B; C; D 2 E 3 such that AB = AD and CD = BC. The closed broken line ABCDA is symmetric with respect to a plane p (it is the plane containing the line AC and bisecting the dihedral angle between ABD and CBD). Consider two more points Q and Q0 which are symmetric with respect to p. Then the closed simplicial complex generated by the collection of triangles fABQ; BC Q; C DQ; ADQ; ABQ0 ; BCQ0 ; CDQ0 ; ADQ0 g is called the Bricard’s second °exible octahedron. These complexes are °exible indeed: for each of them, there exists a non-trivial (not generated by a mutual movement of triangles) continuous °ex which preserves the combinatorics and the inner metric of the complex [16]. We consider below some particular cases of Bricard’s octahedra. For each of these cases, there exists a virtual polytope K such that the collection of closures of supports of its facets coincides with the above collection of triangles. Thus we obtain °exible virtual polytopes with connected nets. Note also that not every Bricard’s octahedron can be turned into a virtual polytope. Example 6.3. Consider a Bricard’s octahedron B (the ¯rst or the second one) as in Fig.3. It means that the collection of triangles fABQ; BCQ; CDQ; ADQg as well as the collection fABQ0 ; BCQ 0 ; CDQ0 ; ADQ0 g forms a saddle. For convenience, we draw only the triangles containing Q. Fix the facet orientations (by choosing a normal vector as shown in Fig.3.) and mark the spherical images of the normal vectors on the sphere . There exists a fan § with vertices in the marked points such that two marked points in S 2 are connected by an edge if and only if the corresponding triangles are adjacent in the complex B. The fan § is, of course, non-convex. The facets of B have ¯xed orientations. Consider the support numbers fhi g of B. According to Theorem2.11, there exists a unique virtual polytope K = K (B) with the fan § and support numbers fhi g. Make sure that it is the required virtual polytope. First notice that the set of vertices of K coincides with the set fA; B; C; D; Q; Q0 g. Take any facet of of K, say the one with the normal ABQ. It has exactly 3 edges lying on the lines AB; BQ and AQ. This follows from the construction of the fan §. Each of these edges either coincides with one of the segments AB; BQ and AQ or equals the virtual segment with the same endpoins (see Remark2.8). Therefore the closure of the facet coincides with the triangle ABQ. The other facets are treated analogously.
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Fig. 3 a) The triangels containing Q. b) The fan of K(B)
Example 6.4. Another example of a °exible virtual polytope arises from a particular case of the Bricard’s ¯rst °exible octahedron considered in [15]. Fix a Cartesian system (x; y; z) and consider a circle S lying in the plane (x; y) and centered at O. Choose four points A; B; C and D lying on S such that jABj = jCDj (see Fig.4). Let Q = (0; 0; 1); Q0 = (0; 0; ¡ 1). The collection of triangles C = fABQ; BCQ; CDQ; ADQ; ABQ0 ; BCQ 0 ; CDQ0 ; ADQ0 g form a Bricard’s ¯rst °exible octahedron. Analogously to the previous example, we need to choose orientations of these triangles such that there exists a fan with vertices in the choosen points and consistent with the combinatorics of C . Such a fan § is shown in Fig. 4. As in the previous example, there exists, according to Theorem2.11 a virtual polytope such that the collection of closures of supports of its facets coincides with the above collection of triangles.
Fig. 4 a) The upper part of (C ) and orientation of triangles. b) The fan of K(C )
As mentioned in [15], no other choice would make the fan convex.
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167
As was kindly pointed out by Y. Martinez-Maure, the upper part of K is the simplest polytopal version of a smooth cross-cap, which is used in [10] as a crucial tool for constructing a counter-example to an old hypothesis. This notion appeared originally in [18] ( see also [2]) where some special types of saddle surfaces were discussed.)
Acknowledgments Partly supported by the RFBR grant N 02 ¡
01 ¡
00908.
References [1] A.D. Alexandrov: Konvexe Polyeder, Berlin, Akademie-Verlag (1958). [2] Yu. Burago: \Theory of surfaces", in Encyclopaedia of Math. Sc., Vol. 48, (1992), Geometry 3. [3] R. Conelly: \Rigidity", in: Handbook of convex geometry, Gruber and Wills, (1993), pp. 223{271. [4] V.Danilov: \The geometry of toric varieties", Russian Math. Surveys, Vol. 33, (1978), pp. 97{154. [5] A. Khovanskii, A. Pukhlikov: \Finitely additive measures of virtual polytopes", St. Petersburg Math. J., Vol. 4, (1993), pp. 337{356. [6] K.Leichtweiss: Konvexe Mengen, VEB Deutscher Verlag der Wissenschaften, Berlin, 1980. [7] P.McMullen: \The polytope algebra", Adv.Math., Vol. 78, (1989), pp. 76{130. [8] P.McMullen: \On simple polytopes", Invent. Math., Vol. 113, (1993), pp. 19{111. [9] Y. Martinez-Maure: "De nouvelles in¶egalit¶es g¶eom¶etriques pour les h¶erissons", Arch. Math., Vol. 72, (1999), pp. 444{453. [10] Y. Martinez-Maure: "Contre-exemple µa une caract¶erisation conjectur¶ee de la sphµere", C.R. Acad. Sci. Paris, Vol. 332, (2001), pp.41{44. [11] G.Panina: \Mixed volumes for non-convex bodies", Isv. Akad. Nauk Armenii, Matematika, Vol. 28, (1993), pp. 51{59. [12] G.Panina: "The structure of virtual polytope group related to cylinders subgroups", St. Petersburg Math. J., Vol. 13, (2001), pp. 471{484. [13] G.Panina: \Virtual polytopes and some classical problems", St. Petersburg Math. J., Vol. 14, (2002), pp.152{170. [14] H.RadstrÄom: \An embedding theorem for spaces of convex sets", Proc. AMS, Vol. 3, (1952), pp. 165{169. [15] L.Rodriguez and H. Rosenberg: \Rigidity of certain polyhedra in R3 ", Comment. Math. Helv., Vol. 75, (2000), pp. 478{503. [16] I.Sabitov: \The local theory of bendings of surfaces", in: Encyclopaedia of Math. Sc., Vol. 48, (1992), Geometry 3, pp. 179{250.
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[17] O.Ya.Viro: \Some integral calculus based on Euler characteristic", Topology and Geometry - Rokhlin Seminar, Lecture Notes in Math., 1346, Springer-Verlag, BerlinNew York, 1988, pp.127{138. [18] O.Ya.Viro: \Some integral calculus based on Euler characteristic", Topology and Geometry - Rokhlin Seminar, Lecture Notes in Math., 1346, Springer-Verlag, BerlinNew York, 1988, pp.127{138.
CEJM 2 (2003) 169{183
Sets with two associative operations Teimuraz Pirashvili A.M. Razmadze Mathematical Inst. Aleksidze str. 1, Tbilisi, 380093, Republic of Georgia
Received 7 January 2003; revised 3 March 2003 Abstract: In this paper we consider duplexes, which are sets with two associative binary operations. Dimonoids in the sense of Loday are examples of duplexes. The set of all permutations carries a structure of a duplex. Our main result asserts that it is a free duplex with an explicitly described set of generators. The proof uses a construction of the free duplex with one generator by planary trees. c Central European Science Journals. All rights reserved. ® Keywords: Trees, permutations, free algebras MSC (2000): 05E99, 20M05, 08B20
1
Introduction
Jean-Louis Loday introduced the notion of a dimonoid and a dialgebra. Let us recall, that a dimonoid is a set equipped with two associative operations satisfying 3 more axioms (see [2, 3] or Section 6.2), while a dialgebra is just a linear analog of a dimonoid. In this paper we drop these additional axioms and we consider sets equipped with two associative binary operations. We call such an algebraic structure as a duplex. Thus dimonoids are examples of duplexes. The set of all permutations, gives an example of a duplex which is not a dimonoid (see Section 3). In Section 4 we construct a free duplex generated by a given set via planar trees and then we prove that the set of all permutations forms a free duplex on an explicitly described set of generators. In the last section we consider duplexes coming from planar binary trees and vertices of the cubes as in [5]. We prove that these duplexes are free with one generator in appropriate variety of duplexes.
2
Graded sets
` A graded set is a set X¤ together with a decomposition X ¤ = n2N X n . Here and elsewhere ` denotes the disjoint union of sets. A map of graded sets is a map f : X ¤ ! Y ¤ such
170
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that f (Xn ) » Yn , n ¶ 0. We let ¤ be the category of graded sets. If x 2 X n we write O(x) = n. In this way we obtain a map O : X ¤ ! N and conversely, if such a map is given then X¤ is graded with X n = O ¡1 (n): If X ¤ and Y¤ are graded sets, then the Cartesian product X ¤ £ Y ¤ is also a graded set with O(x; y) := O(x) + O(y); x 2 X¤ ; y 2 Y ¤ : ` ` The disjoint union of two graded sets X ¤ and Y¤ is also graded by (X ¤ Y ¤ )n = X n Yn . A graded set X¤ is called locally ¯nite if Xn is ¯nite for any n 2 N. In this case we put X¤ (T) :=
1 X
(Xn )Tn 2 Z[[T]]:
n=0
An n-ary operation [ on a graded set X¤ is homogeneous if [ : X¤n ! X ¤ is a map of graded sets, in other words [(X k1 £ ¢ ¢ ¢ £ X kn ) » Xk1+¢¢¢+kn : Free semigroups are examples of graded sets. Let us recall that a semigroup is a set equipped with an associative binary operation. For any set S the free semigroup on S is the following graded set a n (S) = n (S); n (S) := S ; n ¶ 0 n¸1
while the multiplication is given by S n £ S m ! S n+m ; ((x1 ; ¢ ¢ ¢ ; xn ); (y1 ; ¢ ¢ ¢ ; ym )) 7! (x1 ; ¢ ¢ ¢ ; xn ; y1 ; ¢ ¢ ¢ ; ym ): It is clear that (S)(T) =
3
(S)T : (S)T
1¡
Two operations on permutations
For any n ¶ 1 we let n be the set f1; 2; ¢ ¢ ¢ ; ng. Furthermore, we let n be the set of all bijections n ! n. An element f 2 n , called a permutation, is speci¯ed by the sequence (f (1); ¢ ¢ ¢ ; f (n)). The composition law ¯ yields the group structure on n . We put :=
1 a
n:
n=1
We are going to consider as a graded set with respect of this grading. Thus O(f ) = n means that f : n ! n is a bijection. On we introduce two homogeneous associative operations. The ¯rst one is the concatenation, which we denote by ]. Thus ]:
n
£
m
!
n+m ;
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171
is de¯ned by if 1 µ i µ n;
(f ]g)(i) = f (i); (f ]g)(i) = n + g(i ¡
n);
if n + 1 µ i µ n + m:
Here O(f ) = n and O(g) = m. For example if f 2 permutations
then f ]g 2
6
3
and g 2
3
are the following
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
4
5
6
1
2
3
4
5
6
is given by
The second operation \:
n
£
m
!
n+m ;
is de¯ned by
For example if f 2
(f \g)(i) = m + f (i);
if 1 µ i µ n;
(f \g)(i) = g(i ¡
if n + 1 µ i µ n + m:
3
and g 2
3
n);
are as above, then f \g 2
1
2
3
4
5
6
1
2
3
4
5
6
6
is given by
These operations appear also in [5] under the name `over’ and `under’.
3.1 Duplexes One easily checks that both operations ]; \ de¯ned on 1.10 in [5]). This suggests the following de¯nition
are associative (see also Lemma
De¯nition 3.1. A duplex is a set D equipped with two associative operations ¢ : D £D ! D and ¤ : D £ D ! D. A map f : D ! D 0 from a duplex D to another duplex D 0 is a homomorphism, provided f (x ¢ y) = f (x) ¢ f (y) and f (x ¤ y) = f (x) ¤ f (y). We let be the category of duplexes. As usual one can introduce the notion of free duplex as follows.
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De¯nition 3.2. A duplex F is called free if there exists a subset X » F , such that for any duplex D and any map f : X ! D there exists a unique homomorphism of duplexes g : F ! D such that g(x) = f (x) for all x 2 X . If this holds, then we say that F is a free duplex on X . Thus, duplexes generalize the notion of dimonoids introduced by Jean-Louis Loday in [3].Our main result is the following Theorem 3.3. The set
equipped with ] and \ is a free duplex on the set
The description of the set
2
2
.
and the proof of Theorem 3.3 is given in Section 5.
3.2 Semigroups ( ; ]) and ( ; \) In this section we prove the fact that the semi-group ( ; ]) is free. It is easy to see that ( ; ]) and ( ; \) are isomorphic semi-groups (see Lemma 3.4) and therefore ( ; \) is free as well. Lemma 3.4. Let !n 2
n
be the permutation de¯ned by !n (i) = n + 1 ¡
and let ¹
n
:
n
!
n
i; i 2 n
be the map given by ¹ n (f ) = !n ¯ f . Then ¹
n
¯¹
n
= Id
n
and the diagram n
£
] n+m
m
»n£»m n
commutes. Therefore the map ¹ = ( ; ]) to the semi-group ( ; \).
`
»n+m
£
m
n¹ n
:
\
!
n+m
is an isomorphism from the semi-group
Proof. Since ! ¯ ! = I dn it follows that ¹ n (¹ n (f )) = !n ¯ (!n ¯ f ) = f: Furthermore, for any f 2 n , g 2 m and 1 µ i µ n + m, both ¹ n+m (f ]g)(i) and (¹ n (f )\¹ m (g))(i) are equal to n + m + 1 ¡ f (i) or m + 1 ¡ g(i ¡ n) depending whether i 2 n or not. An element f 2 n is called ]-decomposable, provided f = g]h for some g 2 k ; h 2 m . If such type decomposition is impossible, then f is called ]-indecomposable. For each f 2 n we put ¯ (f ) := i fi 2 n j f (i) » ig: In particular f (f1; ¢ ¢ ¢ ; ¯ (f )g) » f1; ¢ ¢ ¢ ; ¯ f g. We let f± 2 on f1; ¢ ¢ ¢ ; ¯ (f )g.
±(f )
be the restriction of f
T. Pirashvili / Central European Journal of Mathematics 2 (2003) 169{183
Lemma 3.5. i) If f = g]h, then ¯ (f ) = ¯ (g) and f± = g± . ii) If f 2 then there is a unique f¯ 2 n¡±(f) such that
173
n
is ]-decomposable,
f = f± ]f¯ : iii) An element f 2
n
is ]-indecomposable i® f = f± .
Proof. i) It is su±cient to note that if f = g]h and O(g) = k, then f (k) » f (k). To show ii) and iii) let us assume that f = g]h. Then by i) we have ¯ (f ) = ¯ (g) µ O(g) < O(f ): 6 f± , then f = f± ]f¯ ; where f¯ 2 k is given by Thus f 6= f± . Conversely, if f = f¯ (i) = f (¯ (f ) + i) ¡ ¯ (f ). Here k = O(f ) ¡ ¯ (f ) and 1 µ i µ k. The permutation f¯ is well-de¯ned since f maps the subset f¯ (f ) + 1; ¢ ¢ ¢ ; O(f )g to itself. We let ] be the set of ]-indecomposable elements of T ] ] n : For example we have n = n ] 2
= f(2; 1)g;
] 3
We let un be the cardinality of the set
. We have
]
=
`
n
] n,
where
= f(2; 3; 1); (3; 1; 2); (3; 2; 1)g ] n.
Here are the ¯rst values of un :
1; 1; 3; 13; 71; 461; 3447; ¢ ¢ ¢ which can be deduced from Corollary 3.7. Jean-Louis Loday informed me that the integer un is the number of permutations with no global descent [1]. Theorem 3.6. The semigroup ( ; ]) is free on
]
.
Proof. First we show that the set ] generates . Take f 2 . We have to prove that f lies in the subsemigroup generated by ] . We may assume that f 62 ] . Thus f is ]-decomposable and we can write f = g]h. Since O(g); O(h) < O(f ) we may assume by induction that g and h lie in the subsemigroup generated by ] . Thus the same is true for f as well. The fact that any f can be written uniquely in the form f = g1 ] ¢ ¢ ¢ ]g k with gi 2 ] follows from Lemma 3.5. Corollary 3.7. One has (
] n)
= n! ¡
X
p!q! +
p+q=n
X
p!q!r! ¡
¢¢¢
p+q+r=n
Here p; q; r; ¢ ¢ ¢ runs over all strictly positive integers. Proof. We have an isomorphism of graded sets X X ¡X n!tn = n¸1
n¸1
m¸1
¹= `
n¸1
¢ ] m n t m
] n
and therefore
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T. Pirashvili / Central European Journal of Mathematics 2 (2003) 169{183
and the result follows. By transportation of structures we see that ( ; \) is a free semigroup on the set \ , which is by de¯nition the set of all \-indecomposable permutations. Here a permutation f is called \-indecomposable if !n ¯ f is ]-indecomposable, in other words if f (f1; ¢ ¢ ¢ ; ig) 6» fn ¡ i + 1; ¢ ¢ ¢ ; ng for all 1 µ i µ n ¡ 1.
4
Free duplexes
4.1 Planar trees By tree we mean in this paper a planar rooted tree. Such trees play an important role in the recent work of Jean-Louis Loday and Maria Ronco [6]. We let be the set of trees. It is a graded set a = n; n¸1
where n is the set of trees with n leaves. The number of elements of n are known as super Catalan numbers and they are denoted by Cn . Here are the ¯rst super Catalan numbers: C1 = 1 = C2 , C3 = 3, C4 = 11, C5 = 45. In general one has the following well-known relation (see for example Section 8.2 of [4]). f (T) :=
1 X
C n Tn =
n¸1
1¡ T+1¡ 4
So 1 has only one element j and similarly has tree elements
2
p
T2 ¡
6T + 1
¢
has also only one element
(1) , while
3
, , Any vertex v of a tree t 2 n with n ¶ 2, has a level, which is equal to the number of edges in the path connecting v to the root. Thus the root is the unique vertex of level 0. For example, if u1 = , u2 = then the tree u1 has only one vertex (which is of course the root), while u2 has two vertices the root and a vertex of level one. Let us also recall that on trees there exists an important operation which is called grafting. The grafting de¯nes a map :
n1
£ ¢¢¢£
nk
!
n;
n = n1 + ¢ ¢ ¢ + nk :
For example, if u1 and u2 are as above, then we have:
(u1 ; u2 ) =
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175
Let us also note, that (u1 ; u2 ) has 4 vertices, two of them of the level one and one of the level two. It is clear that any tree from n , n > 1 can be written uniquely as (t1 ; ¢ ¢ ¢ ; tk ). Here k is the number of incoming edges at the root. We will say that t is constructed by the grafting of (t1 ; ¢ ¢ ¢ ; tk ). We need also the following construction on trees. Let t1 ; ¢ ¢ ¢ ; tk be trees and let I be a subset of k. We consider t = (t1 ; ¢ ¢ ¢ ; tk ). The tree obtained by contracting the edges of t which connect the root of t with the roots of ti , i 2 I is denoted by I (t1 ; ¢ ¢ ¢ ; tk ): If I = ;, then I = . For example, if u1 and u2 are as above, then we have
1 (u1 ; u2 )
=
12 (u1 ; u2 )
=
4.2 The free duplex with one generator We consider now two di®erent copies of the set n for n ¶ 2, which are denoted by :n S S and ¤n , n ¶ 2. We put ¤ := n¸2 ¤n and ¢ := n¸2 ¢n : If t 2 :, then we assign ¢ to any vertex of t of even level and ¤ to any vertex of t of odd level. Similarly, if t 2 ¤ , then we assign ¤ to any vertex of t of even level and ¢ to any vertex of t of odd level. So, for example . *
.
2
*
¤ 6
We call such trees as decorated trees. To be more precise a decorated tree is an element of the set 1 [ = (n); n=1
¢ n[
¤ n;
where (1) = 1 and (n) = n ¶ 2: We are going to de¯ne a duplex structure on decorated trees in such a way, that the above tree can be expressed (e ¢ e ¢ e) ¤ (e ¢ (e ¤ e)); where e =j. More formally, the operations ¢:
£
can be de¯ned as follows. If t1 ; t2 2 t1 ¤ t2 := if t1 ; t2 2
¤
! 1
; ¤:
£
!
then
(t1 ; t2 ) 2
¤ 2;
t1 ¢ t2 :=
(t1 ; t2 ) 2
¢ 2
then t1 ¤ t2 :=
12 (t1 ; t2 )
2
¤
;
t1 ¢ t2 :=
(t1 ; t2 ) 2
¢
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T. Pirashvili / Central European Journal of Mathematics 2 (2003) 169{183
if t1 ; t2 2
¢
then t1 ¤ t2 :=
if t1 2
¢
and t2 2
¤
¤
and t2 2
¢
t1 ¢ t2 :=
;
12 (t1 ; t2 )
2
¢
1 (t1 ; t2 )
2
¢
, then
t1 ¤ t2 := if t1 2
¤
(t1 ; t2 ) 2
2 (t1 ; t2 )
¤
2
;
t1 ¢ t2 :=
, then
t1 ¤ t2 :=
1 (t1 ; t2 )
¤
2
;
t1 ¢ t2 :=
This concludes the construction of operations on
2 (t1 ; t2 )
2
¢
:
. Let us observe that in all cases
t1 ¤ t2 =
I (t1 ; t2 )
2
¤
t1 ¢ t2 =
J (t1 ; t2 )
2
¢
and where I » f1; 2g consists of i 2 2 such that ti 2 that ti 2 ¢:
¤
;
and J » f1; 2g consists of i 2 2 such
Lemma 4.1. Both operations ¤ and ¢ are associative. Proof. One observes that for any decorated trees t1 ; t2 ; t3 in (t1 ¤ t2 ) ¤ t3 =
I (t1 ; t2 ; t3 )
¤
one has
= t1 ¤ (t2 ¤ t3 )
where I » f1; 2; 3g consists of the indices i such that ti 2
¤
, i = 1; 2; 3. Similarly for ¢.
Lemma 4.2. The semigroup ( ; ¤) is a free semigroup on the set [ ¢ S ¢ := 1 [ n » n¸1
and similarly, the semigroup ( ; ¢) is a free semigroup on the set [ ¤ S ¤ := 1 [ n » n¸1
Proof. It su±ces to note that if t 2 , but t 62 S ¢, then t can be written uniquely as (t1 ; ¢ ¢ ¢ ; tk ). If we consider t1 ; ¢ ¢ ¢ ; tk as elements from S ¢ , then we have t = t1 ¤ ¢ ¢ ¢ ¤ tk : Similarly for ( ; ¢). Theorem 4.3. The duplex
is a free duplex with one generator j 2
1
»
.
Proof. For simplicity we let e be the tree j. Then e generates the duplex . This can be proved by induction. Indeed, (2) = ¢2 [ ¤2 has two decorated trees, which are
T. Pirashvili / Central European Journal of Mathematics 2 (2003) 169{183
177
equal respectively to e ¢ e and e ¤ e. Assume we already proved that any decorated tree from (m) lies in the subduplex generated by a for any m < n and let us prove that the same holds for n = m. Take t 2 (n). Assume t 2 ¤n . Then there exists unique trees t1 ; ¢ ¢ ¢ ; tk such that t = (t1 ; ¢ ¢ ¢ ; tk ). We consider t1 ; ¢ ¢ ¢ ; tk as elements of the set 1 [ ¢ . Then we have t = t1 ¤ ¢ ¢ ¢ ¤ tk and by the induction assumption t can be expressed in the terms of e. Similarly, if t 2 ¢ . Now we will show that is free on e. We take a duplex D and an element a 2 D. We have to show that there exists a unique ! D such that f (e) = a. We construct such an f by induction. homomorphism f : For the tree e we already have f (e) = a. Assume f is de¯ned for all decorated trees from (m), m < n and take a decorated tree t 2 n . As above we can write t = t1 ¤ ¢ ¢ ¢ ¤ tk with unique t1 ; ¢ ¢ ¢ ; tk and then we put f (t) = f (t1 ) ¤ ¢ ¢ ¢ ¤ f (tk ): Similarly, if t 2 ¢ . Let us prove that f is a homomorphism. The equations f (x ¤ y) = f (x) ¤ f (y);
f (x ¢ y) = f (x) ¢ f (y)
is clear if x; y 2 1 . Let us prove only the ¯rst one, because the proof of the second one is completely similar. If x; y 2 ¢ then we have x ¤ y = (x; y). Thus by de¯nition f (x ¤ y) = f (x) ¤ f (y). If x 2 ¤ and y 2 ¢. Then x¤y =
1 (x; y)
(x1 ; ¢ ¢ ¢ ; xk ; y) = x1 ¤ ¢ ¢ ¢ ¤ xk ¤ y
=
where x = (x1 ; ¢ ¢ ¢ ; xk ) and x1 ; ¢ ¢ ¢ ; xk are considered as elements of that f (x ¤ y) = f (x1 ) ¤ ¢ ¢ ¢ ¤ f (xk ) ¤ f (y) = f (x) ¤ f (y): Similarly if x 2 x; y 2 ¤ . Then x¤y =
12 (x; y)
=
1
[ ¢ . It follows ¢ and y 2 ¤ . If
(x1 ; ¢ ¢ ¢ ; xk ; y1 ; ¢ ¢ ¢ ; yl ) = x1 ¤ ¢ ¢ ¢ ¤ xk ¤ y1 ¤ ¢ ¢ ¢ ¤ yl
where x = (x1 ; ¢ ¢ ¢ ; xk ), y = (y1 ; ¢ ¢ ¢ ; yl ) and x1 ; ¢ ¢ ¢ ; xk ; y1 ; ¢ ¢ ¢ ; yl are considered as elements of 1 [ ¢. It follows that f (x ¤ y) = f (x1 ) ¤ ¢ ¢ ¢ ¤ f (xk ) ¤ f (y1 ) ¤ ¢ ¢ ¢ ¤ f (yl ) = f (x) ¤ f (y): h
4.3 Free duplexes For a set S we consider the set (S) :=
a
n (S)
n¸1
where
n (S)
=
(n) £ S n . We de¯ne maps ¢; ¤ :
n (S)
£
m (S)
!
n+m (S)
by (t1 ; x1 ; ¢ ¢ ¢ ; xn ) ¤ (t2 ; y1 ¢ ¢ ¢ ; ym) := (t1 ¤ t2 ; x1 ¢ ¢ ¢ xk ; y1 ¢ ¢ ¢ ; ym ):
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(t1 ; x1 ; ¢ ¢ ¢ ; xn ) ¢ (t2 ; y1 ¢ ¢ ¢ ; ym) := (t1 ¢ t2 ; x1 ¢ ¢ ¢ xk ; y1 ¢ ¢ ¢ ; ym ): We leave as an exercise to prove that ¯nite set S one has
(S) is a free duplex on S. It is clear that for a
(S)(T) = T + 2
1 X
Cn
n¸2
5 5.1
¡
¢n (S)T :
(2)
Proof of Theorem 3.3 2 -indecomposable
permutations
A permutation f : n ! n is called
2 -indecomposable
f (i) 6» i; and f (i) 6» fn ¡
if for any i = 1; ¢ ¢ ¢ ; n ¡
1 one has
i + 1; ¢ ¢ ¢ ; ng
It is clear that a permutation is 2 -indecomposable if it is simultaneously ]- and \2 indecomposable and therefore for the set of 2 -indecomposable permutations we have \ \ 2 = ] : (3) It is clear that
2
1
One checks that
=
1;
2
2
=;=
2
3
;
2
4
= f(2; 4; 1; 3); (3; 1; 4; 2)g
has 22 elements. In general for d n = n 2 we have X X d n = n! ¡ 2 p!q! + 2 p!q!r! ¡ ¢ ¢ ¢ = 2un ¡ n! 5
2
p+q=n
(4)
p+q+r=n
Here p; q; r; ¢ ¢ ¢ runs over all strictly positive integers. This follows from the formula for ] un = n and from Lemma 5.1. Here are the ¯rst values of dn : 1; 0; 0; 2; 22; 202; 1854; ¢ ¢ ¢ Lemma 5.1. If a permutation is ]-decomposable then it is \-indecomposable. Similarly, if a permutation is \-decomposable then it is ]-indecomposable. Proof. If not, then there exists a permutation f : n ! n and integers 1 µ i µ n ¡ 1 µ j µ n ¡ 1 with properties f (i) » i; and f (j) 6» fn ¡
1 and
j + 1; ¢ ¢ ¢ ; ng:
Without loss of generality we may assume that j µ i (otherwise we take !n ¯ f ). Thus for any k = 1; ¢ ¢ ¢ ; j we have n ¡ j < f (k) µ i. Therefore the interval ]n ¡ j; i] contains at least j integers. But this is impossible, because i ¡ n + j < j.
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179
5.2 Proof of Theorem 3.3 2 2 We will prove that is free on as a duplex. First we show that generates . 2 Indeed, if f 62 then either f = g]h or f = g\h and it is impossible to have both cases. Without loss of generality we may assume that one has the ¯rst possibility. Since ( ; ]) is a free semigroup, there exist uniquely de¯ned f1 ; ¢ ¢ ¢ ; fk 2 ] such that f = f1 ] ¢ ¢ ¢ ]fk . 2 If all fi 2 we stop, otherwise for some i we have fi 62 \ and therefore one can write fi in unique way as fi = g 1 \ ¢ ¢ ¢ \g m , with gj 2 \ . Since O(g i ) < O(f ) this process stops 2 after a few steps. This shows that generates as a duplex. Since in each step there 2 were unique choices we conclude that elements of are free generators.
Corollary 5.2. Between the numbers d n and super Catalan numbers Cn there is the following relation X X X ¡X ¢m n!tn = d n tn + 2 Cm d n tn n¸1
n¸1
m¸2
n¸1
Furthermore one has
Ã2 + ¹ Ã ¡ Ã + ¹ = 0 P P where Ã(T) = n¸1 n!Tn and ¹ (T) = n¸1 d n Tn :
Proof. The ¯rst part follows from Theorem 3.3 and Equation (2). To get the second part one rewrites the same relation as Ã(T) = 2f (¹ (T)) ¡ ¹ (T) P and then use the formula (1) for f (T) = n¸1 Cn Tn .
6
Duplexes satisfying some additional identities
6.1 Duplex of binary trees We let
1
be the category of duplexes satisfying the identity (a ¢ b) ¤ c = a ¢ (b ¤ c)
(5)
We prove that the free object in 1 is given via binary trees. Let us recall that for us all trees are planar and rooted. A tree is called binary if any vertex is trivalent. We let be the set of all binary trees: a := n; n¸1
where n is the set of planar binary trees with n + 1 leaves. We have n » n+1. The number of elements of n are known as Catalan numbers and they are denoted by cn . Here are the ¯rst Catalan numbers: c1 = 1, c2 = 2, c3 = 5. In general one has cn =
(2n)! n!(n + 1)!
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So
T. Pirashvili / Central European Journal of Mathematics 2 (2003) 169{183
1
has only one element
while
2
has two elements
, By our de¯nition the tree j 2 1 is not a binary tree, but it will play also an important role for binary trees as well. We let t be this particular tree. Let us note that any binary tree u can be written in the unique way as gr(ul ; ur ), where ul ; ur 2 [ ftg: The map gr : £ ! is no longer homogeneous, because we changed the grading, but it is of degree one: gr : n £ m ! n+m+1 . We introduce two homogeneous operations ¢; ¤ :
n
£
m
!
n+m
by u ¢ v := gr(u ¢ vl ; v r ) u ¤ v := gr(ul ; ur ¤ v): Here we use induction and the convention u ¢ t = t ¢ u = u ¤ t = t ¤ u = u for t = j. These associative operations under the name ’over’ and ’under’ ¯rst appeared in [5]. One observes that (see [5]) u ¢ v (resp. u ¤ v) is the tree obtained by identifying the root of u (resp. v) with the left (resp. right) most leaf of v (resp. u). Theorem 6.1. The duplex 1
satis¯es the identity (5). Moreover it is a free object in
generated by e = gr(t; t) =
.
Proof. We have (u ¢ v)l = u ¢ v l ; (u ¢ v)r = vr ; and (u ¤ v)l = ul ; (u ¢ v)r = ur ¤ v: Therefore (a ¢ b) ¤ c = gr(a ¢ bl ; br ¤ c) = a ¢ (b ¤ c): Thus
2
1.
Now we prove that
is generated by e. First one observes that
u ¢ e = gr(u; t); e ¤ u = gr(t; u): We claim that for any a; b 2
[ ftg one has (a ¢ e) ¤ b = gr(a; b)
Indeed, we can write (a ¢ e) ¤ b = gr(a; t) ¤ b, which is the same as gr(a; t ¤ b) thanks to the de¯nition of the operation ¤. By our convention t ¤ b = b and the claim is proved. Take any element u 2 . By the claim we have u = (ul ¢ e) ¤ ur = ul ¢ (e ¤ ur )
(6)
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181
Based on this equality one easily proves by induction that e generates the duplex . Let D be any duplex satisfying the equality (5) and take any element a 2 D. We have to show that there exists an unique homomorphism f : ! D such that f (e) = a. Since e generates the uniqueness is obvious. We construct recursively f by f (e) = a; and f (u) = (f (ul ) ¢ a) ¤ f (ur ) and an obvious induction shows that f is indeed a homomorphism.
6.2 Dimonoids We let
be the full subcategory of
satisfying two more identities:
1
(a ¤ b) ¢ c = (a ¢ b) ¢ c
(7)
a ¤ (b ¤ c) = a ¤ (b ¢ c)
(8)
We refer the reader to [2] for the extensive information on dimonoids and dialgebras which are just linear analogs of dimonoids.
6.3 Duplex of vertices of cubes We let
2
be the full subcategory of
1
satisfying the identity
(a ¤ b) ¢ c = a ¤ (b ¢ c)
(9)
Such types of algebras were ¯rst considered in [7] under the name \Doppelalgebren". Free objects in n be 1 are given via the following construction. For all n ¶ 2 we let n¡1 the set of vertices of the n ¡ 1 dimensional cube. So n = f¡ 1; 1g and elements of n are sequences a = (a 1 ¢ ¢ ¢ ; an¡1 ), where ai = ¡ 1 or ai = 1. Moreover we let 1 to the singleton feg and a := n: n¸1
Following [5] we de¯ne two homogeneous operations on e ¢ e := ¡ 1 2 e¤e = 12
by
2; 2;
e ¢ a := (¡ 1; a1 ; ¢ ¢ ¢ ; an¡1 ); a ¢ e := (a1 ; ¢ ¢ ¢ ; an¡1 ; ¡ 1); e ¤ a := (1; a1 ; ¢ ¢ ¢ ; an¡1 ); a ¤ e := (a1 ; ¢ ¢ ¢ ; an¡1 ; 1); a ¢ b := (a1 ; ¢ ¢ ¢ ; an¡1 ; ¡ 1; b1 ; ¢ ¢ ¢ ; bm¡1 ); a ¤ b := (a1 ; ¢ ¢ ¢ ; an¡1 ; 1; b1 ; ¢ ¢ ¢ ; bm¡1 ): One checks that
is a duplex, which satis¯es both identities (5) and (9).
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Theorem 6.2. [7] The duplex
is the free object in
2
generated by e.
Proof. We ¯rst show that e generates . For a = (a1 ; ¢ ¢ ¢ ; an ) 2 n , we put ¯i = ¢ if ai = ¡ 1 and ¯i = ¤ if ai = 1. Then a = e¯1 e¯2 ¢ ¢ ¢¯n¡1 e. The fact that this expression does not depend on parentheses follows from the associativity of ¢; ¤ and from the identities (5), (9). Let D be an object of 2 and let x 2 D. One can use the expression of a in terms of e and ¯i to de¯ne the map f : ! D by f (a) = x ¯1 x ¯2 ¢ ¢ ¢ ¯n¡1 x. Then by induction one shows that f is in fact a homomorphism with f (e) = x.
6.4 Remarks Our results can be used to describe maps between combinatorial objects. It is not di±cult to check that the map à : ! constructed in [5] is a homomorphism of duplexes. On the other hand, since is a free duplex with one generator there is a unique homogeneous ! map ¬ : which is also a homomorphism of duplexes. To specify this map it su±ces to know the image of the generator: ¬ (j) = I d1 . Similarly, if one considers as ! an object of , then there exists a unique surjective homomorphism % : which takes the generator j of to e = gr(t; t) 2 1 . Thus we have % = à ¯ ¬ . Since 2 » 1 , the free object on a set X in 2 is a quotient of the free object on a set X in 1 . In particular for X = feg we obtain the canonical quotient ! . If one forgets the corresponding algebraic structures this homomorphism ¿ : map from binary trees to vertices of cubes coincides with one considered in [5]. The composite map ¿ ¯ % : ! from the decorated trees to the vertices has the following description. Take a decorated tree u 2 n and label the leaves of u by the numbers 1; ¢ ¢ ¢ ; n. Then ¿ ¯ » (u) = (a1 ; ¢ ¢ ¢ ; an¡1 ), where ai is +1 if the sign at the end of the edge coming from the (i + 1)-th leaf is ¤ and is ¡ 1 if the corresponding sign is ¢.
Acknowledgments This paper is in°uenced by the work of Jean-Louis Loday, who started to investigate algebras with two associative operations (see [2, 3]) and initiated me to the beauty of the combinatorics of trees during several informal discussions. This work was written during my visit at UniversitÄat Bielefeld. I would like to thank Friedhelm Waldhausen for the invitation to Bielefeld. Thanks to my wife Tamar Kandaurishvili for various helpful discussions on the subject and encouraging me to write this paper. The author was partially supported by the grant INTAS-99-0081 and TMR Network ERB FMRX CT-97-0107.
References [1] M. Aguiar and F. Sottile: Structures of the Malvenuto-Reutenauer Hopf algebra of permutations, Preprint (2002), http://front.math.ucdavis.edu/math.CO/0203282.
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[2] J.-L. Loday: \Alg¶ebras ayant deux op¶erations associatives (dig¶ebres)", C. R. Acad. Sci. Paris S¶er. I Math., Vol. 321, (1995), pp. 141{146. [3] J.-L. Loday: \Dialgebras", In: Dialgebras and related operads, Lecture Notes in Math. 1763, Springer, Berlin, 2001, pp. 7{66. [4] J.-L. Loday: \Arithmetree", J. of Algebra, Vol. 258, (2002), pp. 275{309. [5] J.-L. Loday and M. O. Ronco: \Order structure on the algebra of permutations and of planar binary trees", J. Algebraic Combin., Vol. 15, (2002), pp. 253{270 [6] J.-L. Loday and M. O. Ronco: Trialgebras and families of polytopes, Preprint (2002), http://front.math.ucdavis.edu/math.AT/0205043. [7] B. Richter: Dialgebren, Doppelalgebren und ihre Homologie, Diplomarbeit, UniversitÄat Bonn, 1997, http://www.math.uni-bonn.de/people/richter/.
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Realization of Primitive Branched Coverings over Closed Surfaces Following the Hurwitz Approach Semeon Bogatyi¤1 , Daciberg L. Gon»calvesy2 , Elena Kudryavtsevaz1, Heiner Zieschangx1;3 1
Mechanics-Mathematics Faculty, Moscow State Lomonossov-University, 119992 Moscow - Russia 2 Departamento de Matem¶atica - IME-USP, Caixa Postal 66281- Ag^encia Cidade de S~ao Paulo, 05311-970 - S~ao Paulo - SP - Brasil 3 Institut fÄur Mathematik, Ruhr-UniversitÄat Bochum, 44780 Bochum-Germany
Received 7 January 2003; revised 3 March 2003 Abstract: Let V be a closed surface, H ³ º 1 (V ) a subgroup of nite index ` and D = [A1 ; : : : ; Am ] a collection of partitions of a given number d ¶ 2 with positive defect v(D). When does there exist a connected branched covering f : W ! V of order d with branch data D and f# (º 1 (W )) = H? It has been shown by geometric arguments [4] that, for ` = 1 and a surface V di¬erent from the sphere and the projective plane, the corresponding branched covering exists (the data D is realizable) if and only if the data D ful lls the Hurwitz congruence v(D) ² 0 mod 2. In the case ` > 1, the corresponding branched covering exists if and only if v(D) ² 0 mod 2, the number d=` is an integer, and each partition Ai 2 D splits into the union of ` partitions of the number d=`. Here we give a purely algebraic proof of this result following the approach of Hurwitz [11]. The realization problem for the projective plane and ` = 1 has been solved in [7,8]. The case of the sphere is treated in [1, 2, 12, 7]. c Central European Science Journals. All rights reserved. ®
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z
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Keywords: covering, branched covering of surfaces, branching order, Hurwitz problem, representations to the symmetry groups §d MSC (2000): 55M20, 57M12, 20F99 E-mail: E-mail: E-mail: E-mail:
[email protected] [email protected] [email protected] [email protected], E-mail: [email protected]
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1
185
Introduction
Branched coverings of surfaces have been studied by Hurwitz since 1891, see [11]. He gave a result for the existence of branched coverings and also for the classi¯cation of them. For the existence he postulated a condition on the branch data { that is the collection of the branching orders at the di®erent branch points { and the connectedness of the covering surface. Very little has been done for the more re¯ned, yet still natural question of existence of branched coverings between surfaces p : W ! V under the additional condition that the image of the fundamental group º 1 (W ) under p# is a given subgroup H » º 1 (V ) of ¯nite index; in particular, for the main, or primitive, case where H = º 1 (V ). In this paper we will extend results of the literature and give a full solution of this problem assuming that the target V is neither the sphere nor the projective plane. The question is transformed to a simple arithmetic one, see Theorem 2.4. Our arguments in the proofs are of purely algebraic nature following the approach of Hurwitz [11]. The next section contains a more detailed presentation of the problem, its history and the main results in an introductory form. In section 3, we consider the reduction to an algebraic group theoretical problem. In section 4, we introduce a \gluing" operation of homomorphisms to symmetric groups, which is used to answer the algebraic question in sections 5 (for primitive branched coverings over the torus), 6 (over the Klein bottle), and 7 (for the general case).
2
On the classi¯cation of branched coverings
Let us ¯rst recall some basic notions and facts. Let f : W ! V be a branched covering of ¯nite degree d between closed connected surfaces and let x1 ; : : : ; xm 2 V be the points over which the branching occurs. Assume that over xi there are ri points with branching orders d i1 ; : : : ; diri where these numbers form a partition of d, that is, d = di1 + : : : + d iri ;
1 µ d ij 2 Z; i 2 f1; : : : ; mg:
We denote this partition by Ai = [di1 ; : : : ; d iri ] and call D = [A 1 ; : : : ; A m ] the branch data of the branched covering. The number à ! m X ri m ri m m X X X X X ¡ ri + v(D) = (dij ¡ 1) = d ij = (d ¡ ri ) = md ¡ ri i=1 j=1
i=1
j=1
i=1
i=1
is non-negative and is called the defect of the branched covering. It has the following important property: v(D) ² 0 mod 2 which is called the Hurwitz congruence [11]. A proper branching happens if and only if v(D) > 0. We choose to call a system D = [A 1 ; : : : ; Am ], with Ai = [d i1 ; : : : ; diri ], of partitions of d with the aforementioned properties virtual branch data of order d. Finally, the branched covering f : W ! V de¯nes the subgroup H = f# (º 1 (W )) < º 1 (V ) of
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¯nite index; in fact, its system of conjugate subgroups is a suitable invariant of f . The covering is called primitive if H = º 1 (V ). Given a connected closed surface V , virtual branch data D, and a subgroup H of º 1 (V ) the following questions arise: Problems 2.1. (a) Does there exist a connected branched covering f : W ! V with the branch data D? (b) Does there exist a primitive connected branched covering f : W ! V with the branch data D? (c) Does there exist a connected branched covering f : W ! V with the branch data D and f# (º 1 (W )) = H ? (d) How many \di®erent" connected branched coverings solve the considered problem? Edmonds, Kulkarni and Stong [7] gave a positive answer to the ¯rst question for any surface V 6= S 2 and the full answer to questions (a) and (b) for the projective plane. Positive answers to questions (b) and (c) are given for all closed surfaces di®erent from the sphere and the projective plane in [4]. The proof in [4] consists of constructing the corresponding branched coverings by ¯gures for \small" cases and using a gluing procedure for the general case. Here we follow the Hurwitz approach [11] of constructing branched coverings using representations of the fundamental groups in symmetric groups. The suitable representations were found by looking at the ¯gures from [4], but the formal algebraic proof of the claim given in this article appears simpler and more algorithmic than the geometric one. Let us also remark that, according to the theorem of Gabai-Kazez [9], Problem 2.1 (c) is not only of interest in itself, but also plays an important role in the Nielsen theory to ¯nd the minimal number of roots in the homotopy class of the given mapping [3]. Two branched coverings fi : W i ! V are considered as equivalent if there exists a homeomorphism h : W 1 ! W 2 such that f2 = f1 ¯ h. A lower bound for the number asked for in Problem 2.1 (d) could be found from a solution of the following problem: What is the maximal number m such that the branched covering is the composition of m branched coverings of degree ¶ 2? It would also be of interest to ¯nd other invariants of branched coverings of geometric nature. By the Hurwitz approach, with each branched covering f of degree d over a closed surface V with the set of branch points Bf » V one associates a homomorphism ’f : º 1 (V n Bf ) ! §d called the Hurwitz representation realized by f , see 3.1. We ¯rst ¯nd out the necessary and su±cient algebraic conditions to guarantee that a representation is realized by a branched covering admitting the prescribed subgroup H , see Theorem 3.2. Then, for given virtual branch data and a subgroup of ¯nite index in the fundamental group of the closed surface, we construct representations of the fundamental groups to the symmetric groups which provide the desired branched coverings. Our main algebraic results are the following two implying the existence of primitive branched coverings over the torus and the Klein bottle. By the commutator and the
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quasi-commutator of two elements a; b we mean [a; b] = aba¡1 b¡1 and [a; b]¡ = abab¡1 . Theorem 2.2. For each partition A = [d1 ; : : : ; d r ] of the number d with a positive even defect v(A) = (d1 ¡ 1) + : : : + (d r ¡ 1) = d ¡ r > 0 there are permutations a^; ^b 2 §d with the following properties: (a) (b) (c) (d)
The The The The
subgroup of §d generated by ^a; ^b acts transitively on f1; : : : ; dg. commutator [^a; ^b] consists of cycles of the lengths d 1 ; : : : ; dr . symbol 1 is ¯xed under the action of ^a. symbol 1 is ¯xed under ^b[^a; ^b] or under ^b[^a2 ; ^b].
Theorem 2.3. For each partition A = [d1 ; : : : ; d r ] of the number d with a positive even defect v(A) = (d 1 ¡ 1) + : : : + (d r ¡ 1) = d ¡ r > 0 there are permutations a^; ^b 2 §d and a natural number q with the following properties: (a) (b) (c) (d)
The The The The
subgroup of §d generated by ^a; ^b acts transitively on f1; : : : ; dg. quasi-commutator [^a; ^b]¡ consists of cycles of the lengths d 1 ; : : : ; dr . symbol 1 is ¯xed under the action of ^a. symbol 1 is ¯xed under ^b[^a; ^b]q¡ .
These two results of special nature can easily be joined to a geometric result on branched coverings over surfaces of arbitrary genus. We say that the subgroup H of º 1 (V ) corresponds to the branched covering f : W ! V if H = f# (º 1 (W )). Theorem 2.4. [4, Theorem 4.2 ] Let V be a closed surface di®erent from the sphere and the projective plane, H » º 1 (V ) a subgroup, and let D = [A 1 ; : : : ; Am ] be some virtual branch data of order d. Then the following two assertions are equivalent. (1) The subgroup H corresponds to some connected branched covering between closed surfaces realizing the branch data D. (2) H is a subgroup of ¯nite index ` such that `jd. For each i 2 f1; : : : ; mg there exist ` partitions Bi1 = [di11 ; : : : ; di1ri1 ]; : : : ; Bi` = [di`1 ; : : : ; di`ri` ] of the number d=` such that A i = Bi1 t : : : t Bi` = [d i11 ; : : : ; d i1ri1 ; : : : ; di`1 ; : : : ; di`ri` ]: The algebraic version of this theorem follows. For a homomorphism ’ : º ! §d of a group º , consider the corresponding action of º on the set f1; : : : ; dg. Denote by Stab’ (k), 1 µ k µ d the stabilizer of the symbol k under this action. By hhx1 ; : : : ii we denote the smallest normal subgroup containing the elements x1 ; : : : . Q¤ ¢(s1 : : : sm )i where n ¶ 2, m ¶ 1, and Theorem 2.5. Let º = ha1 ; : : : ; an ; s1 ; : : : ; sm j Q¤ Qn=2 Q¤ 2 2 = i=1 [a2i¡1 ; a2i ] or = a1 ¢ : : : ¢ an , and let H < º =hhs1 ; : : : ; sm ii be a subgroup.
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Furthermore let ¼ 1 ; : : : ; ¼ m 2 §d , where ¼ i 6= id for at least one i, be some permutations Q such that m i=1 ¼ i is an even permutation. Denote by Ai the collection of the orders of the cycles of ¼ i . Then the following two assertions are equivalent: (1) There exists a homomorphism ’ : º ! §d such that (a) the group ’(º ) < §d acts transitively on f1; : : : ; dg, (b) ’(si ) is conjugate to ¼ i in §d , (c) the image of the composition Stab’ (1) ,! º ! º =hhs1 ; : : : ; sm ii is H . (2) H is a subgroup of ¯nite index ` such that `jd. For each i 2 f1; : : : ; mg there exist ` partitions Bi1 = [di11 ; : : : ; di1ri1 ]; : : : ; Bi` = [di`1 ; : : : ; di`ri` ] of the number d=` such that A i = Bi1 t : : : t Bi` = [d i11 ; : : : ; d i1ri1 ; : : : ; di`1 ; : : : ; di`ri` ]:
3
A reduction to algebra
In this section we transform the problem of constructing (primitive) branched coverings over surfaces into algebraic terms. First we describe the Hurwitz representation associated to a branched covering. Hurwitz Representation 3.1. Let f : W ! V be a d-fold branched covering of a connected closed surface W over V , let Bf » V denote the set of branch points of f and ¤V 2 V n Bf , ¤W 2 f ¡1 (¤V ) the basepoints. Take a small disk U arround ¤V such that p¡1 (U ) consists of disjoint disks each of which is mapped homeomorphically to U . Enumerate the disks by f1; : : : ; dg where the disk with label 1 contains ¤W in its interior. Moreover let ¤W i be the point over ¤V in the i-th disk, in particular, ¤W = ¤W 1 . A closed path ® in V n Bf starting in ¤V admits, for each i 2 f1; : : : ; dg, a uniquely determined lift ® ~i that starts in ¤W i . Adjoining to i the label of the endpoint of ®~i , we obtain a permutation ’f (® ) lying in the symmetric group §d . This permutation remains the same when ® is continuously deformed in V n Bf such that the start and end of ® always stay at the basepoint ¤V . Thus ’f induces a homomorphism of the fundamental group of V n Bf to the symmetric group which we also denote by ’f ; now ’f : º 1 (V n Bf ; ¤V ) ! §d is called the Hurwitz representation associated to f . We can interpret this as an action of º 1 (V n Bf ; ¤V ) on f1; : : : ; dg. This action is transitive since W is connected. For details see [11], [5], [15, 6.7.2]. In the following we use the notation hii¼ = j if the permutation ¼ maps i to j. By (i1 ; i2 ; : : : ; ik ) we denote the cyclic permutation sending ij to ij+1 , 1 µ j µ k ¡ 1 and ik to i1 . As above, let Stab’f (k) denote the stabilizer of the symbol k under the action of º 1 (V n Bf ; ¤V ) on f1; : : : ; dg corresponding to the Hurwitz representation ’f , that is, Stab’f (k) = fa 2 º 1 (V n Bf ; ¤V ) j hki’f (a) = kg:
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Theorem 3.2. Let if : (V n Bf ; ¤V ) ! (V; ¤V ) be the inclusion. Under the hypotheses from above, f# (º 1 (W; ¤W )) = if# (Stab ’f (1)): Proof. Consider the restriction g = f jW nf ¡1(Bf ) and the inclusion jf : (W n f ¡1 (Bf ); ¤W ) ! (W; ¤W ). From 3.1, see also [13, x58], it follows that g# (º 1 (W n f ¡1 (Bf ); ¤W )) = Stab’f (1) » º 1 (V n Bf ; ¤V ): Clearly, the homomorphisms if # : º 1 (V n Bf ; ¤V ) ! º 1 (V; ¤V );
jf# : º 1 (W n f ¡1 (Bf ); ¤W ) ! º 1 (W; ¤W )
are surjective; hence, f# (º 1 (W; ¤W )) = f# ¯ jf # (º 1 (W n f ¡1 (Bf ); ¤W )) = if # ¯ g# (º 1 (W n f ¡1 (Bf ); ¤W )) = if# (Stab ’f (1)):
From the de¯nitions and Theorem 3.2, the following corollaries are direct consequences. Corollary 3.3. Let f : W ! V be a branched covering of order d between two connected closed surfaces, Bf » V the set of branch points, and ’f : º 1 (V n Bf ; ¤V ) ! §d the Hurwitz representation for f . Then the following conditions are equivalent: (a) f is primitive; (b) the composition Stab’f (k) ,! º 1 (V n Bf ; ¤V ) ! º 1 (V; ¤V ) is surjective for each symbol k 2 f1; : : : ; dg; (c) the composition Stab’f (k) ,! º 1 (V n Bf ; ¤V ) ! º 1 (V; ¤V ) is surjective for some symbol k 2 f1; : : : ; dg: Corollary 3.4. Let f : W ! V be a branched covering of order d between two connected closed surfaces, Bf » V the set of branch points, ’f : º 1 (V n Bf ; ¤V ) ! §d the Hurwitz representation for f , and H < º 1 (V; ¤V ) a subgroup. Then the following conditions are equivalent: (a) the subgroup H corresponds to the branched covering f ; (b) the image of the composition Stab’f (1) ,! º 1 (V n Bf ; ¤V ) ! º 1 (V; ¤V ) is H . Remark that the fundamental groups º 1 (V n Bf ; ¤V ) and º 1 (V; ¤V ) are isomorphic to the groups º and º =hhs1 ; : : : ; sm ii considered in Theorem 2.5 respecting the projections º 1 (V n Bf ; ¤V ) ! º 1 (V; ¤V ) and º ! º =hhs1 ; : : : ; sm ii. It follows from Corollary 3.4 that the Theorems 2.4 and 2.5 are equivalent.
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A Gluing Operation on Homomorphisms to Symmetric Groups
From two representations ’1 : G ! §n and ’2 : G ! §m we easily construct the direct sum ’1 £ ’2 : G ! §n+m , but it is not of geometric interest since in the corresponding covering the source consists of two connected components corresponding to the two given representations. To get a connected source we use a gluing procedure [4, Section 2] for the two covering surfaces and, thus, have to ¯nd in both surfaces a non-separating simple loop such that both curves are mapped to the same power of a loop of the target. We describe an algebraic version of the gluing operation, but only for the groups G = G § where G + = ha; b; c j [a; b]c¡1 i and G ¡ = ha; b; c j abab¡1 c¡1 i, the fundamental groups of the torus and the Klein bottle minus a \small" disk. We also assume that the two loops are mapped homeomorphically to the same loop of the standard homotopy class a. Notation 4.1. The image of an element g 2 G under a representation ’ to §d is denoted by g^; similarly, g^i denotes ’i (g) for i = 1; 2. To denote a permutation we write it either explicitly or by adding a \^". Construction of a Gluing Operation 4.2. Let the permutations a^1 and ^b1 generate a subgroup transitively acting on f1; : : : ; ng and assume that the symbol i1 stays invariant under a^1 . Similarly, let a^2 ; ^b2 generate a subgroup acting transitively on fn+1; : : : ; n+mg and let i2 be ¯xed under a^2 . Clearly, for n ¶ 2 it follows from the transitivity that ^b1 does not ¯x i1 . The element a^ 2 §n+m is de¯ned as the direct sum of a^1 and ^a2 , that is, it operates on the ¯rst n symbols like a^1 and on the last m like a^2 . In the following the direct sum of a^1 and a^2 is denoted by a^1 £ a^2 . The permutation ^b 2 §n+m is de¯ned as the direct sum ^b1 £ ^b2 , followed by the transposition (i1 ; i2 ) of the symbols i1 and i2 , that is, ^b = (^b1 £ ^b2 ) ¯ (i1 ; i2 ). For elements c^; c^1 ; ^c2 related to the a^; ^b; : : : as in the presentations of G + or G ¡ , it will be shown below (Proposition 4.4) that c^ = c^1 £ c^2 is the permutation corresponding to c = [a; b] or c = abab¡1 . The result of the gluing operation on the representations ’1 and ’2 is the representation ’ : G ! §n+m with ’(a) = a^ = ^a1 £ a^2 , ’(b) = ^b = (^b1 £ ^b2 ) ¯ (i1 ; i2 ). Proposition 4.3. If the subgroups generated by a^1 ; ^b1 and a^2 ; ^b2 transitively act on f1; : : : ; ng and fn + 1; : : : ; n + mg, respectively, then the subgroup generated by a ^; ^b transitively acts on f1; : : : ; n + mg. Proof. Observe that the orbit of any symbol i µ n contains i2 . In fact, there is a word in a^1 ; ^b1 which maps i to the symbol hi1 i^b¡1 1 . Take the shortest word with this property. ^ If we replace in this word a^1 by ^a and b1 by ^b then the obtained word in a^ and ^b also ^ maps the symbol i to hi1 i^b¡1 1 . If we next apply once more b then we obtain the symbol i2 . A consequence is that there is a word in a^; ^b that maps i into i2 . Analogously, there exists a word in ^a; ^b that transforms a symbol j ¶ n + 1, in particular i2 , into the symbol i1 . Hence, every symbol can be sent to i1 and this shows the transitivity. In the study of branched coverings over nonorientable surfaces an important role is
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played by the quasi-commutator [a; b]¡ = abab¡1 ; now a corresponds to a two-sided, but b to a one-sided curve. To unify the notions of the commutator and the quasi-commutator, let us consider a more general analog of the commutator. For integers r and s, let [a; b]rs = ar bas b¡1 . Then [a; b]1;¡1 = [a; b] and [a; b]1;1 = [a; b]¡ . Proposition 4.4. For any integers r and s, [^ a; ^b]rs = [^ a1 ; ^b1 ]rs £ [^ a2 ; ^b2 ]rs : Proof. Consider a symbol i µ n. We identify ^a1 ; ^b1 ; a^2 ; ^b2 with the elements a^1 = a^1 £ id; ^b1 = ^b1 £ id; a^2 = id £ ^a2 ; ^b2 = id £ ^b2 of §m+n . Then hii^ar^b^ as^b¡1 = hii^ar1^b1 ¯ (i1 ; i2 ) ¯ a^s^b¡1 : Suppose that hii^ar1^b1 6= i1 , thus hii^ ar^b^ as^b¡1 = hii^ ar1^b1 a^s1^b¡1 . Since the symbol i1 is ¯xed under ^a1 it follows that hii^ ar1^b1 ^as1 6= i1 . Hence, under the action of ^b¡1 the transposition (i1 ; i2 ) is not applied to the obtained element and, thus, hii^ ar1^b1 ^as1^b¡1 = hii^ ar1^b1 ^as1^b¡1 1 . For r^ hii^ a1 b1 = i1 we obtain hii^ ar^b^as^b¡1 = hii^ ar1^b1 (i1 ; i2 ) ^as^b¡1 = hi1 i(i1 ; i2 ) ^as^b¡1 = hi2 i^as^b¡1 = hi2 i^b¡1 = hi1 i^b¡1 as1^b¡1 1 = hi1 i^ 1 = hii^ ar1^b1 ^as1^b¡1 1 : For i ¶ n + 1 a similar consideration takes place.
5
Realization of Primitive Branched Coverings over the Torus
Proof of Theorem 2.2 by induction on r. We assume that d 1 ¶ d2 ¶ : : : ¶ dr . Case r = 1: From the condition that the defect v(A) is even and > 0 it follows that the number d is odd and > 2; hence, d = 2k + 1 with k > 0. Put 0 1 B 1 : : : k k + 1 : : : 2k 2k + 1 C ^a = @ A; 1 : : : k k + 2 : : : 2k + 1 k + 1 0 1 ^b = B @
1
:::
k
k + 1 k + 2 : : : 2k + 1 C A: 2k + 1 : : : k + 2 k + 1 1 : : : k
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Next we check the claims (a) { (d). (b): By a direct calculation it follows that [^ a; ^b] = (1; 2; : : : ; 2k; 2k + 1):
(¤)
(a) is a direct consequence of (¤), claim (c) is obvious. (d): h1i^b[^ a; ^b] = h2k + 1i[^ a; ^b] = 1
by (¤).
Case r ¶ 2: If there is an odd number dj then we consider the partitions A 0 = [d1 ; : : : ; d j¡1 ; dj+1; : : : ; dr ] and A00 = [dj ] of the numbers d0 = d ¡
d j = d1 + ¢ ¢ ¢ + dj¡1 + dj+1 + ¢ ¢ ¢ + dr
If dr = 1 we put j = r. Since the defect v([d j ]) = dj ¡
and
d 00 = d j :
1 of the partition [d j ] is even and
v(A) = v(A 0 ) + v(A00 ); both considered partitions have an even defect. Furthermore, v(A 0 ) > 0 and v(A00 ) = dj ¡ 1 ¶ 0, thus the induction hypothesis can be applied to both partitions except for the case d 00 = d j = 1 where v(A 00 ) = 0. But for the trivial permutation the properties (a) { (d) are easily checked. By induction hypothesis, there are pairs of permutations a^1 ; ^b1 and a^2 ; ^b2 realizing the corresponding partitions A0 and A00 of d0 and d00 . Since these pairs of permutations have the property (c), we can apply the gluing operation 4.2 to them. As the result of this operation, we obtain a pair of permutations a^; ^b. Let us check the properties (a) { (d) for them. The property (a) follows from Proposition 4.3; (b) follows from Proposition 4.4; and (c) is a consequence of h1i^ a = h1i^a1 = 1. To check the truth of (d) it su±ces to use i1 = 1 for the gluing operation. In fact, using Proposition 4.4, we have h1i^b = hh1i^b1 i(1; i2 ) = h1i^b1 =) ³ ´ h1i^b[^ a; ^b] = hh1i^b1 i [^ a1 ; ^b1 ] £ [^a2 ; ^b2 ] = h1i^b1 [^ a1 ; ^b1 ] = 1 ³ ´ h1i^b[^a2 ; ^b] = hh1i^b1 i [^ a21 ; ^b1 ] £ [^a22 ; ^b2 ] = h1i^b1 [^ a21 ; ^b1 ] = 1;
or
in dependence on the equality from (d) ful¯lled by a^1 ; ^b1 . Now let all d i be even. Then the number d is also even and, thus, r too. Consider the case r = 2. It follows from the hypothesis that d = 2k for some k ¶ 2 and that d1 and d 2 are even. Thus ` = (d 1 ¡ d2 )=2 is an integer with 0 µ ` µ k ¡ 2; now d1 = k + ` and d 2 = k ¡ `. Next we consider a more general situation assuming only that d 2 > 1, that is, ` µ k¡ 2. In other words, for the next steps we do not need that d 1 and d2 are even, but only that d2 > 1.
S. Bogatyi et al. / Central European Journal of Mathematics 2 (2003) 184{197
For 0 µ ` µ k ¡ 0
193
2 we de¯ne 1
B 1 : : : k ¡ 1 k k + 1 : : : 2k ¡ 1 2k C a^ = @ A; ¡ 1 : : : k 1 k + 1 k + 2 : : : 2k k 0 1 1 : : : k ¡ 1 k k + 1 : : : k + ` k + ` + 1 k + ` + 2 : : : 2k C ^b = B @ A: 2k ¡ 1 : : : k + 1 k 1 : : : ` 2k `+ 1 ::: k ¡ 1
The columns with the upper symbols k + 1; : : : ; k + ` in the last formula are ignored if ` = 0. Next we check the claims (a) { (d). (b): By a direct calculation we obtain 0 1 2k B 1 : : : k + ` ¡ 1 k + ` k + ` + 1 : : : 2k ¡ 1 C [^ a; ^b] = @ A 2 ::: k + ` 1 k + ` + 2 : : : 2k k + ` + 1 = (1; 2; : : : ; k + `)(k + ` + 1; : : : ; 2k):
(a): The transitivity follows from the fact that the commutator consists of two cycles and that a^ maps the symbol 2k from the second cycle to the symbol k of the ¯rst cycle. The property (c) is obvious. (d): This follows from h1i^b = 2k ¡ 1 and h2k ¡ 1i[^ a2 ; ^b] = 1. If r ¶ 3 then r ¶ 4 and d r ¶ 2. Since any two permutations from §d are conjugate in §d if they admit (up to a permutation) the same systems of lengths of their cycles, we can apply the gluing operation to pairs of permutations realizing the partitions A0 = [d 1 ; : : : ; dr¡2 ] and A 00 = [d r¡1 ; dr ] of the numbers d 0 = d ¡ d r¡1 ¡ dr = d1 + : : : + dr¡2 and d00 = dr¡1 + d r . Corollary 5.1. Let c^ 2 §d be a non-trivial even permutation. Then there are permutations a^; ^b with the following properties: (a) (b) (c) (d)
The permutations a^; ^b generate a subgroup of §d which transitively acts on f1; : : : ; dg. c^ = [^a; ^b]. The symbol 1 is ¯xed under the action of ^a. The symbol 1 is ¯xed under ^b[^a; ^b] or under ^b[^a2 ; ^b].
Notice, the usual proof [7,8,12] of the existence theorem of a branched covering with given branch data of even defect over the torus and surfaces of higher genus is obtained as the geometric equivalent (according to the Hurwitz criterion) of the algebraic fact that each even permutation c^ is the commutator of two permutations where one of these permutations is a large cycle and the other one admits a ¯xed symbol. Corollary 5.1 of Theorem 2.2 gives a new proof that each even permutation is the commutator of
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two permutations generating a transitive subgroup. In our representation of the even permutation c^ as a commutator of a^ and ^b, the permutation a^ has exactly v(^ c)=2 + e ¯xed symbols, where e is the number of cycles of length 1 of the permutation c^ = [^ a; ^b].
6
Realization of Primitive Branched Coverings over the Klein Bottle
Proof of Theorem 2.3 by induction on r. We assume that d 1 ¶ : : : ¶ d r . Case r = 1. From the condition that the defect v(A) is even it follows that d = 2k + 1 for some integer k ¶ 1. Consider 0 1 B 1 : : : k k + 1 : : : 2k 2k + 1 C ^a = @ A; 1 : : : k k + 2 : : : 2k + 1 k + 1 0 1 1 2 : : : k + 1 k + 2 : : : 2k + 1 C ^b = B @ A: 2k + 1 k + 1 : : : 2k 1 ::: k
(b): By a direct calculation we obtain
[^a; ^b]¡ = (1; 2; : : : ; 2k; 2k + 1): (a) is a direct consequence of (b), claim (c) is obvious. (d): ^b maps the symbol 1 to the symbol 2k + 1 and this is mapped by [^a; ^b]¡ back into 1. Case r ¶ 2. If there is an odd d j , then the assertion is obtained by the same arguments as in the proof of Theorem 2.2. Therefore, we may assume that all numbers d j are even. Then d and thus, r are even. Consider the case r = 2. From the conditions it follows that d = 2k for some k ¶ 2 and that d 1 ; d 2 are even. Put ` = (d 1 ¡ d 2 )=2, thus d 1 = k + ` and d2 = k ¡ `. Now we obtain more general conclusions which are possible for d 2 > 1, that is, ` µ k ¡ 2. In other words, for the following discussion we may only assume that d 2 > 1. For 0 µ ` µ k ¡ 2 we de¯ne 0 1 B 1 : : : k ¡ 1 k k + 1 : : : 2k ¡ 1 2k C ^a = @ A; 1 : : : k ¡ 1 k + 1 k + 2 : : : 2k k 0 1 1 ::: k k + 1 : : : k + ` k + ` + 1 k + ` + 2 : : : 2k C ^b = B @ A: k : : : 2k ¡ 1 1 : : : ` 2k ` + 1 ::: k ¡ 1
The columns with the upper symbols k + 1; : : : ; k + ` in the last formula are ignored if ` = 0.
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(b): A direct calculation gives 0
195
1
2k B 1 : : : k + ` ¡ 1 k + ` k + ` + 1 : : : 2k ¡ 1 C [^ a; ^b]¡ = @ A 2 ::: k + ` 1 k + ` + 2 : : : 2k k + ` + 1 = (1; 2; : : : ; k + `)(k + ` + 1; : : : ; 2k):
Now (a) follows from the facts that the quasi-commutator consists of two cycles and that under the action of a^ the symbol 2k from the second cycle goes into the symbol k from the ¯rst cycle. (c) is obvious. (d): Under the action of ^b the symbol 1 goes into k and this one is mapped by [^a; ^b]`+1 ¡ to 1. The case r ¶ 3, i.e. r ¶ 4 can be handled in the same way as at the end of the proof of the Theorem 2.2. Corollary 6.1. Let c^ 2 §d be a non-trivial even permutation. Then there are permutations a^; ^b and a natural number q with the following properties: (a) (b) (c) (d)
The permutations a^; ^b generate a subgroup of §d which transitively acts on f1; : : : ; dg. c^ = [^a; ^b]¡ . The symbol 1 is ¯xed under the action of ^a. The symbol 1 is ¯xed under ^b[^a; ^b]q¡ .
Thus, each non-trivial even permutation c^ 2 §d is the quasi-commutator of two permutations which generate a subgroup of §d acting transitively on f1; : : : ; dg. Since [^a; ^b]¡ = a ^21^b¡2
with
a^1 = a^^b
we also obtain that c^ is the product of the squares of two permutations which generate a subgroup of §d acting transitively on f1; : : : ; dg.
7
The General Case
Proof of Theorem 2.5 for the primitive case H = º =hhs1 ; : : : ; sm ii. Remark that, in this case, the condition (2) of Theorem 2.5 is always true. So, we must prove that (1) is always true too. First consider the case º = ha; b; s1 ; : : : ; sm j [a; b] ¢ (s1 : : : sm )i where m ¶ 1. Assume m = 1 and denote A 1 = [d1 ; : : : ; d r ]. According to Corollary 5.1 there are two permutations ^a; ^b 2 §d such that the commutator [^a; ^b] consists of cycles of lengths d1 ; : : : ; d r . The Hurwitz representation ’ of the group º = ha; b; c j [a; b]c¡1 i is given by a 7! ^a, b 7! ^b, c 7! [^ a; ^b]. Now the properties (a) and (b) follow from the assertions (a) and (b) of Corollary 5.1. By the assertions (c) and (d) of Corollary 5.1, the symbol 1 is ¯xed under the actions of a^ and ^b[^aq ; ^b] for an appropriate integer q, thus a; b[aq ; b] 2 Stab’ (1).
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Therefore, the composition Stab’ (1) ,! º ! º =hhcii is surjective. So, the property (c) is ful¯lled. Let m ¶ 2 and denote A i = [d i1 ; : : : ; dir ]. We take some permutations s^i 2 §d consisting of cycles of lengths d i1 ; : : : ; diri , respectively. If the permutation c^ = s^1 ¢ : : : ¢ s^m 6= e^, e^ the identity, we apply Corollary 5.1 to it. If s^1 ¢ : : : ¢ s^m = e^ then one of the following three cases is possible: (1) in some s^i there exists a cycle of length ¶ 3; (2) all cycles have length µ 2, but d ¶ 3; (3) d = 2. In the ¯rst case, replace s^i by s^¡1 ^i 6= e^ one symbol i . In the second case, permute in s appearing in a cycle of length 2 with a symbol appearing in another cycle. In both cases, the new s^i have cycles of the same length, but the product of s^i is not the identity, so Corollary 5.1 can be applied. Now the Hurwitz representation ’ : º ! §d is given by a 7! ^b, b 7! ^a, si 7! s^i , thus [a; b] ¢ (s1 : : : sm ) 7! [^ a; ^b]¡1 s^1 : : : s^m = e^. As for the case m = 1, the assertions (a) { (d) of Corollary 5.1 imply the required properties (a) { (c). In the third case, there is a s^i 6= e^. Take arbitrary permutations a^; ^b 2 §2 . The required properties (a) { (c) are easily checked. Q¤ Q ¢(s1 : : : sm )i where m ¶ 1, and ¤ = Assume that º = ha1 ; : : : ; a2g ; s1 ; : : : ; sm j Qg ! §d , we map a1 ; a2 , i=1 [a 2i¡1 ; a2i ]. For de¯nition of the Hurwitz representation º s1 ; : : : ; sm as above and map a3 ; : : : ; a2g to the identity permutation. Q¤ ¢(s1 : : : sm )i, where m ¶ 1, n ¶ 2, and For the case º = ha1 ; : : : ; an ; s1 ; : : : ; sm j Q¤ 2 2 = a1 ¢ : : : ¢ an , we proceed as before, but using Corollary 6.1. Proof of Theorems 2.5 and 2.4 in the general case. Since the Theorems 2.5 and 2.4 are equivalent, we have obtained Theorem 2.4 for the primitive case H = º 1 (V ) and it remains to prove it for the general case. (1) =) (2): Let f : W ! V be a branched covering with f# (º 1 (W )) = H . Consider the unbranched covering p : V¹ ! V corresponding to the subgroup H . Then f lifts to f¹: W ! V¹ . Now, for any branch point x 2 Bf , the union of the branch data (with respect to f¹) of ` points fy1 ; : : : ; y` g = p¡1 (x) gives the branch data for f at x. (2) =) (1): Let p : V¹ ! V be the unbranched covering which corresponds to the ¹ = [B11 ; : : : ; B1` ; : : : ; Bm1 ; : : : ; Bm` ]. subgroup H . Consider the virtual branch data D Since À (V¹ ) = ` ¢ À (V ) µ 0, the surface V¹ is di®erent from the sphere and the projective plane. It follows from Theorem 2.4 for the primitive case that there is a connected ¹ Therefore p ¯ h is the required primitive branched covering h : W ! V¹ which realizes D. covering.
References [1] I. Berstein and A.L. Edmonds: \On the construction of branched coverings of lowdimensional manifolds", Trans. Amer. Math. Soc., Vol. 247, (1979), pp. 87{124. [2] I. Berstein and A.L. Edmonds: \On the classi¯cation of generic branched coverings of surfaces", Illinois. J. Math., Vol. 28, (1984), pp. 64{82 .
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[3] S. Bogatyi, D.L. Gon»calves, E. Kudryavtseva, H. Zieschang: \Minimal number of roots of surface mappings", Matem. Zametki, Preprint (2001). [4] S. Bogatyi, D.L. Gon»calves, E. Kudryavtseva, H. Zieschang: \Realization of primitive branched coverings over closed surfaces", Kluwer Academic Publishers, Preprint (2002). [5] S. Bogatyi, D.L. Gon»calves, H. Zieschang: \The minimal number of roots of surface mappings and quadratic equations in free products", Math. Z., Vol. 236, (2001), pp. 419{452. [6] A.L. Edmonds: \Deformation of maps to branched coverings in dimension two", Ann. Math., Vol. 110, (1979), pp. 113{125. [7] A.L. Edmonds, R.S. Kulkarni, R.E. Stong: \Realizability of branched coverings of surfaces", Trans. Amer. Math. Soc., Vol. 282, (1984), pp. 773{790. [8] C.L. Ezell: \Branch point structure of covering maps onto nonorientable surfaces", Trans. Amer. Math. Soc., Vol. 243, (1978), pp. 123{133. [9] D. Gabai and W.H. Kazez: \The classi¯cation of maps of surfaces", Invent. math., Vol. 90, (1987), pp. 219{242. [10] D.L. Gon»calves and H. Zieschang: \Equations in free groups and coincidence of mappings on surfaces", Math. Z., Vol. 237, (2001), pp. 1{29. Ä [11] A. Hurwitz: \Uber Riemannische FlÄache mit gegebenen Verzweigungspunkten", Math. Ann., Vol. 39, (1891), pp. 1{60. [12] D.H. Husemoller: \Rami¯ed coverings of Riemann surfaces", Duke Math. J., Vol. 29, (1962), pp. 167{174. [13] H. Seifert and W. Threlfall: Lehrbuch der Topologie, Teubner, Leipzig, 1934. [14] R. Skora: \The degree of a map between surfaces", Math. Ann., Vol. 276, (1987), pp. 415{423. [15] R. StÄocker and H.Zieschang: Algebraische Topologie, B.G. Teubner, Stuttgart, 1994.
CEJM 2 (2003) 198{207
Standard monomials for q-uniform families and a conjecture of Babai and Frankl G¶abor Heged} us2 , Lajos R¶onyai12¤ 1
2
Computer and Automation Institute Hungarian Academy of Sciences H-1111 Budapest, L¶agym¶anyosi u. 11, Hungary Budapest University of Technology and Economics H-1111 Budapest, M^ uegyetem rkp. 3{9, Hungary
Received 28 October 2002; revised 11 March 2003 Abstract: Let n; k; ¬ q-uniform family
be integers, n; ¬
> 0, p be a prime and q = p . Consider the complete
F (k; q) = fK ³ [n] : jKj ² k (mod q)g: We study certain inclusion matrices attached to F (k; q) over the eld Fp . We show that if ` µ q ¡ 1 and 2` µ n then µ ¶ µ ¶ [n] n rank p I(F (k; q); )µ : µ` ` This extends a theorem of Frankl [7] obtained for the case ¬ = 1. In the proof we use arguments involving Grobner bases, standard monomials and reduction. As an application, we solve a problem of Babai and Frankl related to the size of some L-intersecting families modulo q. c Central European Science Journals. All rights reserved. ® Keywords: Grobner basis, inclusion matrix, set family MSC (2000): 05D05, 13P10, 05B20
1
Introduction
¤
Throughout the paper n will be a positive integer and [n] stands for the set f1; 2; : : : ; ng. The family of all subsets of [n] is denoted by 2[n] . For an integer 0 µ d µ n we denote by ¡[n]¢ ¡ [n]¢ ¡[n]¢ ¡ ¢ the family of all d element subsets of [n], and ·d = 0 [ : : : [ [n] the subsets of d d size at most d. E-mail: [email protected]
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199
Babai and Frankl conjectured the following in [3], p. 115. Theorem 1.1. Let k be an integer and q = p , ¬ ¶ 1, a prime power. Suppose that 2(q ¡ 1) µ n. Assume that F = fA1 ; : : : ; Am g is a family of subsets of [n] such that (a) jAi j ² k (mod q ) for i = 1; : : : ; m (b) jAi \ Aj j 6² k (mod q ) for 1 µ i; j µ m; i 6= j: Then mµ
µ
q¡
n 1
¶
:
In the proof we combine the linear algebra bound method with an argument involving GrÄobner-standard monomials and the corresponding reduction. For families F; G ³ 2[n] the inclusion matrix I(F ; G ) is a (0,1) matrix of size jF j £ jG j whose rows and columns are indexed by the elements of F and G , respectively. The entry at position (F; G) is 1 if G ³ F and 0 otherwise (F 2 F ; G 2 G ). Let p be a prime and k an integer. Let q = p , ¬ ¶ 1. Let F(k; q) = fK ³ [n] : jK j ² k (mod q)g: In Theorem 1:1 of [7] Frankl proved the following Theorem. Theorem 1.2. Let p be a prime and k an integer. If ` µ p ¡ 1 and 2` µ n, then µ ¶ µ ¶ [n] n rank p I (F(k; p); )µ : µ` ` Our proof of Theorem 1.1 relies on the following generalization of Theorem 1.2. Theorem 1.3. Let p be a prime and k an integer. Let q = p > 1. If ` µ q ¡ 2` µ n, then µ ¶ µ ¶ [n] n rank p I (F (k; q); )µ : µ` `
1 and
We give an equivalent form of Theorem 1.3 in Section 2 with the aid of standard monomials. The bound in Theorem 1.3 is sharp. We have equality here if ` µ k µ n ¡ ` or if k + ` + q µ n. Indeed, by a theorem of Wilson [11] (see also Corollary 3.1 in [10]) ¡ ¢ ¡[n]¢ ¡n ¢ the sub-matrix I ( [n] ; ) has rank , where m = k if ` µ k µ n ¡ ` and m = k + q ·` m ` if k + q + ` µ n.
2
Preliminaries
2.1 Polynomials, GrÄobner bases, standard monomials, reduction Let F be a ¯eld. As usual, F[x1 ; : : : ; xn ] denotes the ring of polynomials in variables x1 ; : : : ; xn over F. Let S = F[x1 ; : : : ; xn ]. In this paper F will be a ¯nite prime ¯eld Fp or
200
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the ¯eld of rational numbers Q. We denote by F[x1 ; : : : ; xn ]·s the vector space of all polynomials over F with degree at most s. Q For a subset F ³ [n] we write xF = j2F xj . In particular, x; = 1. We recall now some basic facts concerning GrÄobner bases in polynomial rings. A total order ¿ on the monomials (words) composed from variables x1 ; x2 ; : : : ; xn is a term order, if 1 is the minimal element of ¿ , and uw ¿ vw holds for any monomials u; v; w with u ¿ v. There are many interesting term orders. For the rest of the paper we assume that the term order ¿ we work with is the deglex order. Let u = xi11 xi22 ¢ ¢ ¢ xinn and v = xj11 xj22 ¢ ¢ ¢ xjnn be two monomials. Then u is smaller than v with respect to deglex (u ¿ v in notation) i® either deg u < deg v, or deg u = deg v and ik < jk holds for the smallest index k such that ik 6= jk . Note that we have xn ¿ xn¡1 ¿ : : : ¿ x1 . The leading monomial lm(f ) of a nonzero polynomial f 2 S is the largest (with respect to ¿ ) monomial which appears with nonzero coe±cient in f when written as a linear combination of monomials. Let I be an ideal of S. A ¯nite subset G ³ I is a GrÄobner basis of I if for every f 2 I there exists a g 2 G such that lm(g) divides lm(f ). In other words, the leading monomials of the polynomials from G generate the semi-group ideal of monomials flm(f ) : f 2 Ig. Using the fact that ¿ is a well founded order, it follows that G is actually a basis of I , i.e. G generates I as an ideal of S. It is a fundamental fact (cf. [6, Chapter 1, Corollary 3.12] or [1, Corollary 1.6.5, Theorem 1.9.1]) that every nonzero ideal I of S has a GrÄobner basis. A monomial w 2 S is called a standard monomial for I if it is not a leading monomial of any f 2 I. Let sm(¿ ; I ; F) stand for the set of all standard monomials of I with respect to the term-order ¿ over F. It follows from the de¯nition and existence of GrÄobner bases (see [6, Chapter 1, Section 4]) that for a nonzero ideal I the set sm(¿ ; I ; F) is a basis of the F-vector-space S=I . More precisely every g 2 S can be written uniquely as g = h + f where f 2 I and h is a unique F-linear combination of monomials from sm(¿ ; I ; F). Let vF 2 f0; 1gn denote the characteristic vector of a set F ³ [n]. For a family of subsets F ³ 2[n] , let V (F) = fvF : F 2 F g ³ f0; 1gn ³ Fn . To obtain information on polynomial functions on V (F ), it is natural to consider the ideal I (V (F )): I (V (F)) := ff 2 S : f (v) = 0 whenever v 2 V (F)g: If F ³ 2[n], then x2i ¡ xi 2 I (V (F)), hence x2i is a leading monomial for I (V (F )). It follows that the standard monomials for this ideal are all square-free, i.e. of form xG for G ³ [n]. We put Sm(¿ ; F; F) = fG ³ [n] : xG 2 sm(¿ ; I (V (F )); F)g ³ 2[n] : It is immediate that Sm(¿ ; F ; F) is a downward closed set system. Also, the standard monomials for I (V (F)) form a basis of the functions from V (F ) to F (see Section 4 in [2]), hence jSm(¿ ; F ; F)j = jF j:
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201
It is also easy to see that if H ³ G are arbitrary set systems on [n], then Sm(¿ ; H; F) ³ Sm(¿ ; G ; F). Now we introduce the notion of reduction. Let G be a set of polynomials in F[x1 ; : : : ; xn ] and let f 2 F[x1 ; : : : ; xn ] be a ¯xed polynomial. We can reduce f by the set G with respect to ¿ . This gives a new polynomial h 2 F[x1 ; : : : ; xn ]. Here reduction means that we possibly repeatedly replace monomials in f by smaller ones (with respect to ¿ ) as follows: if w is a monomial occurring in f and lm(g) divides w for some g 2 G (i.e. w = lm(g)u for some monomial u), then we replace w in f with u(lm(g) ¡ g). Clearly the monomials in u(lm(g) ¡ g) are ¿ -smaller than w. It is a fundamental fact that if G is a GrÄobner basis of I, then with G we can reduce every polynomial into a linear combination of standard monomials for I . In particular, f 2 I i® f can be G -reduced to 0. Let I be an ideal of S = F[x1 ; : : : ; xn ]. The Hilbert function of the algebra S=I is the sequence hS=I (0); hS=I (1); : : :. Here hS=I (m) is the dimension over F of the factor-space F[x1 ; : : : ; xn ]·m =(I \ F[x1 ; : : : ; xn ]·m ) (see [5, Section 9.3]). In the case when I = I (V (F )) for some set system F ³ 2[n], the number hF (m) := hS=I (m) is the dimension of the space of functions from V (F ) to F which can be represented as polynomials of degree at most m. In the combinatorial literature this important quantity is expressed in terms of inclusion matrices. It is straightforward to verify that µ ¶ [n] hF (m) = rank I (F ; ): (1) µm On the other hand, hS=I (m) is the number of standard monomials of degree at most m with respect to an arbitrary degree-compatible term order, for instance deglex.
2.2 The polynomials f H;d We introduce a family of polynomials with integer coe±cients. They played an important role in the description of the GrÄobner bases for the complete uniform families given in [10]. Let t be a integer, 0 < t µ n=2. We de¯ne Ht as the set of those subsets H = fs1 < s2 < ¢ ¢ ¢ < st g of [n] for which t is the smallest index j with sj < 2j. Thus, the elements of Ht are t-subsets of [n]. We have H 2 Ht i® s1 ¶ 2; : : : ; st¡1 ¶ 2t ¡ 2 and st < 2t. It follows that st = 2t ¡ 1, and if t > 1, then st¡1 = 2t ¡ 2. For the ¯rst few values of t it is easy to give Ht explicitly: we have H1 = ff1gg and H2 = ff2; 3gg, and H3 = ff2; 4; 5g; f3; 4; 5gg. For a subset J ³ [n] and an integer 0 µ i µ jJj we denote by ¼ J;i the i-th elementary symmetric polynomial of the variables xj , j 2 J: X ¼ J;i := xT 2 Z[x1 ; : : : ; xn ]: T µJ;jT j=i
In particular, ¼
J;0
= 1.
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Now let 0 < t µ n=2, 0 µ d µ n and H 2 Ht . Put H 0 = H [ f2t; 2t + 1; : : : ; ng ³ [n]. We write µ ¶ t X i t¡i d ¡ fH;d = fH;d (x1 ; : : : ; xn ) := (¡ 1) ¼ H 0;i : t¡ i i=0 Speci¯cally, we have ff1g;d = x1 + x2 + ¢ ¢ ¢ + xn ¡ ff2;3g;d = ¼
U;2
¡
d, and µ ¶ d 1)¼ U;1 + ; 2
(d ¡
where U = f2; 3; : : : ; ng. Assume that 0 µ d µ n=2. We write Md := ffs1 < : : : < sj g » [n] : j µ d and si ¶ 2i for 1 µ i µ jg:
(2)
In particular ; 2 Md . The sets Md were studied in [2], [8] and [10] in connection with order shattering and GrÄobner bases for uniform families. In Theorem 1:4 and Lemma 2:3 of [2] it was proved that µ ¶ n jM d j = d holds for 0 µ d µ n=2. The following statement is from [10]. We include a proof for the reader’s convenience. Characteristic vectors are interpreted now as elements of Zn . Proposition 2.1. Assume that 0 < t µ n=2, H 2 Ht and 0 µ d µ n. (a) The degree of fH;d is t, lm(fH;d ) = xH , and the leading coe±cient is 1. (b) If D ³ [n], jDj = d, then fH;d (vD ) = 0. Proof. (a) From the de¯nition of fH;d it is immediate that deg fH;d µ t. The coe±cient ¡ ¢ of xH in fH;d is (¡ 1)t¡t d¡t = 1, giving also that deg fH;d = t. Also, H is the lexicot¡t graphically largest among the subsets of H 0 with at most t elements. This implies that xH is the leading monomial of fH;d . (b) Write v = vD . Recall that H 0 = H [ f2t; : : : ; ng has n ¡ t + 1 elements. From jDj = d this gives that jD \ H 0 j 2 fd; d ¡
1; : : : ; d ¡
t + 1g:
(3)
We have fH;d (v) =
t X
(¡ 1)
k=0
t¡k
µ ¶ d¡ k ¼ t¡ k
H 0;k (v)
=
t X
(¡ 1)
k=0
t¡k
µ
d¡ k t¡ k
¶µ ¶ jD \ H 0 j : k
We use the following identity involving binomial coe±cients µ
x¡
d+t¡ t
1
¶
=
t X k=0
(¡ 1)
t¡k
µ ¶µ ¶ x d¡ k ; k t¡ k
(4)
G. Heged} us, L. R´onyai / Central European Journal of Mathematics 2 (2003) 198{207
203
valid for every x 2 C, d 2 Z and t 2 Z+ . From (4) we infer that µ ¶ jD \ H 0 j ¡ d + t ¡ 1 fH;d (v) = ; t which is indeed 0 because of (3). It remains to prove (4). We consider ¯rst the Vandermonde identity ([9], pp. 169-170) µ ¶ X ¶ t µ ¶µ x+s x s = ; (5) t k t¡ k k=0
which holds for all x; s 2 C and t 2 Z+ . By negating the upper index s on the right-hand side we obtain µ ¶ X µ ¶ t µ ¶ x+s x s¡ k¡ 1 t¡k t ¡ = (¡ 1) : ¡ k t k t k=0 Finally the substitution s = t ¡
d¡
1 gives (4).
h
The next statement is a slight generalization of an argument from [10]. Proposition 2.2. Assume that 0 µ ` µ n=2 and let G
`
= fg H : H 2 Ht ; 0 < t µ `g [ fx21 ¡
x1 ; : : : ; x2n ¡
xn g » F[x1 ; : : : ; xn ]
be a collection of polynomials such that the degree of gH is t and lm(gH ) = xH . Let f 2 F[x1 ; : : : ; xn ] be a polynomial, deg f µ `, which is irreducible with respect to G ` and ¿ . Then f is an F-linear combination of monomials from N
`
:= fxG : G 2 M` g » F[x1 ; : : : ; xn ]:
Remark. The term irreducible means here that no reduction of f by any element of G is possible.
`
Proof. Assume for contradiction that f contains a monomial w not in N ` . We have deg w µ `. Also, w is square-free; otherwise we could reduce it further by some binomial x2i ¡ xi . We have therefore w = xU for some U = fu1 < : : : < um g with m µ `. The condition U 2 = M` means now that there is an index i µ m µ ` such that ui < 2i. Let t be the smallest such index i. Then by the de¯nition of t we have ui ¶ 2i for 1 µ i < t, hence fu1 < : : : < utg := H 2 Ht . Also, we see that H ³ U . But then the leading term xH of g H divides w, hence f is reducible with respect to G ` , a contradiction. This proves the statement.
3
The main results
3.1 A generalization of Frankl’s Theorem In view of (1) and the subsequent remark, Theorem 1.2 follows from the next statement.
204
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Theorem 3.1. Let ¿ and 2` µ n, then
be the deglex order, p be a prime and k an integer. If ` µ p ¡
1
µ
¶ µ ¶ [n] n jSm(¿ ; F (k; p); Fp) \ jµ : µ` `
We give a generalization to q-uniform families. It is slightly stronger than Theorem 1.3. Theorem 3.2. Let ¿ be the deglex order, p be a prime and q = p > 1. Suppose that k; ` 2 N for which 0 µ k; ` < q, and 2` µ n. Then µ ¶ [n] ³ M` ; Sm(¿ ; F (k; q); Fp ) \ µ` hence
µ
¶ µ ¶ [n] n jSm(¿ ; F (k; q); Fp) \ jµ : µ` `
Proof. We intend to use Proposition 2.2. We exhibit a set of polynomials G ` » Fp [x1 ; : : : ; xn ] satisfying the conditions of the Proposition, with the additional property that all elements of G ` vanish on the characteristic vectors of the family F(k; q). This su±ces, because the standard monomials for V (F(k; q)) must be irreducible with respect to any set of polynomials which vanish on the set, in particular, with respect to G ` . On the other hand, the G ` -irreducible monomials are in N ` by Proposition 2.2. These allow us to conclude that the standard monomials of degree at most ` for the complete q-uniform family are in N ` . We thus turn to the construction of G ` . Obviously the binomials x2i ¡ xi 2 Fp[x1 ; : : : ; xn ] vanish on V (F (k; q)). For 0 < t µ ` and H 2 Ht we de¯ne gH 2 Fp [x1 ; : : : ; xn ] as the modulo p reduction of the polynomial (with integer coe±cients) fH;k . By Proposition 2.1 (a) the degree of gH is t and the leading term of gH is xH . It su±ces to verify that gH (vD ) = 0 for the characteristic vectors vD of elements D 2 F (k; q). We recall the following simple fact. Lemma 3.3. Let q = p > 1 a prime power. Let x; j be integers, 0 µ j < q. Then µ ¶ µ ¶ x+q x ² (mod p): j j Proof. The congruence follows from the Vandermonde identity (5) with s = q and t = j, ¡¢ by noting that the binomial coe±cients qi vanish modulo p for 1 µ i < q. Now let D 2 F(k; q), and write v = vD . Then jDj = k 0 for some k 0 such that 0 µ k 0 µ n and k ² k 0 (mod q). We observe that fH;k ² fH;k 0 (mod p), i.e., the coe±cients of the two polynomials are the same modulo p. This holds because for 0 µ i µ t we have µ ¶ µ 0 ¶ k¡ i k ¡ i ² (mod p); t¡ i t¡ i
G. Heged} us, L. R´onyai / Central European Journal of Mathematics 2 (2003) 198{207
a consequence of 0 µ t ¡ We conclude that
iµq¡
205
1 and Lemma 3.3.
gH (v) ² fH;k (v) ² fH;k 0(v) = 0
(mod p):
Here the last equality follows from Lemma 2.1 (b). The proof is complete.
3.2 The conjecture of Babai and Frankl We prove Theorem 1.1 by a combination of the linear algebra argument presented in Theorem 5.30 of [3] with Theorem 3.2. We make ¯rst some preparations. The following fact was proved in Proposition 5:31 of [3]. Proposition 3.4. Let q = p , p a prime, and ¬ ¶ 1. For any integer r, the binomial ¡ ¢ coe±cient r¡1 is divisible by p i® r is not divisible by q. h q¡1
Let f (x1 ; : : : ; xn ) 2 Q[x1 ; : : : ; xn ] be a polynomial. The square-free reduction f 0 of f is obtained by reducing f with respect to the set of polynomials fx21 ¡ x1 ; : : : ; x2n ¡ xn g. In other words, we replace x2i with xi as long as it is possible. Clearly f 0 is a Q-linear combination of monomials xU , U ³ [n]. It is immediate that deg f 0 µ deg f and f (v) = f 0 (v) for every vector v 2 f0; 1gn. Lemma 3.5. Let f 2 Q[x1 ; : : : ; xn ] be a polynomial such that f (v) 2 Z for every v 2 f0; 1gn . Let f 0 be the square-free reduction of f . Then f 0 2 Z[x1 ; : : : ; xn ]: Proof. We have f 0 (x1 ; : : : ; xn ) =
X
¬
H
¢ xH ;
(6)
Hµ[n]
where ¬ H 2 Q. Suppose for contradiction that f 0 62 Z[x1 ; : : : ; xn ]. Then there exists G ³ [n] such that ¬ G 2 Q n Z. Let K be minimal with respect to inclusion among P those subsets G. Obviously ¬ K 62 Z. Then f (vK ) = f 0 (vK ) = Y µK ¬ Y xY (vK ) = P P 2 Z. Also Y ½K ¬ Y 2 Z, by the minimality of K . These imply that Y ½K ¬ Y + ¬ KP 0 ¬ K = f (vK ) ¡ Y ½K ¬ Y is also in Z, a contradiction. This proves the claim. Proof (of Theorem 1.1). Let vi denote the characteristic vector of the set Ai . Let us consider the polynomials µ ¶ x ¢ vi ¡ k ¡ 1 fi (x1 ; : : : ; xn ) = q¡ 1
in n rational variables x = (x1 ; : : : ; xn ) 2 Qn (i = 1; : : : ; m); where a ¢ b denotes the scalar product in Qn . Denote by fi0 the square-free reduction of fi for i = 1; : : : ; m. Then fi0 2 Z[x1 ; : : : ; xn ], because fi (v) 2 Z for each v 2 f0; 1gn , and hence Lemma 3.5 applies. Let gi 2 Fp [x1 ; : : : ; xn ] denote the reduction of fi0 modulo p and hi 2 Fp[x1 ; : : : ; xn ] the reduction of gi by a deglex GrÄobner basis for the ideal I := I(V (F (k; q))) of polynomials
206
G. Heged} us, L. R´onyai / Central European Journal of Mathematics 2 (2003) 198{207
vanishing on V (F (k; q)) (actually reduction by G fi (vj ) = fi0 (vj ) ² gi (vj ) ² hi (vj )
q¡1
su±ces here). Obviously we have
(mod p) for 1 µ j µ m:
(7)
Here the (¯rst) equality is valid for any 0,1-vector, while the second congruence holds because Aj 2 F (k; q). Next note that by Proposition 3.4 the integer µ ¶ jAi \ Aj j ¡ k ¡ 1 fi (vj ) = q¡ 1 6 j. Then (7) implies that hi (vj ) will be 0 in Fp i® i 6= j. We will be divisible by p i® i = thus found that the m £ m matrix H = (hi (vj ))m i;j=1 is a diagonal matrix over Fp with no zeroes in the diagonal. From Proposition 2:7 of [3] (Determinant Criterion) it follows that the polynomials h1 ; : : : ; hm are linearly independent over Fp. Moreover, being reduced polynomials with respect to a GrÄobner basis, the hi are linear combinations of standard monomials for I and deg hi µ q ¡ 1 because deg fi = q ¡ 1, and the reductions (modulo p, and deglex) involved can not increase the degree. By Theorem 3.2 we infer that the linearly independent polynomials h1 ; : : : ; hm are in the Fp -space spanned by N q¡1 , and hence µ ¶ n jF j = m µ jN q¡1 j = ; q¡ 1 which was to be proved.
4
A concluding remark
¡[n]¢ Here we gave an upper bound on the rank of inclusion matrices I(F (k; q); ·` ) over Fp , where 0 µ ` µ q ¡ 1. In fact, for n su±ciently large, we determined the rank precisely. It would be interesting to describe the set of standard monomials Sm(¿ ; F (k; q); Fp) in a fashion similar to the uniform case given in [10]. ¡ ¢ This could give bounds on the rank of inclusion matrices I (F(k; q); [n] ·` ) over Fp , where ` > q ¡ 1.
Acknowledgment Research supported in part by OTKA and NWO-OTKA grants, and the EU-COE Grant of MTA SZTAKI.
References [1] W.W. Adams and P. Loustaunau: An Introduction to GrÄobner Bases, American Mathematical Society, 1994. [2] R.P. Anstee, L. R¶onyai, A. Sali: \Shattering news", Graphs and Combinatorics, Vol. 18, (2002), pp. 59{73.
G. Heged} us, L. R´onyai / Central European Journal of Mathematics 2 (2003) 198{207
207
[3] L. Babai and P. Frankl: Linear algebra methods in combinatorics, September 1992. [4] D.A. Barrington, R. Beigel, S. Rudich: \Representing boolean functions modulo composite numbers", In: Proc. 24th Annual ACM Symposium on Theory of Computing, Victoria, BC, Canada, 1992, pp. 455{461. [5] T. Becker and V. Weispfenning: GrÄobner bases - a computational approach to commutative algebra, Springer-Verlag, Berlin, Heidelberg, 1993. [6] A.M. Cohen, H. Cuypers, H. Sterk (eds.): Some Tapas of Computer Algebra, Springer-Verlag, Berlin, Heidelberg, 1999. [7] P. Frankl: \Intersection theorems and mod p rank of inclusion matrices", J. Combin. Theory A , Vol. 54, (1990), pp. 85{94. [8] K. Friedl and L. R¶onyai: Mathematics, to appear.
\Order-shattering and Wilson’s theorem", Discrete
[9] R.L. Graham, D.E. Knuth, O. Patashnik: Concrete Mathematics, Addison-Wesley, Reading, Massachusetts, 1989. [10] G. Heged} us and L. R¶onyai: \GrÄobner bases for complete uniform families", J. of Algebraic Combinatorics, Vol. 17, (2003), pp. 171{180. [11] R.M. Wilson: \A diagonal form for the incidence matrices of t-subsets vs. k-subsets", European Journal of Combinatorics, Vol. 11, (1990), pp. 609{615.
CEJM 2 (2003) 208{220
On Macbeath - Singerman Symmetries of Belyi Surfaces with PSL(2,p) as a Group of Automorphisms Ewa Tyszkowska¤ Institute of Mathematics, University of Gda¶ nsk Wita Stwosza 57, 80-952 Gda¶ nsk, Poland
Received 13 December 2002; revised 4 March 2003 Abstract: The famous theorem of Belyi states that the compact Riemann surface X can be de ned over the number eld if and only if X can be uniformized by a nite index subgroup ¡ of a Fuchsian triangle group ¤. As a result such surfaces are now called Belyi surfaces. The groups PSL(2; q); q = pn are known to act as the groups of automorphisms on such surfaces. Certain aspects of such actions have been extensively studied in the literature. In this paper, we deal with symmetries. Singerman showed, using acertain result of Macbeath, that such surfaces admit a symmetry which we shall call in this paper the Macbeath - Singerman symmetry. A classical theorem by Harnack states that the set of xed points of a symmetry of a Riemann surface X of genus g consists of k disjoint Jordan curves called ovals for some k ranging between 0 and g+1. In this paper we show that given an odd prime p, a Macbetah - Singerman symmetry of Belyi surface with PSL(2; p) as a group of automorphisms has at most #=
8 > < 3(p ¡
1)=4 if p ² 1 (4);
> : 3(p + 1)=4 if p ² 3 (4);
ovals and that these bounds are exact for arbitrary p ² 1 (8) and p ² 3 (8), respectively. Furthermore we show that a Macbeath - Singerman symmetry of a surface having maximal genus among Belyi surfaces with PSL(2; p) as a group of automorphisms has at most (p + 1)=2 ovals. c Central European Science Journals. All rights reserved. ®
¤
Keywords: Riemann surface, automorphism, symmetry, ovals, minimum genus action, ¯nite projective special linear groups MSC (2000): Primary: 30F20, 30F50; Secondary:14H37, 20H30, 20H10
E-mail: [email protected]
E. Tyszkowska / Central European Journal of Mathematics 2 (2003) 208{220
1
209
Introduction
It is known that projective complex algebraic curves bijectively and functorially correspond to compact Riemann surfaces. Belyi [1] proved that under this correspondence ¹ if and the smooth complex algebraic curve X can be de¯ned over the number ¯eld Q only if the corresponding surface is representable as the orbit space H=¡ of the upper half plane, where ¡ is a ¯nite index subgroup of a Fuchsian triangle group. The groups PSL(2; q); q = pn are known to act as groups of automorphisms on such surfaces and certain aspects of such actions have been extensively studied in a number of papers [3, 5, 6, 9, 10, 12, 13]. In [5] and [6] Glower and Sjerve found the minimum genus among the surfaces on which PSL(2; q) acts and in [9] and [10] Rosenberger with Langer and Levin respectively described all such actions. Certain important results have been obtained also by Sah in [12]. In this paper, we deal with the symmetries of such surfaces. A symmetry of a Riemann surface X is an antiholomorphic involution %. Under the above mentioned correspondence between curves and surfaces the fact that a surface X is symmetric means that the corresponding curve can be de¯ned over the reals. Furthermore the non-conjugate, in the group of all automorphisms of X , symmetries correspond to non-isomorphic, over the reals, real curves. Finally, if X has genus g then the set of ¯xed points Fix(%) of % consists of k disjoint Jordan curves called ovals, where k varies between 0 and g + 1. This follows from the classical Harnack Theorem [8]. This set is homeomorphic to a smooth projective real model of the corresponding curve. The genus g of a compact Riemann surface X can be understood as the topological genus of the underlying topological surface. Topologically each symmetry is conjugate to a canonical one which is either induced by a symmetry % with respect to a plane in 3-space or is such % composed with one or two Dehn twists around its ovals. The ¯rst symmetry has k ovals where k can range from 0 to g + 1 and k ² g + 1(2). It is separable in the sense that X nFix(» ) is disconnected. The second is nonseparable; each twist kills the corresponding oval and so the resulting symmetry can have an arbitrary number of ovals between 0 and g. This process is nicely explained in the recent paper of SeppÄalÄa and Silhol [4]. Using certain result of Macbeath, Singerman [13] showed that a Belyi surface with PSL(2; q) as the group of automorphisms admits a certain symmetry which we shall call in this paper the Macbeath - Singerman symmetry. In [3], an algorithm for ¯nding symmetry types of surfaces on which PSL(2; q) acts as a Hurwitz group of automorphisms was given. In this paper we prove that, given an odd prime p, a Macbeath - Singerman symmetry of an arbitrary compact Riemann surface de¯ned over a number ¯eld and having PSL(2; p) as the groups of birational automorphisms admits at most 8 > > < 3(p ¡
#=> >
1)=4 if p ² 1 (4);
: 3(p + 1)=4 if p ² 3 (4);
ovals. We also show that these bounds are exact for arbitrary p ² 1 (8) and p ² 3 (8) respectively. As a by-product of our considerations we obtain that a symmetry of a surface
210
E. Tyszkowska / Central European Journal of Mathematics 2 (2003) 208{220
having maximal genus among Belyi surfaces with PSL(2; p) as the group of automorphisms has at most (p + 1)=2 ovals.
2
Preliminaries
Given a prime p, let K = GF(p) be the Galois ¯eld with p elements. The group SL(2; p) consists of 2 £ 2 matrices with determinant 1 and entries in K . The projective special linear group PSL(2; p) is obtained by factoring out the centre from the group SL(2; p). For a matrix A from SL(2; p) let !A denote its trace. The characteristic polynomial of A is ¶ 2 ¡ !A ¶ + 1. The matrix A is called hyperbolic if its characteristic polynomial has two distinct zeros. If it has no zero in K but two zeros a; ap in GF (p2 ) then A is called elliptic. Finally, if !A = §2, then A is called parabolic. Let H denote the upper-half plane, let Aut+ (H) be the group of conformal automorphisms of H. These the transformations of the form z¡ !
az + b : cz + d
Let Aut§ (H) be the group of conformal and anticonformal homeomorphisms, where the latter are the transformations a¹ z+b z¡ ! c¹ z+d for some reals a; b; c; d for which ad ¡ bc = 1 and ad ¡ bc = ¡ 1 respectively. An NEC group is a discrete subgroup of Aut§ (H) and a Fuchsian group is a discrete subgroup of Aut+ (H). Details of the theory are can be found in [2]. Given a ¯nite group G generated by two elements A and B of order k and l whose product has order m, take a Fuchsian triangle group ¤ with the signature [k; l; m]. It has the presentation with generators x1 ; x2 ; x3 subject to the relations xk1 = xl2 = xm 3 = x1 x2 x3 = 1. Consider an epimorphism ³ : ¤ ! G induced by the map which sends x1 and x2 to A and B respectively. Then ¡ = ker ³ is a Fuchsian surface group and G acts as a group of automorphisms on the Riemann surface X = H=¡. We shall say in such case that the action of G on X is determined by the pair A; B. Singerman [13] showed that such X is symmetric if and only if either A 7! A ¡1 ; B 7! B ¡1 or A 7! B ¡1 ; B 7! A ¡1
(1)
induce an automorphism of G. On the other hand, Macbeath [11] showed that two couples (A; B) and (A1 ; B1 ) of matrices from SL(2; K ) for which trace A = trace A 1 and ¹ ). As a result the ¯rst trace B = trace B1 are conjugate within a larger group SL(2; K map (1) induces an automorphism of PSL(2; K ) and therefore the corresponding surface admits a symmetry [13] which we call here a Macbeath - Singerman symmetry. In terms e with of NEC groups the existence of such symmetry means that there is an NEC group ¤ signature (0; +; [¡ ]; (k; l; m)), which we shall abbreviate as (k; l; m), containing ¤ as a subgroup of index 2 and ¡ as a normal subgroup [13]. In such a case it is also known
E. Tyszkowska / Central European Journal of Mathematics 2 (2003) 208{220
211
e = ¤=¡ e e = PGL(2; p) [13] even though no immediate criteria that G = PSL(2; p) £ Z2 or G exist for deciding which case is actually present in concrete situations exists. e ! ¤=¡ e e be the canonical epimorphism and let an involution % be the Let ³ e : ¤ =G e n ¤. If d cannot be chosen as a re°ection then % has image under ³ e of an element d from ¤ e and the number no ovals and otherwise d is conjugate to a canonical re°ection c from ¤ e = ³ e¡1 (h%i). We of ovals jj%jj equals the number of empty period cycles in the group ¡ can count it using the following formula
jj%jj =
X
e ³ e(c ))j=j³ e(C(¤; e c ))j; jC(G; i i
(2)
e whose images are where ci runs over pairwise non-conjugate canonical re°ections in ¤ e that appear in the above formula are also known. conjugate to % [7]. The centralizers in G
Lemma 2.1. Let p be an odd prime and let A be an involution from PSL(2; p) or PGL(2; p). Then
e Aj = jC(G;
8 > > > 2(p ¡ > > > > > > > <
e = PSL(2; p) £ Z ; 1) if p ² 1 mod 4; G 2
e = PSL(2; p) £ Z ; 2(p + 1) if p ² 3 mod 4; G 2
> > e = PGL(2; p); > 2(p + 1) if p ² 1 mod 4; G > > > > > > > e = PGL(2; p): : 2(p ¡ 1) if p ² 3 mod 4; G
The remaining ingredient to calculate the right hand side in (2) can be found with the help of Singerman’s paper [14], (cf. [7] for explicit form). e be an NEC group with signature (k 0 ; l0 ; m0 ) and let c ; c ; c be a Lemma 2.2. Let ¤ 0 1 2 e Then according to the parity of k 0 ; l 0 ; m 0 we have canonical system of generators of ¤.
(i) for k 0 = 2k + 1; l0 = 2l + 1; m0 = 2m + 1; c0 ¹ c1 ¹ c2 and e c ) = hc i © (h(c c )m (c c )l (c c )k i), C(¤; 0 0 2 0 1 2 0 1 0 0 0 (ii) for k = 2k; l = 2l + 1; m = 2m + 1; c0 ¹ c1 ¹ c2 and e c ) = hc i © (h(c c )k i ¤ h(c c )m (c c )l (c c )k (c c )l (c c )m i), C(¤; 0 0 0 1 2 0 1 2 1 0 2 1 0 2 0 0 0 (iii) for k = 2k; l = 2l; m = 2m + 1; c0 ¹ c2 and e c ) = hc i © (h(c c )k i ¤ h(c c )m (c c )l (c c )m i), C(¤; 0 0 0 1 2 0 2 1 0 2 k l e C(¤; c1 ) = hc1 i © (h(c0 c1 ) i ¤ h(c1 c2 ) i), (iv) for k 0 = 2k; l 0 = 2l; m0 = 2m e c ) = hc i © (h(c c )k i ¤ h(c c )m i), C(¤; 0 0 0 1 0 2 k e C(¤; c1 ) = hc1 i © (h(c0 c1 ) i ¤ h(c1 c2 )l i), e c ) = hc i © (h(c c )m i ¤ h(c c )l i). C(¤; 2 2 0 2 1 2
3
Orders, traces and extensions of PSL(2; p)
Our goal in this section is to give some auxiliary results concerning the group PSL(2; p). Throughout the paper, we will use !T to denote the trace of the matrix T . Let K = GF (p)
212
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and let Sn (x) 2 K [x] be the Chebyshev polynomials de¯ned recursively by S0 (x) = 0; S1 (x) = 1; Sn+1(x) = xSn (x) ¡
Sn¡1 (x) for n ¶ 2:
The ¯rst lemma is well known (cf. [9]). Lemma 3.1. Given T 2 SL(2; p) let ! = !T . Then T n = Sn (!)T ¡ particular T has order n if and only if Sn (!) = 0.
Sn¡1 (!)I . In
The following is apparently part of the folklore of mathematics. Lemma 3.2. The Chebyshev polynomials satisfy the relations S2k+1 (x) = Sk+1(x)2 ¡
Sk (x)2 and S2k (x) = Sk (x)(Sk+1 (x) ¡
Sk¡1 (x)):
In particular for an element T 2 PSL(2; p) of order jT j and ! = !T we have jT j = 2k + 1 if and only if Sk¡1 (!) = (w ¡ 1)Sk (!) or Sk¡1 (!) = (w + 1)Sk (!) and jT j = 2k if and only if Sk¡1 (!) = 12 wSk (!). Proof. The relations are certainly true for k=1. Let us assume that both are true for some positive integer k. Then S2(k+1) = xS2k+1(x) ¡ =
2 x(Sk+1 (x)
S2k (x) ¡
Sk2 (x)) ¡
= Sk+1(x)(xSk+1 (x) ¡
Sk (x)(Sk+1 (x) ¡
= Sk+1(x)Sk+2(x) ¡
Sk (x)) ¡
Sk¡1 (x))
Sk (x)(xSk (x) ¡
Sk¡1 (x))
Sk (x)Sk+1(x)
= Sk+1(x)(Sk+2(x) ¡
Sk (x)):
Similarly S2(k+1)+1 (x) = xS2(k+1) (x) ¡
S2k+1 (x)
= xSk+1(x)(Sk+2(x) ¡
Sk (x)) ¡
2 Sk+1 (x) + Sk2 (x)
= xSk+1(x)Sk+2(x) ¡
Sk (x)(xSk+1 (x) ¡
= xSk+1(x)Sk+2(x) ¡
Sk (x)Sk+2(x) ¡
= (xSk+1(x) ¡
Sk (x))Sk+2(x) ¡ 2
Sk (x)) ¡ 2 Sk+1 (x)
2 2 Sk+1 (x) = Sk+2 (x) ¡
3
6a b7 7 we get the following 5
Combining the above lemmata for T = 6 4 Corollary 3.3.
2
c d
6¡ d § 1
(i) If jT j = 2k + 1 then T k = Sk (!T ) 6 4
c
2 Sk+1 (x)
3
b7
¡ a§1
7; 5
2 Sk+1 (x):
E. Tyszkowska / Central European Journal of Mathematics 2 (2003) 208{220 2 6
(ii) If jT j = 2k then T k = Sk (!T ) 6 4
1 (a 2
¡
3
d) 1 (d 2
c
213
¡
b7
a)
7: 5
Lemma 3.4. Given a non-parabolic matrix T 2 SL(2; p), let ! = x + x¡1 be the unique decomposition in the quadratic ¯eld extension GF (p2 ) of the trace ! = !T . Then (i) jT j = 2k + 1 if and only if, for every 0 µ i µ k, Sk+1+i (!) = Sk¡i (!) and x2k+1 = ¡ 1 or, for every 0 µ i µ k, Sk+1+i (!) = ¡ Sk¡i (!) and x2k+1 = 1, (ii) jT j = 2k if and only if Sk+i (!) = Sk¡i (!) for every 0 µ i µ k and x2k = ¡ 1. Proof. We shall prove (i) only; the proof of (ii) is similar and we omit it. The previous lemma shows the su±ciency and the case i = 0 in the proof of the necessity. Let us assume that it is also true for some 0 µ i < k: Then Sk+1+(i+1) (!) = wSk+1+i (!) ¡ Sk+i(!) = §[wSk¡i (!) ¡ Sk¡i+1(!)] = §[wSk¡i (!) ¡ wSk¡i(!) + Sk¡i¡1 (!)] = §Sk¡(i+1) (!): Thus, by induction, Sk+1+i (!) = §Sk¡i (!) for every 0 µ i µ k. Now conjugating T if necessary, we can assume without loss of generality that it is diagonal. So 2
6x
T 2k+1 = 6 4
0
3
2k+1
x
07
¡(2k+1)
7: 5
If Sk+1+i (!) = +Sk¡i (!) then S2k (!) = S1 (!) = 1, S2k+1 (!) = S0 (!) = 0 and thus, by lemma 3.1, T 2k+1 = ¡ 1 which gives x2k+1 = ¡ 1. Similarly if Sk+1+i (!) = ¡ Sk¡i (!) for every 0 µ i µ k, we get x2k+1 = 1. e from Lemma 2.1 is We still need the following lemma to decide when the group G isomorphic to PSL(2; p) £ Z2 .
Lemma 3.5. Let X be a Belyi surface with the action of G = PSL(2; p) determined by a pair A, B having Macbeath { Singerman symmetry. Let ¬ = !A ; = !B ; ® = !AB and · = ¬ ® ¡ ¬ 2 ¡ 2 ¡ ® 2 + 4. If at least one of A; B is non-parabolic and · is a square in e = PSL(2; p) £ Z : GF(p2 ) then G 2
Proof. Let ’ be an automorphism of G = PSL(2; p) satisfying ’(A) = A ¡1 and ’(B) = B ¡1 . If ’ is an inner automorphism then there exists an element T of order 2 e = PSL(2; p) £ Z from PSL(2; p) for which T AT ¡1 = A¡1 and T BT ¡1 = B ¡1 and then G 2 e [13]. Otherwise, T can be taken as an element of PGL(2; p) and so G = PGL(2; p). Without loss of generality we can assume that A is not a parabolic element and that matrices 2
6x 6 4
3 2
3 2
3
0 7 6 b1 b2 7 6 e f 7 7;6 7;6 7 5 4 5 4 5 ¡1 0 x b3 b4 g¡ e
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E. Tyszkowska / Central European Journal of Mathematics 2 (2003) 208{220
represent A; B and T respectively. Then T A = A ¡1 T gives 2
6 ex 6 4
fx
¡1
gx ¡ ex¡1
3
2
7 6 ex 7=¯ 6 5 4
¡1
fx
¡1
¡ xe
xg
3 7 7 5
where ¯ = §1. If ¯ = ¡ 1 then f = g = 0 and x2 = ¡ 1. Now T B = B ¡1 T gives b3 = b2 = 0 or b1 = b4 . The former is impossible because then A; B generate a diagonal subgroup which is a contradiction. The condition b1 = b4 implies ® = 0. Thus jAj = 2 = jABj and so A; B generate a dihedral group, which is again a contradiction. For ¯ = +1 we get e = 0. Now T B = B ¡1 T gives 2
6 f b3 6 4
¡ f
3
¡1
b1 ¡ f
f b4 7
¡1
b2
2
6f 7 = "6 5 4
¡1
b2
¡ b1 f
¡1
3
f b4 7
¡ b3 f
7: 5
If " = ¡ 1 then we get b1 = b4 = 0. Thus = ® = 0, which again implies that the group generated by elements A; B is dihedral, a contradiction. Finally for " = +1 we get 2 b2 b¡1 3 = f . But then we can choose T in the form 2
60 6 4
3
f7 7
¡ f ¡1 0
5:
e = PSL(2; p)£Z . Let us write the condition for So ’ is an inner automorphism and thus G 2 ¡1 ¡1 b2 b3 to be a square in terms of traces x + x = ¬ ; b1 + b4 = and xb1 + x¡1 b4 = ® : Since ¡1 ¡1 detB = 1, b2 b¡1 3 is a square if and only if b1 b4 ¡ 1 is a square. Clearly (® ¡ x )=(x¡ x ) = b1 and straightforward calculus gives b1 b4 ¡ 1 = (¬ ® ¡ ¬ 2 ¡ 2 ¡ ® 2 + 4)=(x ¡ x¡1 )2 . Thus b2 b¡1 ¬ 2 ¡ 2 ¡ ® 2 + 4 is a square as well. 3 is a square if and only if · = ¬ ® ¡
4
Symmetries of Belyi surfaces with PSL(2; p) as a group of automorphisms
We see that the Macbeath - Singerman symmetry type of Belyi surface on which the group PSL(2; p) acts as the group of orientation preserving automorphisms is determined by an epimorphism ³ : ¤ ! PSL(2; p) with torsion free kernel ¡, where ¤ is a Fuchsian group with signature [k; l; m]. Following Singerman we shall refer to such epimorphism as a surface kernel epimorphism. As we mentioned above, the action PSL(2; p) on X is determined by a pair of generators A; B of orders k and l for which C = (AB)¡1 has order m. Fuchsian triangle groups with the minimum possible area of the fundamental region which admit a surface kernel epimorphisms onto a given PSL(2; p) were found in [5]. All Fuchsian triangle groups allowing such epimorphisms were found by Langer and Rosenberger in [9]. Before we prove the main result we need to ¯nd the maximal number of ovals for the symmetry of the surface corresponding to a couple of parabolic elements whose product is also parabolic.
E. Tyszkowska / Central European Journal of Mathematics 2 (2003) 208{220
215
Lemma 4.1. A Macbetah { Singerman symmetry of a Belyi surface corresponding to a pair of parabolic generators A; B for which C = (AB)¡1 is also parabolic has at most 8 > > <
(p ¡
#=> > :
1)=2 if p ² 1 (4);
(p + 1)=2 if p ² 3 (4);
ovals and these bounds are exact.
Proof. Using the results from [9], an element of PSL(2; p) is parabolic if and only if it has order p and the group PSL(2; p) can be generated by two parabolic elements A; B whose product is also parabolic. Let p = 2k + 1. By (2), the symmetry with ¯xed points e ³ e(c ))j=2jEj ovals, where E = C k B k A k , and of the corresponding surface has jj%jj = jC (G; i according to lemma 2.1, jj%jj µ (p +1)=jEj. The only possibilities for the traces of A; B; C are (¡ 2; ¡ 2; ¡ 2); (¡ 2; 2; 2); (2; ¡ 2; 2); (2; 2; ¡ 2) since in the remaining cases A; B; C generate a singular subgroup of PSL(2; p), [6]. We shall consider the case (2; 2; ¡ 2) , the other cases are similar. Every parabolic element of SL(2; p) is conjugate to the matrix 2
3
61 x7 7; 5
§6 4
0 1
for some x 2 GF(p). So assume that it represents A. Since !B = 2 and detB = 1, the matrices representing B and C have to be of the form 2
6a 6 4
¡
1 (a 4 4 x
¡
2
3 2
1) x 7 6 2 ¡ 7;6 5 4 4 2¡ a x
1 (a 4
a¡
3
2
¡
3) x 7
a¡
4
7; 5
for some a in GF(p). Now !C = ¡ 2 and Sn (¡ 2) = (¡ 1)n+1n for every n 2 N . So from lemma 3.1, we obtain 2
6 3k ¡
C k = (¡ 1)k+1 6 4
ak ¡
1 k(a 4
1¡
4k x
¡
3) x 7
¡ 3k + ka ¡
1
Similarly, since !B = 2, Sn (2) = n and 2k = ¡ 1 in GF(p), 2
6 ka ¡
B k Ak = 6 4
k+1
1 (2a 4
¡
¡ 4 kx
3
2
7: 5 3
2
3 + ka + k)x 7 k¡
ka
7 :: 5
Now !E = (¡ 1)k+1(2k 2 a + 4k 2 ¡ 1 + ka) = 0 and so E has order 2. It remains to e is isomorphic to PSL(2; p) £ Z . As in lemma 3.5, let decide when the group G 2 2
6e
T =6 4
3
f7
g¡ e
7 5
216
E. Tyszkowska / Central European Journal of Mathematics 2 (2003) 208{220
be an element of order 2 in PSL(2; p) for which T AT ¡1 = A¡1 and T BT ¡1 = B ¡1 . The ¯rst condition implies g = 0 and therefore e must be a solution of the equation u2 = ¡ 1. For p ² 1 (4), ¡ 1 is a square in GF(p) and we can choose T in the form 2
1 xe(a 2
6e 6 4
3
¡
1) 7
¡ e
0
7: 5
e = PSL(2; p) £ Z . For p ² 3 (4) let e be So ’ is an inner automorphism and thus G 2 a solution of the equation u2 = ¡ 1 in the quadratic ¯eld extension GF(p2 ). The group PSL(2; p) is isomorphic to PSU(2; p) which consists of elements 2
6a 6 4
3
b7
¡ bp ap
7; 5
where a; b in GF(p2 ) satisfy ap+1 + bp+1 = 1. Thus we can choose T of the form 2
3
2
6e 07 6e 6 7=6 4 5 4
0e
p
3
07
0¡ e
7: 5
But then the condition T BT ¡1 = B ¡1 implies 2e(a ¡ 1) = 0. So if a = 1 then ’ is inner e = PSL(2; p) £ Z and otherwise G e = PGL(2; p). and G 2 The above arguments demonstrate that the above bounds are sharp since the group PSL(2; p) is a surface kernel factor of the triangle Fuchsian group ¤ generated by three elements of orders p (see theorem 4.1 in [9]).
Corollary 4.2. A Macbeath { Singerman symmetry of a surface having maximal genus among Belyi surfaces with PSL(2; p) as the group of automorphisms has at most (p + 1)=2 ovals. Proof. Indeed in such case ¤ is a Fuchsian group with the signature [p; p; p] since there are no elements in PSL(2; p) of order exceeding p. But then the corresponding generators A; B; C are parabolic and the corollary follows from the above lemma. Theorem 4.3. Let % be a Macbeath - Singerman symmetry of a Belyi surface on which PSL(2; p) acts as a group of automorphisms. Then % has at most 8 > > <
#=> >
3(p ¡
1)=4 if p ² 1 (4);
: 3(p + 1)=4 if p ² 3 (4);
ovals and these bounds are exact for arbitrary p ² 1 (8) and p ² 3 (8), respectively.
E. Tyszkowska / Central European Journal of Mathematics 2 (2003) 208{220
217
Proof. Let jj%jj denote the number of ovals of %. We can ¯nd it using (2). The counters are given in lemma 2.1 and are equal to 2(p § 1). So it remains only to decide when the sign can be + and to ¯nd lower bounds for the denominators. Let A; B; C be arbitrary generators of PSL(2; p) of orders k 0 ; l 0 ; m 0 and traces ¬ ; ; ® respectively, for which ABC = 1. We shall split the proof into parts according to the parity of k 0 ; l0 ; m0 . From the proof of lemma 4.1, jj%jj µ (p + 1)=2 < # when all generators of PSL(2; p) are parabolic. e has a So we can assume that at least one of them is non-parabolic. As claimed before ¤ signature (k 0 ; l 0 ; m 0 ). Let k 0 = 2k + 1; l 0 = 2l + 1; m 0 = 2m + 1. From lemmata 2.1 and 2.2, jj%jj = 2(p § 1)=2jEj, where E = C m B l Ak . We shall show that E 6= I . Every non-parabolic element from SL(2; p) is conjugate to a diagonal matrix and so we can assume without loss of generality that A; B, C are represented by matrices 2
3 2
3 2
3
¡1
0 7 6 a b 7 6 e d ¡ eb 7 7;6 7;6 7 5 4 5 4 5 ¡1 ¡1 ¡ e c ea 0e c d
6e 6 4
(3)
respectively, where e2k+1 = §1 and en 6= §1 for any n < 2k + 1. In addition S2l+1 ( ) = 0 and S2m+1 (® ) = 0. If E = 1, then A ¡k = C m B l , which by corollary 3.3 gives 2
6 ¡ ea § 1
A¡k = · 6 4
¡1
¡ e c
32
¡1
¡ eb 7 6 ¡ d § 1
¡ e d§1
76 54
c
3
b7
¡ a§1
7 5
where · = §Sm (® )Sl ( ). So b(1§e) = 0 and c(1§e¡1 ) = 0 and therefore b = c = 0 which is impossible since in such case the triple A; B; C would generate a diagonal subgroup. Thus jEj ¶ 2 and therefore jj %jj µ (p + 1)=2 < #. For k 0 = 2k; l0 = 2l + 1; m 0 = 2m + 1, jj %jj = 2(p § 1)=4jEj µ (p + 1)=2 < #: Now let k 0 = 2k; l 0 = 2l; m0 = 2m + 1. Then jj %jj = 2(p § 1)=4jE1 j + 2(p § 1)=4jE2 j, where E1 = A k B k and E2 = Ak C m B l C ¡m : First we shall see that E1 is nontrivial and next we shall ¯nd the conditions for E1 to be of order 2 and E2 to be trivial. Since A has even order it cannot be parabolic and we can assume that A; B; C are represented by matrices (3). If E1 = 1 then A k = B l and from corollary 3.3 we get 2
6e 6 4
k
0 e
3
2
07
¡k
6 7 = §S ( ) 6 l 5 4
1 (a 2
¡
3
d) 1 (d 2
c
¡
b7
a)
7: 5
But then b = c = 0 which is a contradiction. So jE1 j ¶ 2. If E2 = 1 then 2
6 ¡ ea § 1
Ak = C m B l C ¡m = ¹ 6 4
¡1
¡ e c
32
¡1
¡ eb 7 6
¡ e d§1
76 54
1 (a 2
c
¡
32
d) 1 (d 2
¡
¡1
b76¡ e d § 1
a)
76 54
¡1
e c
3
eb 7
¡ ea § 1
7; 5
218
E. Tyszkowska / Central European Journal of Mathematics 2 (2003) 208{220
2 where ¹ = §Sm (® )Sl ( ). Thus eb(e + e¡1 § (a + d)) = 0 and e¡1 c(e + e¡1 § (a + d)) = 0 which gives b = c = 0 or ¬ = § . The former is impossible so if E2 = 1 then A and B have equal orders. Next, let us consider what other conditions should be satis¯ed for E1 to be of order 2. If C is non-parabolic then we conjugate the generators in such a way that the resulting triple is 2 3 2 3 2 3 ¡1 6r f 7 6x h 6 7;6 4 5 4
¡ xf 7 6 x 0 7 7;6 7; 5 4 5 ¡ x¡1 g xr 0 x¡1
g h
(4)
where x2m+1 = §1, xn 6= §1 for n < 2m + 1 and S2k (¬ ) = S2k ( ) = 0. Now 2
1 (r 2
6
E1 = §Sk (¬ )Sl ( ) 6 4
¡
32
h) 1 (h 2
g
¡
f 76
r)
76 54
1 (x¡1 h 2
¡
3
xr)
¡1
1 (xr 2
¡ x g
¡
¡ xf 7 ¡1
x h)
7: 5
If jE1 j = 2 then 0 = !E1 = 12 (r ¡ h)(x¡1 h ¡ xr) ¡ gf (x + x¡1 ), while the latter implies that 2® = ¬ . This means that jE2 j = 2 if and only if 2® = ¬ . Summing up, if jE1 j = 2 and E2 = 1 then 2® = §¬ 2 . If 2® = ¬ 2 then · = ¬ ® ¡ ¬ 2 ¡ 2 ¡ ® 2 + 4 = (® ¡ 2)2 , but if e = PSL(2; p)£Z 2® = ¡ ¬ 2 then · = (® +2)2 . So, according to the lemmata 3.5 and 2.1, G 2 and the number of ovals of % does not exceed 8 > > <
#=> >
3(p ¡
1)=4 if p ² 1 mod 4;
: 3(p + 1)=4 if p ² 3 mod 4:
If C is parabolic then we can assume that matrices 2
3 2
¡ b¡
6a b7 6d 6 7;6 4 5 4
¡ c
c d
3 2
3
xd 7 6 1 x 7 7;6 7 5 4 5 a + xc 0 1
represent A; B and C. We have shown that E1 cannot be the identity and if E2 = 1 then ¬ = or ¬ = ¡ . If ¬ = then we get xc = 0 which leads to a contradiction. So it remains to consider the case ¬ = ¡ in which xc = ¡ 2¬ . Now 2
6
E1 = Sk (¬ )Sl ( ) 6 4
1 (a 2
c
¡
32
d) 1 (d 2
¡
b76
a)
76 54
1 (d 2
¡ c
¡
a¡
¡ b¡
xc) 1 (a 2
+ xc ¡
3
xd 7 d)
7: 5
If E1 has order 2 then !E2 = 0 and thus (a ¡ d)(d ¡ a ¡ xc) = 4bc + 2xdc which gives ¬ 2 = ¡ 4. Therefore · = ¬ ® ¡ ¬ 2 ¡ 2 ¡ ® 2 + 4 = 16 is a square and so jj%jj µ # again. Finally let k 0 = 2k; l0 = 2l; m 0 = 2m. Here jj%jj = 2(p § 1)=4jE1 j + 2(p § 1)=4jE2 j + 2(p §1)=4jE3 j, where E1 = A k C ¡m ; E2 = Ak B l ; E3 = C ¡m B l . We shall show that none of E1 ; E2 ; E3 can be trivial and that they cannot have orders 2 simultaneously which means
E. Tyszkowska / Central European Journal of Mathematics 2 (2003) 208{220
219
that jj%jj < #. Once more we can assume that the generators A; B; C are represented by matrices (3). Then 2
6e
E1 = §Sm (® ) 6 4
32
k
0 e
076
¡k
76 54
1 (ea 2
3
¡
¡1
e d)
¡1
1 ¡1 (e d 2
e c
eb 7
¡
ea)
7 5
and so E1 = 1 gives b = c = 0, a contradiction. Assuming that E2 or E3 is diagonal we can show by similar arguments that they, too, cannot be trivial. Next we shall show that E1 , E2 and E3 cannot have order 2 simultaneously. Indeed if jE1 j = 2 then !E1 = §ek Sm (® )(ea ¡ e¡1 d) = 0. But then d = e2 a and therefore 2
6e
E2 = §Sl ( ) 6 4
k
32
076
0 e¡k
76 54
1 (a 2
c
¡
3
2
e a) 1 2 (e a 2
¡
b7
a)
7: 5
Thus jE2 j = 2 if and only if a = 0 or e2 = 1. We claim that neither is possible. Indeed, if a = 0 then = ® = 0 and so A; B; C have orders k 0 ; 2; 2 respectively and thus generate a dihedral subgroup while if e2 = 1 then A is trivial. Similarly if we represent A; B; C by matrices (4) then the condition wE1 = wE3 = 0 leads to a contradiction. Thus jj%jj < #. It remains to give examples of surfaces for which the bounds # are reached. From the proof of the ¯rst part of theorem we see that it may be the case for l 0 = k 0 = 2k; m0 = 2m + 1, jE1 j = 2 and E2 = 1 only. For an arbitrary odd prime p and the three positive integers k 0 = l 0 = 2k, m0 = 2m + 1 which divide (p + 1)=2 or (p ¡ 1)=2 we can choose generators A; B; C for which E2 = 1. Namely, for w; x 2 GF(p) satisfying S2k (!) = 0 and x2m+1 = §1 we can take A; B; C in the form 2
6 w=(x + 1) 6 4
g
if x2m+1 = ¡ 1 and 2
6 ¡ w=(x ¡ 6 4
g
3 2
3 2
3
¡ xf 7 6 x 0 7 f 7 6 w=(x + 1) 7;6 7;6 7; 5 4 5 4 5 ¡1 ¡1 ¡ x g wx=(x + 1) wx=(x + 1) 0 x 3 2
1) wx=(x ¡
3 2
3
¡ xf 7 6 x 0 7 f 7 6 w=(x ¡ 1) 7;6 7;6 7; 5 4 5 4 5 ¡ x¡1 g ¡ wx=(x ¡ 1) 1) 0 x¡1
if x2m+1 = 1. It is easy to check that E2 = 1. If in addition p ² §1 (8) then the group PSL(2; p) can be generated by two elements of order 4 whose product has order 3 [10]. Remember that the element of order 3 has the trace §1 and the square of the trace of an element of order 4 equals 2 [5]. So generators of orders 4; 4; 3 respectively satisfy an extra condition §w 2 = 2(x + x¡1 ) which implies that jE1 j = 2 and thus a symmetry of the corresponding surface has # ovals.
Acknowledgement The author would like to thank professor Grzegorz Gromadzki for suggesting the problem and his helpful comments concerning the paper.
220
E. Tyszkowska / Central European Journal of Mathematics 2 (2003) 208{220
References [1] G.V. Belyi: \On Galois extensions of maximal cyclotomic ¯eld" (English translation), Math. USSR Izvestiya, Vol. 14, (1980), pp. 247{256. [2] E. Bujalance, J. Etayo, J. Gamboa, G. Gromadzki: \Automorphisms Groups of Compact Bordered Klein Surfaces. A Combinatorial Approach", Lecture Notes in Math., Vol. 1439, Springer Verlag, 1990. [3] S.A. Broughton, E. Bujalance, A.F. Costa, J.M. Gamboa, G. Gromadzki: \Symmetries of Riemann surfaces on which PSL(2; q) acts as a Hurwitz automorphism group", J. Pure Appl. Alg., Vol. 106, (1996), pp. 113{126. [4] P. Buser and M. SeppÄalÄa: \Real structures of TeichmÄ uller spaces, Dehn twists, and moduli spaces of real curves", Math. Z., Vol. 232, (1999), pp. 547{558. [5] H. Glover and D. Sjerve: \Representing PSL(2; p) on a Riemann surface of least genus", L’Enseignement Mathematique, Vol. 31, (1985), pp. 305{325. [6] H. Glover and D. Sjerve: \The genus of PSL(2; q)", Jurnal fur die Reine und Angewandte Mathematik, Vol. 380, (1987), pp. 59{86. [7] G. Gromadzki: \On a Harnack-Natanzon theorem for the family of real forms of Rieamnn surfaces", J. Pure Appl. Alg., Vol. 121, (1997), pp. 253{269. [8] A. Harnack: \Uber die Vieltheiligkeit der ebenen algebraischen Kurven", Math. Ann., Vol. 10, (1876), pp. 189{199. [9] U. Langer and G. Rosenberger: \Erzeugende endlicher projectiver linearer Gruppen", Result in Mathematics, Vol. 15, (1989), pp. 119{148. [10] F. Levin and G. Rosenberger: \Generators of ¯nite projective linear groups, Part 2", Results of mathematics, Vol. 17, (1990), pp. 120{127. [11] A.M. Macbeath: \Generators of the linear fractional groups", Proc. Symp. Pure Math, Vol. 12, (1967), pp. 14{32. [12] C.H. Sah: \Groups related to compact Riemann surfaces", Acta Math, Vol. 123, (1969), pp. 13{42. [13] D. Singerman: \Symmetries of Riemann surfaces with large automorphism group", Math. Ann., Vol. 210, (1974), pp. 17{32. [14] D. Singerman: \On the structure of non-euclidean crystallographic groups", Proc. Camb. Phil. Soc., Vol. 76, (1974), pp. 233{240.
CEJM 2 (2003) 221{237
On the lattice of deductive systems of a BL-algebra Dumitru Bu»sneag ¤, Dana Piciu y University of Craiova, 1100 Craiova, Romania
Received 9 January 2003; revised 13 March 2003 Abstract: For a BL-algebra A we denote by Ds(A) the lattice of all deductive systems of A. The aim of this paper is to put in evidence new characterizations for the meet-irreducible elements on Ds(A). Hyperarchimedean BL-algebras, too, are characterized. c Central European Science Journals. All rights reserved. ® Keywords: BL-algebra, Boolean algebra, Archimedean, Hyperarchimedean BL-algebra, Deductive system, Irreducible element, Prime deductive system, Maximal deductive system. MSC (2000): 03G10
1
Introduction
y
¤
The origin of BL-algebras is in Mathematical Logic; they where invented by H¶ ajek in [8] in order to study the ,,Basic Logic" (BL, for short) arising from the continuous triangular norms, familiar in the framework of fuzzy set theory. They play the role of Lindenbaum algebras from classical Propositional calculus. Apart from their logical interest, BLalgebras have interesting algebraic properties (see [9], [10], [12]). The paper is organized as follows. In section 2 we recall the basic de¯nitions and put in evidence many rules of calculus in BL-algebras which we need in the rest of paper. Section 3 contains some results relative to the lattice of deductive systems of a BLalgebra (Theorem 3.13 characterizes the BL-algebras for which the lattice of deductive systems is a Boolean lattice). Section 4 contains new characterizations for prime and completely meet-irreducible deductive systems of a BL-algebra (see Proposition 4.9, Corollary 4.11, Theorem 4.12, Theorem 4.19, Theorem 4.20 and Corollary 4.21). E-mail: [email protected] E-mail: [email protected]
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D. Bu¹sneag, D. Piciu / Central European Journal of Mathematics 2 (2003) 221{237
In section 5 we introduce the notions of archimedean and hyperarchimedean BLalgebra and we prove a theorem of Nachbin type for BL-algebras (see Theorem 5.15). These results are in the general spirit of algebras of logic, as exposed in [11].
2
De¯nitions and ¯rst properties
De¯nition 2.1. A BL-algebra ([8]{[10],[12]) is an algebra (A; ^; _; ; !; 0; 1) of type (2,2,2,2,0,0) satisfying the following: (a1 ) (A; ^; _; 0; 1) is a bounded lattice, (a2 ) (A; ; 1) is a commutative monoid, (a3 ) and ! form an adjoint pair, i.e. c µ a ! b i® a c µ b for all a; b; c 2 A (where µ is the lattice ordering on A), (a4 ) a ^ b = a (a ! b), (a5 ) (a ! b) _ (b ! a) = 1, for all a; b 2 A. Examples (E1 ) De¯ne on the real unit interval I = [0; 1] binary operations and ! by x y = maxf0; x + y ¡ x ! y = minf1; 1 ¡
1g
x + yg:
Then (I; µ; min; max; ; !; 0; 1) is a BL-algebra (called Lukasiewicz structure). (E2 ) De¯ne on the real unit interval I x y = minfx; yg x ! y = 1 i® x µ y and y otherwise. Then (I; µ; min; max; ; !; 0; 1) is a BL-algebra (called G}odel structure). (E3 ) Let be the usual multiplication of real numbers on the unit interval I and x ! y = 1 i® x µ y and y=x otherwise. Then (I; µ; min; max; ; !; 0; 1) is a BLalgebra (called product structure or Gaines structure). Remark 2.2. Not every residuated lattice, however, is a BL-algebra (see [12], p.16). Consider, for example a residuated lattice de¯ned on the unit interval, for all x; y; z 2 I, such that x y = 0; i® x + y µ
1 and x ^ y elsewhere 2
1 x ! y = 1 if x µ y and maxf ¡ 2
x; yg elsewhere.
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223
Let 0 < y < x; x + y < 12 : Then y < 12 ¡ x and 0 6= y = x ^ y; but x (x ! y) = x ( 12 ¡ x) = 0: Therefore (a4 ) does not hold. (E4 ) If (A; ^; _; :; 0; 1) is a Boolean algebra, then (A; ^; _; ; !; 0; 1) is a BL-algebra where the operation coincides with ^ and x ! y = :x _ y for all x; y 2 A: (E5 ) If (A; ^; _; !; 0; 1) is a relative Stone lattice (see [1], p.176), then (A; ^; _; ; !; 0; 1) is a BL-algebra where the operation coincide with ^. (E6 ) If (A; ©;¤ ; 0) is a MV-algebra (see [2], [3], [12]), then (A; ^; _; ; !; 0; 1) is a BLalgebra, where for x; y 2 A : x y = (x¤ © y ¤ )¤ ; x ! y = x¤ © y; 1 = 0¤ ; x _ y = (x ! y) ! y = (y ! x) ! x and x ^ y = (x¤ _ y ¤ )¤ : 6 1. For any BL-algebra A, the reduct L(A) = A BL-algebra is nontrivial if 0 = (A; ^; _; 0; 1) is a bounded distributive lattice. For any a 2 A , we de¯ne a¤ = a ! 0 and denote (a¤ )¤ by a¤¤ : We denote the set of natural numbers by ! and de¯ne a0 = 1 and an = an¡1 a for n 2 !nf0g: The order of a 2 A; a 6= 1, in symbols ord(a) is the smallest n 2 ! such that an = 0; if no such n exists, then ord(a) = 1: A BL-algebra is called locally ¯nite if all non unit elements in it are of ¯nite order. In [8]-[10], [12] it is proved that if A is a BL-algebra and a; b; c; bi 2 A, ( i 2 I) then we have the following rules of calculus: (c1 ) a b µ a; b; hence a b µ a ^ b and a 0 = 0; (c2 ) a µ b implies a c µ b c; (c3 ) a µ b i® a ! b = 1; (c4 ) 1 ! a = a; a ! a = 1; a µ b ! a; a ! 1 = 1; (c5 ) a a¤ = 0; (c6 ) a b = 0 i® a µ b¤ ; (c7 ) a _ b = 1 implies a b = a ^ b; (c8 ) a ! (b ! c) = (a b) ! c = b ! (a ! c); (c9 ) (a ! b) ! (a ! c) = (a ^ b) ! c; (c10 ) a ! (b ! c) ¶ (a ! b) ! (a ! c); (c11 ) a µ b implies c ! a µ c ! b; b ! c µ a ! c and b¤ µ a¤ ; (c12 ) a µ (a ! b) ! b , ((a ! b) ! b) ! b = a ! b; (c13 ) a (b _ c) = (a b) _ (a c); (c14 ) a (b ^ c) = (a b) ^ (a c); (c15 ) a _ b = ((a ! b) ! b) ^ ((b ! a) ! a); (c16 ) (a ^ b)n = an ^ bn ; (a _ b)n = an _ bn ; hence a _ b = 1 implies an _ bn = 1 for any n 2 !; (c17 ) a ! (b ^ c) = (a ! b) ^ (a ! c); (c18 ) (b ^ c) ! a = (b ! a) _ (c ! a); (c19 ) (a _ b) ! c = (a ! c) ^ (b ! c);
224
(c20 ) (c21 ) (c22 ) (c23 ) (c24 ) (c25 ) (c26 ) (c27 ) (c28 ) (c29 ) (c30 ) (c31 )
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a ! b µ (b ! c) ! (a ! c); a ! b µ (c ! a) ! (c ! b); a ! b µ (a c) ! (b c); a (b ! c) µ b ! (a c); (b ! c) (a ! b) µ a ! c; (a1 ! a2 ) (a2 ! a3 ) : : : (an¡1 ! an ) µ a1 ! an ; a; b µ c and c ! a = c ! b implies a = b; a _ (b c) ¶ (a _ b) (a _ c); hence am _ bn ¶ (a _ b)mn , for any m; n 2 !; 0 0 0 0 (a ! b) (a ! b ) µ (a _ a ) ! (b _ b ); 0 0 0 0 (a ! b) (a ! b ) µ (a ^ a ) ! (b ^ b ); (a ! b) ! c µ ((b ! a) ! c) ! c;
a (
V
i2I
a (
W
(
W
bi ) =
V
bi ) =
i2I
(a bi );
V
(a ! bi );
i2I
bi ) ! a =
V
(bi ! a)
i2I
(bi ! a) µ (
i2I
W
W
i2I
i2I
W
(a bi );
i2I
i2I
a!(
V
bi ) µ
V
bi ) ! a;
i2I
(a ! bi ) µ a ! (
i2I
a^(
W
i2I
bi ) =
W
W
bi );
i2I
(a ^ bi ); if A is an BL-chain then a _ (
i2I
(whenever the arbitrary meets and unions exist);
V
i2I
bi ) =
V
(a _ bi )
i2I
(c32 ) a µ a¤¤ , 1¤ = 0 , 0¤ = 1; a¤¤¤ = a; a¤¤ µ a¤ ! a; (c33 ) (a ^ b)¤ = a¤ _ b¤ and (a _ b)¤ = a¤ ^ b¤ ; (c34 ) (a ^b)¤¤ = a¤¤ ^ b¤¤ , (a_b)¤¤ = a¤¤ _ b¤¤ ; (ab)¤¤ = a¤¤ b¤¤ , (a ! b)¤¤ = a¤¤ ! b¤¤ ; (c35 ) If a¤¤ µ a¤¤ ! a; then a¤¤ = a; (c36 ) a = a¤¤ (a¤¤ ! a); (c37 ) a ! b¤ = b ! a¤ = a¤¤ ! b¤ = (a b)¤ ; (c38 ) (a¤¤ ! a)¤ = 0; (a¤¤ ! a) _ a¤¤ = 1; (c39 ) b¤ µ a implies a ! (a b)¤¤ = b¤¤ : For any BL-algebra A, B(A) denotes the Boolean algebra of all complemented elements in L(A) (hence B(A) = B(L(A))): Proposition 2.3. ([8]-[10], [12])For e 2 A; the following are equivalent: (i) e 2 B(A); (ii) e e = e and e = e¤¤ ; (iii) e e = e and e¤ ! e = e;
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(iv) e _ e¤ = 1: Remark 2.4. If a 2 A; and e 2 B(A); then e a = e ^ a; a ! e = (a e¤ )¤ = a¤ _ e; if e µ a_ a¤ , then e a 2 B(A): Proposition 2.5. For e 2 A; the following are equivalent: (i) e 2 B(A); (ii) (e ! x) ! e = e; for every x 2 A: Proof. (i) ) (ii) If x 2 A; then from 0 µ x we deduce e¤ µ e ! x hence (e ! x) ! e µ e¤ ! e = e: Since e µ (e ! x) ! e we obtain (e ! x) ! e = e: (ii) ) (i) If x 2 A; then from (e ! x) ! e = e we deduce (e ! x) [(e ! x) ! e] = (e ! x) e; hence (e ! x) ^ e = e ^ x: For x = 0 we obtain that e¤ ^ e = 0: Also, from hypothesis (for x = 0) we obtain e¤ ! e = e: So, from (c15 ) we obtain e _ e¤ = [(e ! e¤ ) ! e¤ ] ^ [(e¤ ! e) ! e] = [(e ! e¤ ) ! e¤ ] ^ (e ! e) = [(e ! e¤ ) ! e¤ ] ^ 1 = (e ! e¤ ) ! e¤ = [e (e ! e¤ )]¤ (by (c37 )) = (e ^ e¤ )¤ = 0¤ = 1; hence e 2 B(A): De¯nition 2.6. Following Diego [5], by Hilbert algebra we mean an algebra (A; !; 1) of type (2; 0) satisfying the following identities: (a6 ) x ! (y ! x) = 1; (a7 ) (x ! (y ! z)) ! ((x ! y) ! (x ! z)) = 1; (a8 ) If x ! y = y ! x = 1, then x = y: Proposition 2.7. For a BL- algebra (A; ^; _; ; !; 0; 1) the following are equivalent: (i) (A; !; 1) is a Hilbert algebra, (ii) (A; ^; _; !; 0; 1) is a relative Stone lattice. Proof. (i) ) (ii) Suppose that (A; !; 1) is a Hilbert algebra, then for every x; y; z 2 A we have x ! (y ! z) = (x ! y) ! (x ! z) From (c8 ) and (c9 ) we have x ! (y ! z) = (x y) ! z
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and (x ! y) ! (x ! z) = (x ^ y) ! z; so we obtain (x y) ! z = (x ^ y) ! z hence x y = x ^ y; that is (A; ^; _; !; 0; 1) is a relative Stone lattice. (ii) ) (i) If (A; ^; _; !; 0; 1) is a relative Stone lattice, then (A; ^; _; !; 0; 1) is a Heyting algebra, so (A; !; 1) is a Hilbert algebra.
3
The lattice of deductive systems of a BL-algebra
In the rest of this paper by A we denote a BL-algebra. De¯nition 3.1. A non empty subset D ³ A is a deductive system of A, ds for short, if the following conditions are satis¯ed: (a6 ) 1 2 D; (a7 ) If x; x ! y 2 D, then y 2 D: Clearly f1g and A are ds; a ds of A is called proper if D 6= A: Remark 3.2. A ds D is proper i® 0 2 = D i® no element a 2 A holds a; a¤ 2 D: Remark 3.3. ([12])A non empty subset D ³ A is a ds of A, i® for all a; b 2 A : (a8 ) a; b 2 D implies a b 2 D; (a9 ) a 2 D and a µ b implies b 2 D: Deductive systems of A and congruence relations ¹ x¹
D
D
on A:
y i® (x ! y) (y ! x) 2 D
are in one-to-one correspondence ([12], p.21). Starting from a ds D, the quotient algebra A=D becomes a BL-algebra with the natural operations induced from those of A. We let x=D be the congruence class of x modulo vD , x 2 A: Then, for x; y 2 A, x=D µ y=D i® x ! y 2 D and x=D = 1=D i® x 2 D: Remark 3.4. Deductive systems are called also implicative ¯lters in literature. To avoid confusion we reserve, however, the name ¯lter to lattice ¯lters in this paper. From (c1 ) and Remark 3.3 we deduce that every ds of A is a ¯lter for L(A), but ¯lters of L(A) are not, in general, deductive systems for A; in example (E1 ); the only proper ds is f1g: We denote by Ds(A) the set of all deductive systems of A.
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For a nonempty subset M ³ A we denote by [M ) the ds of A generated by M (that is, [M ) = \fD 2 Ds(A) : M ³ Dg): If M = fag with a 2 A; we denote by [a) the ds generated by fag ([a) is called principal). For D 2 Ds(A) and a 2 AnD; we denote by D(a) = [D [ fag): Proposition 3.5. ([8]-[10], [12]) (i) If M ³ A is a nonempty subset of A, then: [M ) = fa 2 A : x1 : : : xn µ a; for some x1; :::; xn 2 M g: In particular, for a 2 A; [a) = fx 2 A : x ¶ an ; for some n 2 !g: (ii) If D 2 Ds(A) and a 2 AnD; then D(a) = fx 2 A : x ¶ y an ; with y 2 D and n 2 !g: (iii) If x; y 2 A; and x µ y, then [y) ³ [x): (iv) If x; y 2 A; then [x) \ [y) = [x _ y): Remark 3.6. ([12], p.17) If D 2 Ds(A) and a 2 A; then a 2 D; i® an 2 D; for any n 2 !: For D 1 ; D2 2 Ds(A) we put D 1 ^ D2 = D1 \ D2 and D 1 _ D2 = [D 1 [ D2 ) = fa 2 A : a ¶ x y; for some x 2 D1 and y 2 D2 g: Then (Ds(A); ^; _; f1g; A) is a complete Brouwerian lattice; we recall that a complete W W lattice is Brouwerian if it satis¯es the identity a^( bi ) = (a^bi ), whenever the arbitrary unions exists.
i
i
De¯nition 3.7. ([6], p.93) Let L be a complete lattice and let a be an element of L. Then a is called compact if a µ _X for some X ³ L implies that a µ _X1 for some ¯nite X1 ³ X: A complete lattice is called algebraic if every elements is the join of compact elements (in the literature, algebraic lattices are also called compactly generated lattices). Proposition 3.8. The lattice (Ds(A); ³) is an algebraic lattice. Proof.
We know that (Ds(A); ³) is complete. We claim that for a 2 A; [a) is a
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compact element of Ds(A): Let X ³ Ds(A) and let [a) ³
W
D: We have
D2X
_
D = fx 2 A : x1 : : : xn µ x for some xi 2 Di ; Di 2 X g:
D2X
Therefore, a ¶ x1 : : : xn , xi 2 Di ; Di 2 X; 1 µ i µ n: Then, with W W X1 = fD1 ; :::; Dn g, [a) ³ D: Since for any D 2 Ds(A) we have D = [a), we see D2X1
a2D
that Ds(A) is algebraic.
Lemma 3.9. If x; y 2 A; then [x) _ [y) = [x y): Proof. Since x y µ x; y; then [x); [y) ³ [x y); hence [x) _ [y) ³ [x y): If z 2 [x y); then for some natural number n; z ¶ (x y)n = xn y n 2 [x) _ [y); hence z 2 [x) _ [y); that is [x y) ³ [x) _ [y); so [x) _ [y) = [x y): For D 1 ; D2 2 Ds(A) we put D1 ! D2 = fa 2 A : D1 \ [a) 2 D2 g: Lemma 3.10. If D1 ; D2 2 Ds(A) then (i) D1 ! D 2 2 Ds(A); (ii) If D 2 Ds(A); then D1 \ D ³ D2 i® D ³ D1 ! D2 (that is, D1 ! D 2 = supfD 2 Ds(A) : D1 \ D ³ D2 g): Proof. (i) Since [1) = f1g and [1) \ D1 = f1g ³ D2 we deduce that 1 2 D1 ! D 2 . Let x; y 2 A such that x µ y and x 2 D1 ! D 2 , that is [x) \ D1 ³ D2 : Then [y) ³ [x); so [y) \ D1 ³ [x) \ D1 ³ D2 ; hence [y) \ D1 ³ D2 that is y 2 D1 ! D 2 . To prove that (a8 ) is veri¯ed, let x; y 2 A such that x; y 2 D1 ! D 2 , hence [x) \ D1 ³ D2 and [y) \ D1 ³ D2 : We deduce ([x) \ D1 ) _ ( [y) \ D1 ) ³ D2 ; hence ([x)_ [y)) \ D1 ³ D2 : By Lemma 3.9 we deduce that [x y) \ D1 ³ D2 ; that is x y 2 D1 ! D2 ; hence D1 ! D 2 2 Ds(A) (by Remark 3.3): (ii) Suppose D1 \ D ³ D2 and let x 2 D: Then [x) ³ D; hence [x) \ D1 ³ D \ D1 ³ D2 : So x 2 D1 ! D 2 , that is D ³ D1 ! D 2 : Suppose D ³ D1 ! D 2 and let x 2 D1 \ D. Then x 2 D, hence x 2 D1 ! D 2 , that is [x) \ D1 ³ D2 : Since x 2 [x) \ D1 ³ D2 we obtain x 2 D2 ; that is D1 \ D ³ D2 . Remark 3.11. From Lemma 3.10 we deduce that (Ds(A); _; ^; !; f1g; A) is a Heyting algebra; for D 2 Ds(A), D ¤ = D ! 0 = D ! f1g = fx 2 A : [x) \ D = f1gg and so, for a 2 A, [a)¤ = fx 2 A : [x) \ [a) = f1gg = fx 2 A : [x _ a) = f1gg = fx 2 A : x _ a = 1g: Proposition 3.12. If x; y 2 A; then [x y)¤ = [x)¤ \ [y)¤ . Proof. If a 2 [x y)¤ , then a _ (x y) = 1: Since x y µ x; y, then a _ x = a _ y = 1;
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hence a 2 [x)¤ \ [y)¤ ; that is [x y)¤ ³ [x)¤ \ [y)¤ : Let now a 2 [x)¤ \ [y)¤ ; that is a _ x = a _ y = 1: By (c27 ) we deduce a _ (x y) ¶ (a _ x) (a _ y) = 1; hence a _ (x y) = 1; that is a 2 [x y)¤ . It follows that [x)¤ \ [y)¤ ³ [x y)¤ ; hence [x y)¤ = [x)¤ \ [y)¤ . Theorem 3.13. If A is a BL-algebra, then the following assertions are equivalent: (i) (Ds(A); _; ^;¤ ; f1g; A) is a Boolean algebra, (ii) Every ds of A is principal and for every x 2 A;there is n 2 ! such that x _ (xn )¤ = 1: Proof. (i) ) (ii) Let D 2 Ds(A) ; since Ds(A) is Boolean algebra, then D _ D¤ = A: So, for 0 2 A; there exist a 2 D; b 2 D¤ such that a b = 0: Since b 2 D¤ , by Remark 3.11, it follows that a _ b = 1: By (c7 ) we deduce that a ^ b = a b = 0; that is b is the complement of a in L(A): Hence a; b 2 B(A) = B(L(A)): If x 2 D; since b 2 D¤ we have b_x = 1: Since a = a^(b_x) = (a^b)_(a ^x) = a^x we deduce that a µ x; that is D = [a): Hence every ds of A is principal. Let now x 2 A; since Ds(A) is Boolean algebra, then [x)_[x)¤ = A , [x)¤ (x) = A , fa 2 A : a ¶ c xn ; with c 2 [x)¤ and n 2 !g = A (see Proposition 3.5, (ii)). So, since 0 2 A; there exist c 2 [x)¤ and n 2 ! such that c xn = 0: Since c 2 [x)¤ , then x _ c = 1: By (c6 ), from c xn = 0 we deduce c µ (xn )¤ : So, 1 = x _ c µ x _ (xn )¤ ; hence x _ (xn )¤ = 1: (ii) ) (i) By Remark 3.11, Ds(A) is a Heyting algebra. To prove Ds(A) is Boolean algebra, we must show that for D 2 Ds(A), D ¤ = f1g only for D = A ([1], p. 175). By hypothesis every ds of A is principal, so we have a 2 A such that D = [a): Also, by hypothesis, for a 2 A, there is n 2 ! such that a _ (an )¤ = 1: By Remark 3.11, (an )¤ 2 [a)¤ = f1g; hence (an )¤ = 1; that is an = 0: By Remark 3.6, we deduce that 0 2 D, hence D = A:
4
The spectrum of a BL-algebra
De¯nition 4.1. Let L be a lattice with the least element 0 and the greatest element 1. An element p < 1 is meet-irreducible if p = x ^ y implies p = x or p = y; an element p < 1 is meet-prime if x ^ y µ p implies x µ p or y µ p: Dually are de¯ned the notions of join-irreducible and join-prime. Remark 4.2. If L is distributive meet-irreducible and meet-prime elements are the same. These de¯nitions can be extended to arbitrary meets and we obtain the concepts of completely meet (join)-irreducible and completely meet (join)-prime elements, which are
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no longer equivalent. For the lattice Ds(A) (which is distributive) we denote by Spec(A) the set of all meet-irreducible (hence meet-prime) elements (Spec(A) is called the spectrum of A) and by Irc(A) the set of all completely meet-irreducible elements of the lattice Ds(A): De¯nition 4.3. ([12], p.18) A proper ds D of A is called prime if, for any a; b 2 A; the condition a _ b 2 D implies a 2 D or b 2 D: Remark 4.4. ([12], p.18,19) 1: A non-degenerate BL-algebra contains a prime ds. 2: If D is a prime ds of A then, for any a; b 2 A; either a ! b 2 D or b ! a 2 D: Moreover, A is linear i® any proper ds of A is prime. 3: If P is a prime ds of A and D is a proper ds of A such that P ³ D; then also D is prime. Theorem 4.5. ([10],[12]) For a proper P 2 Ds(A) the following are equivalent: (i) P is prime, (ii) A=P is a chain, (iii) For all x; y 2 A; x ! y 2 P or y ! x 2 P; (iv) The set of proper ds including a prime ¯lter P of A is a chain. Theorem 4.6. (Prime ds theorem [10]) If D 2 Ds(A) and I an ideal of the lattice L(A) such that D \ I = ?; then there is a prime ds P of A such that D ³ P and P \ I = ?: Corollary 4.7. If D 2 Ds(A) is proper and a 2 AnD; then there is P 2 Spec(A) such that D ³ P and a 2 = P: In particular, for D = f1g we deduce that for any a 2 A; a 6= 1; there is Pa 2 Spec(A); such that a 2 = Pa: Proposition 4.8. For a proper P 2 Ds(A) the following are equivalent: (i) P is prime, (ii) P 2 Spec(A); (iii) If a; b 2 A; and a _ b = 1; then a 2 P or b 2 P: Proof. (i) ) (ii) Let D1 ; D2 2 Ds(A) such that D1 \ D2 = P: Since P ³ D1 ; P ³ D2 ; by Theorem 4.5, (iv), D1 ³ D2 or D2 ³ D1 ; hence P = D1 or P = D2 : (ii) ) (i) Let a; b 2 A; such that a _ b 2 P: Since P (a) \ P (b) = (P _ [a)) \ (P _ [b)) = P _ ([a) \ [b)) = P _ [a _ b) = P; then P = P (a) or P = P (b), hence a 2 P or b 2 P; that is P is prime. (i) ) (iii) Clearly, since 1 2 P: (iii) ) (i) Clearly by Theorem 4.5, (iii) (since (a ! b)_(b ! a) = 1 for every a; b 2 A).
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Proposition 4.9. For a proper P 2 Ds(A) the following are equivalent: (i) P 2 Spec(A); (ii) For every x; y 2 AnP there is z 2 AnP such that x µ z and y µ z: Proof. (i) ) (ii) Let P 2 Spec(A) and x; y 2 AnP . If by contrary, for every a 2 A with x µ a and y µ a then a 2 P; since x; y µ x _ y we deduce x _ y 2 P: Hence, x 2 P or y 2 P; a contradiction. (ii) ) (i) I suppose by contrary that there exist D1 ; D2 2 Ds(A) such that D1 \ D2 = P; and P 6= D1 ; P 6= D 2 : So, we have x 2 D1 nP and y 2 D 2 nP: By hypothesis there is z 2 AnP such that x µ z and y µ z: We deduce z 2 D1 \ D2 = P - a contradiction. Corollary 4.10. For a proper P 2 Ds(A) the following are equivalent: (i) P 2 Spec(A); (ii) If x; y 2 A and [x) \ [y) ³ P; then x 2 P or y 2 P: Proof. (i) ) (ii) Let x; y 2 A such that [x) \ [y) ³ P and suppose by contrary that x; y 2 = P: Then by Proposition 4.9 there is z 2 AnP such that x µ z and y µ z: Hence z 2 [x) \ [y) ³ P; so z 2 P , a contradiction. (ii) ) (i) Let x; y 2 A such that x _ y 2 P: Then [x _ y ) ³ P . Since [x _ y) = [x) \ [y) (by Proposition 3.5, (iv)) we deduce that [x) \ [y) ³ P; hence x 2 P or y 2 P; that is P 2 Spec(A). Corollary 4.11. For a proper P 2 Ds(A) the following are equivalent: (i) P 2 Spec(A); (ii) For every x; y 2 A=P; x 6= 1; y 6= 1 there is z 2 A=P; z 6= 1 such that x µ z, y µ z: Proof. (i) ) (ii) Clearly, by Proposition 4.9, since if x = a=P; with a 2 A; then the condition x 6= 1 (in A=P ) is equivalent with a 2 = P: (ii) ) (i) Let x; y 2 A=P: Then x = a=P 6= 1 and y = b=P 6= 1 (in A=P ). By hypothesis there is z = c=P 6= 1 (that is c 2 = P ) such that x; y µ z equivalent with a ! c; b ! c 2 P: If consider d = (b ! c) ! ((a ! c) ! c) then by (c4 ) and (c8 ) we deduce that a; b µ d: Clearly d 2 = P , hence by Proposition 4.9 we deduce that P 2 Spec(A): Theorem 4.12. For a proper P 2 Ds(A) the following are equivalent: (i) P 2 Spec(A); (ii) For every D 2 Ds(A); D ! P = P or D ³ P: Proof. (i) ) (ii) Let P 2 Spec(A): Since Ds(A) is a Heyting algebra (by Remark 3.11) for
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D 2 Ds(A) we have P = (D ! P ) \ ((D ! P ) ! P ) and so P = D ! P or P = (D ! P ) ! P: If P = (D ! P ) ! P then D ³ P: (ii) ) (i) Let D1 ; D2 2 Ds(A) such that D1 \ D2 = P: Then D1 ³ D2 ! P (see Lemma 3.10, (ii)) and so, if D2 ³ P; then P = D2 and if D2 ! P = P; then P = D 1 ; hence P 2 Spec(A): We recall that if L is a pseudocompleted distributive lattice, then two subsets associated with L ([1], p.153) are Rg(L) = fx 2 L : x¤¤ = xg and D(L) = fx 2 L : x¤ = 0g. The elements of Rg(L) are called regular and those of D(L) dense. Note that f0; 1g ³ Rg(L); 1 2 D(L) and D(L) is a ¯lter in L and Rg(L) is a Boolean algebra under the operations induced by the ordering on L ([1], p.157). Corollary 4.13. For a BL-algebra A; Spec(A) ³ D(Ds(A)) [ Rg(Ds(A)): Proof. Let P 2 Spec(A) and D = P ¤ 2 Ds(A); then by Theorem 4.12, D ³ P or D ! P = P equivalent with P ¤ ³ P or P ¤ ! P = P: Since Ds(A) is a Heyting algebra then P ¤ ! P = P ¤¤ ; so P ¤¤ = A or P ¤¤ = P equivalent with P ¤ = f1g or P ¤¤ = P; that is P 2 D(Ds(A)) [ Rg(Ds(A)): Remark 4.14. From Corollary 4.7 we deduce that for every D 2 Ds(A),
D = \fP 2 Spec(A) : D ³ P g and \ fP 2 Spec(A)g = f1g: Relative to the uniqueness of deductive systems as intersection of primes we have: Theorem 4.15. If every D 2 Ds(A) has a unique representation as an intersection of elements of Spec(A); then (Ds(A); _; ^;¤ ; f1g; Ag is a Boolean algebra. Proof. Let D 2 Ds(A) and D 0 = \fM 2 Spec(A) : D * M g 2 Ds(A): By Remark 4.14, D \ D 0 = \fM 2 Spec(A)g = f1g; if D _ D 0 6= A; then by Corollary 4.7 there exists D 00 2 Spec(A) such that D _ D 0 ³ D 00 and D 00 6= A: Consequently, D0 has two representations D 0 = \fM 2 Spec(A) : D * M g = D 00 \ (\fM 2 Spec(A) : D * M g); which is contradictory. Therefore D _ D0 = A and so Ds(A) is a Boolean lattice. Lemma 4.16. If D 2 Ds(A); D 6= A and a 2 = D; then there exists Da 2 Ds(A) maximal with the property that D ³ Da and a 2 = Da : Proof. Let FD;a = fD 0 2 Ds(A) : D ³ D 0 and a 2 = D 0 g; clearly D 2 FD;a : If C is a chain in FD;a then [C 2 FD;a : By Zorn’s lemma there exists a ds Da which is maximal subject to containing D and a 2 = Da : De¯nition 4.17. D 2 Ds(A); D 6= A is called maximal relative to a if a 2 = D and if 0 0 0 0 D 2 Ds(A) is proper such that a 2 = D , and D ³ D ; then D = D :
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If in Lemma 4.16 we consider D = f1g we obtain: Corollary 4.18. For any a 2 A; a 6= 1; there is a ds Da maximal with the property that a2 = Da . Theorem 4.19. For D 2 Ds(A); D 6= A the following are equivalent: (i) D 2 I rc(A); (ii) There is a 2 A such that D is maximal relative to a. Proof. (i) ) (ii) See ([7], p.248) (since by Proposition 3.8, Ds(A) is an algebraic lattice). T (ii) ) (i) Let D 2 Ds(A) maximal relative to a and suppose D = D i with Di 2 Ds(A) i2I
for every i 2 I: Since a 2 = D there is j 2 I such that a 2 = Dj : So, a 2 = Dj and D ³ Dj : By the maximality of D we deduce that D = Dj ; that is D 2 I rc(A):
Theorem 4.20. Let D 2 Ds(A); D 6= A and a 2 AnD: Then the following are equivalent: (i) D is maximal relative to a; (ii) For every x 2 AnD there is n 2 ! such that xn ! a 2 D: Proof. (i) ) (ii) Let x 2 AnD: If a 2 = D(x) = D _ [x); since D » D(x) then D(x) = A (by the maximality of D) hence a 2 D(x)- a contradiction. We deduce that a 2 D(x); hence a ¶ d xn ; with d 2 D and n 2 !: Then d µ xn ! a, hence xn ! a 2 D: (ii) ) (i) I suppose by contrary that there is D0 2 Ds(A); D 0 6= A such that a 2 = D0 and D » D 0 : Then there is x0 2 D 0 such that x0 2 = D; hence by hypothesis there is 0 n n n 2 ! such that x0 ! a 2 D » D : Thus x0 ! a 2 D 0 and xn0 2 D 0 ; hence a 2 D 0 -a contradiction. Corollary 4.21. For D 2 Ds(A); D 6= A the following are equivalent: (i) D 2 I rc(A); (ii) In the set A=Dnf1g we have an element p 6= 1 with the property that for every x 2 A=Dnf1g there is n 2 ! such that xn µ p: Proof. (i) ) (ii) By Theorem 4.19, D is maximal relative to an element a 2 = D; then, if we let p = a=D 2 A=D; p 6= 1 (since a 2 = D) and for every x = b=D; x 6= 1 (that is b 2 = D) by n n Theorem 4.20 there is n 2 ! such that b ! a 2 D; that is x µ p: (ii) ) (i) Let p = a=D 2 A=Dnf1g; (that is a 2 = D) and x = b=D 2 A=Dnf1g; (that is b2 = D). By hypothesis there is n 2 ! such that xn µ p equivalent with bn ! a 2 D: Then by Theorem 4.20, we deduce that D 2 I rc(A): De¯nition 4.22. A ds of A is a minimal prime ds if P 2 Spec(A) and, whenever Q 2 Spec(A) and Q ³ P we have P = Q:
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Proposition 4.23. If P is a minimal prime ds, then for any a 2 P there is b 2 AnP such that a _ b = 1: Proof. Let P be a minimal prime ds and a 2 P: We de¯ne the set S = fx 2 A : there is b 2 AnP such that a _ b ¶ xg: If b 2 AnP then a _ b ¶ b; so AnP ³ S: Moreover, a 2 S because a _ 0 = a ¶ a and 0 2 AnP: We shall prove that S is an ideal of the lattice L(A). Let x; y 2 A such that y 2 S and x µ y: Thus, there is b 2 AnP such that a _ b ¶ y ¶ x; hence a _ b ¶ x; so x 2 S: If x; y 2 S then there are b; c 2 AnP such that a _ b ¶ x and a _ c ¶ y: If we suppose that b _ c 2 P we get b 2 P or c 2 P because P is a prime ds. Thus, b _ c 2 AnP and a _ (b _ c) ¶ x _ y; so x _ y 2 S; hence S is an ideal. Now, we suppose that 1 2 = S: It follows that f1g \ S = ? so, by Theorem 4.6, there is a prime ds Q such that S \ Q = ?: Since AnP ³ S; we get Q ³ P: But Q is prime and P is minimal prime, so P = Q: On the other hand, a 2 S; so a 2 = Q: We get a 2 P nQ, which contradicts the fact that P = Q: Thus, our assumption that 1 2 = S is false. We conclude that 1 2 S and our proof is ¯nished.
5
Maximal deductive systems; archimedean and hyperarchimedean BL-algebras
De¯nition 5.1. A ds of A is maximal if it is proper and it is not contained in any other proper ds. We shall denote by M ax(A) the set of all the maximal ds of A; it is obvious that M ax(A) ³ Spec(A): We have: Theorem 5.2. ([12], p.24) For M 2 Ds(A); M 6= A; the following are equivalent: (i) M 2 M ax(A); (ii) For every x 2 = M there is n 2 ! such that (xn )¤ 2 M; (iii) A=M is locally ¯nite. De¯nition 5.3. If D is a proper ds of A and there exists another proper ds D0 such that D ³ D 0 we say that D can be extended to D 0 : Theorem 5.4. ([12], p.19) (i) Any proper ds D can be extended to a prime ds, (ii) Any proper ds D can be extended to a maximal, prime ds. De¯nition 5.5. The intersection of the maximal ds of A is called the radical of A. It
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will be denoted by Rad(A). It is obvious that Rad(A) is a ds. Proposition 5.6. ([10]) Rad(A) = fa 2 A : (an )¤ µ a; for any n 2 !g: Proposition 5.7. For any a; b 2 Rad(A); a¤ b¤ = 0: Proof. Let a; b 2 Rad(A); to prove a¤ b¤ = 0 is equivalent with (a¤ b¤ )¤ = 1: Suppose that (a¤ b¤ )¤ 6= 1: By Corollary 4.7, there is a prime ds P such that (a¤ b¤ )¤ 2 = P . By (c37 ) we have (a¤ b¤ )¤ = a¤ ! b¤¤ 2 = P; so by Theorem 4.5, ¤¤ ¤ ¤¤ ¤ b ! a 2 P; that is (b a) 2 P: By Theorem 5.4 there is a maximal ds M such that P ³ M: Then b¤¤ a 2 = M: By ¤¤ n ¤ n ¤¤ n ¤ Theorem 5.2, there is n 2 ! such that [(b a) ] = [(b ) a ] 2 M ; so, if we let c = (bn )¤¤ an ; we have c¤ 2 M . Since a; b 2 Rad(A) then we infer that a; b 2 M; hence c = (bn )¤¤ an 2 M . Hence c and c¤ are in M which contradicts the fact that M is a proper ds of A. De¯nition 5.8. An element a of A is called in¯nitesimal if a 6= 1 and an ¶ a¤ ; for any n 2 !: Proposition 5.9. For every nonunit element a of A the following are equivalent: (i) a is in¯nitesimal, (ii) a 2 Rad(A): Proof. (i) ) (ii) Let a 6= 1 be an in¯nitesimal and suppose a 2 = Rad(A): Thus, there is a maximal ds M of A such that a 2 = M: By Theorem 5.2, there is n 2 ! such that n ¤ n (a ) 2 M . By hypothesis a ¶ a¤ hence (an )¤ µ a¤¤ ; so a¤¤ 2 M; hence (a¤¤ )n = (an )¤¤ 2 M: If we let b = (an )¤ we conclude that b; b¤ 2 M which contradicts the fact that M is a proper ds. (ii) ) (i) Let a 2 Rad(A); then (an )¤ µ a for any n 2 !: For n = 1 we obtain that a¤ µ a: Since for any n 2 !; an 2 Rad(A) we deduce that (an )¤ µ an : Since a¤ an µ a¤ a = 0 we obtain that a¤ an = 0 for any n 2 !; hence by (c6 ), a¤ µ (an )¤ : So, for any n 2 !; a¤ µ (an )¤ and (an )¤ µ an ; hence a¤ µ an ; that is a is an in¯nitesimal. Lemma 5.10. If a 2 A; n 2 ! such that a _ (an )¤ = 1 and an ¶ a¤ ; then a = 1: Proof. By (c11 ) we obtain (an )¤ µ a¤¤ ; so 1 = a _ (an )¤ µ a _ a¤¤ = a¤¤ ; hence a¤¤ = 1; that is a¤ = 0: Then a ! (a ! 0) = a ! 0 = 0: From (c8 ) we deduce that (a2 )¤ = 0: Recursively we obtain that (an )¤ = 0: Then a _ 0 = 1; hence a = 1: Lemma 5.11. In any BL-algebra A the following are equivalent: (i) For every a 2 A; an ¶ a¤ for any n 2 ! implies a = 1; (ii) For every a; b 2 A; an ¶ b for any n 2 ! implies a ! b = b and b ! a = a:
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Proof. (i) ) (ii) Let a; b 2 A such that an ¶ b for any n 2 !: We get (a _ b)¤ = a¤ ^ b¤ µ a¤ µ an µ (a _ b)n ; hence (a _ b)n ¶ (a _ b)¤ for any n 2 !. By hypothesis, a _ b = 1: From (c15 ) we deduce (a ! b) ! b = (b ! a) ! a = 1; hence a ! b = b and b ! a = a: (ii) ) (i) Let a 2 A such that an ¶ a¤ for any n 2 !: By hypothesis we get a ! a = a; so a = 1: De¯nition 5.12. A BL-algebra A is called archimedean if the equivalent conditions from Lemma 5.11 are satis¯ed. One can easily remark that a BL-algebra is archimedean i® it has no in¯nitesimals. De¯nition 5.13. Let A be a BL-algebra. An element a 2 A is called archimedean if it satis¯es the condition: there is n 2 !; n ¶ 1; such that a _ (an )¤ = 1: A BL-algebra A is called hyperarchimedean if all its elements are archimedean. From Lemma 5.10 we deduce: Corollary 5.14. Every hyperarchimedean BL-algebra is archimedean. Now, we have a theorem of Nachbin type (see [1], p.73) for BL-algebras: Theorem 5.15. For a BL-algebra A the following are equivalent: (i) A is hyperarchimedean, (ii) For any ds D, the quotient BL-algebra A=D is an archimedean BL-algebra, (iii) Spec(A) = M ax(A); (iv) Any prime ds is minimal prime. Proof. (i) ) (ii) To prove A=D is archimedean, let x = a=D 2 A=D such that xn ¶ x¤ for any n 2 !: By hypothesis, there is m 2 !; m ¶ 1 such that a _ (am )¤ = 1: It follows that x _ (xm )¤ = 1 (in A=D). In particular we have xm ¶ x¤ ; so by Lemma 5.10 we deduce that x = 1; that is A=D is archimedean. (ii) ) (iii) Since M ax(A) ³ Spec(A); we only have to prove that any prime ds of A is maximal. If P 2 Spec(A); then A=P is a chain (see Theorem 4.5). By hypothesis A=P is archimedean. By Theorem 5.2 to prove P 2 M ax(A) it su±ces to prove that A=P is locally ¯nite. Let x = a=P 2 A=P; x 6= 1: Then there is n 2 !; n ¶ 1; such that xn ¤ x¤ : Since A=P is chain we have xn µ x¤ . Thus xn+1 µ x x¤ = 0; hence xn+1 = 0; that is o(x) < 1: It follows that A=P is locally ¯nite. (iii) ) (iv) Let P; Q prime ds such that P ³ Q: By hypothesis, P is maximal, so P = Q:
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Thus Q is minimal prime. (iv) ) (i) Let a be a nonunit element from A. We shall prove that a is an archimedean element. If we denote D = [a)¤ = fx 2 A : a _ x = 1g (by Remark 3.11), then D 2 Ds(A). Since a 6= 1; then a 2 = D and we consider 0 n D = D(a) = fx 2 A : x ¶ d a for some d 2 D and n 2 !g: If we suppose that D0 is a proper ds of A, then by Corollary 4.7, there is a prime ds P such that D0 ³ P: By hypothesis, P is a minimal prime. Since a 2 P; using Proposition 4.23, we infer that there is x 2 AnP such that a _ x = 1: It follows that x 2 D ³ D0 ³ P; hence x 2 P; so we get a contradiction. Thus D 0 is not proper, so 0 2 D 0 ; hence there is n 2 ! and d 2 D such that dan = 0: Thus d µ (an )¤ (by (c6 )): We get a _ d µ a _ (an )¤ : But a _ d = 1 (since d 2 D), so we obtain that a _ (an )¤ = 1; that is a is an archimedean element.
Acknowledgements We would like to express our gratitude for the guidance given by the referee in the elaboration of this paper.
References [1] R. Balbes and Ph. Dwinger: Distributive Lattices, University of Missouri Press, 1974. [2] D. Bu»sneag and D. Piciu: \Meet-irreducible ideals in an MV-algebra", Analele Universit¸a»tii din Craiova, Seria Matematica-Informatica, Vol. XXVIII, (2001), pp. 110{119. [3] D. Bu»sneag and D. Piciu: \On the lattice of ideals of an MV-algebra", Scientiae Mathematicae Japonicae, Vol. 56, (2002), pp. 367{372. [4] R. Cignoli, I.M.L. D’Ottaviano, D. Mundici: Algebraic foundation of many-valued reasoning, Kluwer Academic Publ., Dordrecht, 2000. [5] A. Diego: \Sur les algµebres de Hilbert", In. Ed. Hermann: Collection de Logique Math¶ematique, Serie A, XXI, Paris, 1966. [6] G. GrÄatzer: Lattice theory, W. H. Freeman and Company, San Francisco, 1979. [7] G. Georgescu and M. Plo·s·cica: \Values and minimal spectrum of an algebraic lattice", Math. Slovaca, Vol. 52, (2002), pp. 247{253. [8] P. H¶ajek: Metamathematics of Fuzzy Logic, Kluwer Academic Publ., Dordrecht, 1998. [9] A. Iorgulescu: \Is¶eki algebras. Connections with BL-algebras", to appear in Soft Computing. [10] A. Di Nola, G. Georgescu, A. Iorgulescu: \Pseudo-BL-algebras", to appear in Multiple Valued Logic. [11] H. Rasiowa, An Algebraic Approach to Non-Classical Logics, PWN and NorthHolland Publishing Company, 1974. [12] E. Turunen: Mathematics Behind Fuzzy Logic, Physica-Verlag, 1999.
CEJM 2 (2003) 238{271
Solutions to the XXX type Bethe ansatz equations and ° ag varieties E. Mukhin1¤ , A. Varchenko2y 1
Department of Mathematical Sciences, Indiana University - Purdue University Indianapolis, 402 North Blackford St, Indianapolis, IN 46202-3216, USA 2 Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA
Received 9 December 2002; revised 26 February 2003 Abstract: We consider a version of the AN Bethe equation of XXX type and introduce a reproduction procedure constructing new solutions of this equation from a given one. The set of all solutions obtained from a given one is called a population. We show that a population is isomorphic to the slN+1 ®ag variety and that the populations are in one-to-one correspondence with intersection points of suitable Schubert cycles in a Grassmanian variety. We also obtain similar results for the root systems BN and CN . Populations of BN and CN type are isomorphic to the ®ag varieties of CN and BN types respectively. c Central European Science Journals. All rights reserved. ® Keywords: algebraic Bethe anzatz, Schubert Calculus, descrete Wronskian MSC (2000): 82B23, 14C17, 17B37
1
Introduction
y
¤
In this paper we consider a system of algebraic equations, see below (10), which we consider as an AN version of the X X X Bethe equation. In the simplest case of N = 1, our system does coincide with the famous and much studied Bethe equation for the inhomogeneous X X X -model (see [BIK], [Fd], [FT] and references therein). We call the solutions of our system (10) (with additional simple conditions) h-critical points and study the problem of counting the number of the h-critical points. E-mail: [email protected] E-mail: [email protected]
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The de¯nition of the critical points depends on the complex non-zero step h, complex distinct spectral parameters z1 ; : : : ; zn , slN dominant integral weights ¤1 ; : : : ; ¤n and another slN weight ¤1 . We conjecture that for generic positions of zi and dominant integral ¤1 , the number of h-critical points properly counted equals the multiplicity of L¤1 in L¤1 « ¢ ¢ ¢ « L¤n , where L¤ is the irreducible slN +1 representation of highest weight ¤. In the present paper we propose a way to attack this conjecture. Given an h-critical point with a given weight at in¯nity, ¤1 , we describe a procedure of constructing a set of h-critical points with weights at in¯nity of the form w ¢ ¤1 , where w is in the slN +1 Weyl group and the dot denotes the shifted action, see (14). We call this procedure the reproduction procedure and the resulting set of h-critical points a population. The reproduction procedure makes use of a reformulation of the algebraic system (10) in terms of di®erence equations of the second order, and furthermore in the following fertility property. Under some technical conditions, zeroes of an N -tuple of polynomials (y1 ; : : : ; yN ) form an h-critical point if and only if there exist polynomials y~1 ; : : : ; y~N such that W (yi(x); y~i (x)) = yi¡1 (x + h)yi+1 (x)T i (x); where W (u; v) = u(x + h)v(x) ¡ u(x)v(x + h) is the discrete Wronskian and polynomials Ti are given explicitly in terms of zi and ¤i , see (11). Furthermore, it turns out that in this case the zeros of the tuple (y1 ; : : : ; y~i; : : : ; yN ) also form an h-critical point, and therefore we are able to repeat the same argument. Thus, we get a family of h-critical points, each represented by an N -tuple of polynomials. The space V of the ¯rst coordinates of these N -tuples has dimension N + 1 and is called the fundamental space. The population is identi¯ed with the variety of all full °ags in V . Given a °ag f0 = F0 » F 1 » ¢ ¢ ¢ » FN +1 = V g, the corresponding N -tuple of polynomials is given by yi = W i (F i )=U i , where U i are some explicit polynomials written in terms of Ti , see Lemma 4.12, and W i denotes the discrete Wronskian of order i. We consider the fundamental space of a population as an (N +1)-dimensional subspace of the space Cd [x] of polynomials of su±ciently big degree. Therefore a fundamental space de¯nes a point in the Grassmannian variety of (N + 1)-dimensional subspaces of Cd [x]. The fact that Wronskians of all i-dimensional subspaces in V are divisible by U i results in the conclusion that V belongs to the intersection of suitable Schubert cells in the Grassmannian. These Schubert cells are related to special °ags (zi ) in Cd [x]. These °ags are formed by the subspaces Fj (zi ) in Cd [x] which consist of all polynomials divisible by (x ¡ zi )(x ¡ zi ¡ h) : : : (x ¡ zi ¡ (d ¡ j)h). In addition, the point of the Grassmannian V belongs to a speci¯c Schubert cell related to the °ag (1). The °ag (1) is formed by the subspaces Fj (1) in Cd [x] which consist of all polynomials of degree less than j. Any population contains at most one h-critical point associated to an integral dominant weight at in¯nity. The fundamental spaces corresponding to di®erent populations are di®erent. Therefore we related the problem of counting the h-critical points associated to integral dominant weights at in¯nity to the problem of counting the points in intersections of Schubert cycles. It is well-known that these Schubert cycles have the
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algebraic index of intersection equal to the multiplicity of L¤1 in L¤1 « ¢ ¢ ¢ « L¤n . It is an open question if for generic zi the intersection points are all of multiplicity one. The scheme described above repeats the method in [MV1], where a similar picture was developed for the case of Bethe equations related to the Gaudin model. The paper [MV1] is a smooth version of the present paper; it uses Fuchsian di®erential equations and the usual Wronskians. In the smooth case, the critical points, reproduction procedure and populations are de¯ned for any Kac-Moody algebra. We suggest relevant de¯nitions in the case of root systems of type BN and CN , see also [MV3]. We say that the zeroes of polynomials (y1 ; : : : ; yN ) constitute a BN (resp. CN ) hcritical point if the zeroes of (y1 (x); : : : ; yN (x); yN ¡1 (x + h); yN ¡2 (x + 2h); : : : ; y1 (x + (N ¡
1)h))
(resp. (y1 (x); : : : ; yN (x); yN (x + h=2); yN ¡1 (x + 3h=2); : : : ; y1 (x + N ¡
h=2)))
form an sl2N (resp. sl2N+1 ) h-critical point, see Section 6.1 (resp. Section 7.1) for details. Our de¯nition is motivated by similar properties of critical points in the smooth case, see [MV1]. It turnes out that h-critical points de¯ned in such a way are exactly solutions of some systems of algebraic equations in both cases of BN and CN . We describe these algebraic equations. A space of polynomials V of dimension N + 1 is called h-selfdual if the space of discrete Wronskians W (F )=U , where F runs over N -dimensional subspaces of V and U is the greatest common divisor of all W (F ), coincides with the space of functions of the form v(x ¡ (N ¡ 1)h=2), where v 2 V . Such a space V has a natural non-degenerate form which turns out to be symmetric if the dimension of V is odd, and skew-symmetric if the dimension of V is even. A population of BN (resp. CN ) type is then naturally identi¯ed with the space of isotropic °ags in an h-selfdual space of dimension 2N (resp. 2N + 1). In particular a BN (resp. CN ) population is identi¯ed with the CN (resp. BN ) °ag variety. The present paper deals with the h-analysis and additive shifts. Similar results can be obtained in the case of the q-analysis with multiplicative shifts related to the Bethe equations of the XXZ type. The paper is constructed as follows. We start with the case of sl2 in Section 2. The Sections 3 and 4 are devoted to the case of slN+1. We discuss h-selfdual spaces of polynomials in Section 5 and then deal with the cases of BN and CN in Sections 6 and 7. Appendix A describes in detail the simplest example of an sl3 and \C1 " populations. Appendix B collects identities involving discrete Wronskians. We thank V. Tarasov for interesting discussions.
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The case of sl2
2.1 The Bethe equation Our parameters are distinct complex numbers z1 ; : : : ; zn , positive integers ¤1 ; : : : ; ¤n and a complex non-zero number h. We call zi the rami¯cation points, ¤i the weights and h the step. For given l 2 Z¸0 , the sl2 Bethe equation is the following system of algebraic equations for the complex variables = (t1 ; : : : ; tl ) n Y tj ¡ s=1
zs + ¤ s h Y t j ¡ tj ¡ zs tj ¡ k6=j
tk ¡ h tk + h
= 1;
(1)
where j = 1; : : : ; l. If all ¤i are equal to 1 then the system (1) is the Bethe equation for the inhomogeneous XXX model, [BIK]. Consider the product of Euler gamma functions ©=
l Y n Y ¡((tj ¡ zs + ¤s h)=p) ¡((tj ¡ zs )=p) j=1 s=1
Y ¡((tj ¡ ¡((tj ¡
j;k; j
tk ¡ h)=p) : tk + h)=p)
The function © is called the master function associated to the Yangian of sl2 and ; ¤ . It is used in the construction of integral solutions to the rational quantum KnizhnikZamolodchikov equation, see [TV1], [MV2]. The function © is the sl2 di®erence counterpart of the master function (2.1) in [MV1]. Equation (1) can be rewritten in the form ©(t1 ; : : : ; tj + p; : : : ; tl ) lim = 1; j = 1; : : : ; l: p!0 ©(t1 ; : : : ; tj ; : : : ; tl ) If = (t1 ; : : : ; tl ) is a solution of (1) then any permutation of the coordinates is also a solution. An Sl orbit of solutions of equation (1) such that ti 6= tk , ti 6= tk + h and ti 6= zs ¡ jh for all i; s; k, i 6= k, and j = 1; : : : ; ¤s is called an h-critical point. Not all solutions of the Bethe equation are h-critical points. For instance, if n = 0 and l is even, then t1 = ¢ ¢ ¢ = tl = 0 is a solution of the Bethe equation which is not an h-critical point.
2.2 Second order di®erence equations Q For each h-critical point we write the monic polynomial u(x) = li=1 (x ¡ ti ) and say that a polynomial cu(x) for any non-zero complex number c, represents . In this section we rewrite the Bethe equation in terms of a di®erence equation for u(x). For two functions u(x); v(x), de¯ne the discrete Wronskian W (u; v)(x) by the formula W (u; v)(x) = u(x + h)v(x) ¡
u(x)v(x + h):
Let A(x); B(x); C (x) be given functions. Consider the di®erence equation A(x) u(x + h) + B(x)u(x) + C (x)u(x ¡
h) = 0
(2)
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with respect to the unknown function u(x). Note that the complex vector space of polynomial solutions of equation (2) has dimension at most 2. The following lemma is straightforward. Lemma 2.1. If u(x) and v(x) are two solutions of equation (2), then the discrete Wronskian W (u; v)(x) satis¯es the ¯rst order di®erence equation W (u; v)(x + h) =
C (x + h) W (u; v)(x): A(x + h)
(3)
Given rami¯cation points zi and weights ¤i we now ¯x the choice of the functions A(x) and C (x) as follows A(x) =
n Y
(x ¡
zs );
C (x) =
s=1
n Y
(x ¡
zs + ¤s h):
(4)
s=1
Then the polynomial solutions of equation (3) are constant multiples of the function T (x) =
n Y ¤s Y
(x ¡
zs + ih):
(5)
s=1 i=1
We say that a polynomial u(x) is generic with respect to = (z1 ; : : : ; zn ), ¤ = (¤1 ; : : : ; ¤n ) if the polynomial u(x) has no multiple roots and no common roots with polynomials u(x + h), T (x). Lemma 2.2. Assume that B(x) is a polynomial and that a solution u(x) of (2) is a polynomial generic with respect to ; ¤ . Then u(x) represents an h-critical point. Conversely, if u(x) represents an h-critical point then the polynomial u(x) satis¯es equation (2), where B(x) is a polynomial given by à n ! n Y Y B(x) = ¡ (x ¡ zs )u(x + h) + (x ¡ zs + ¤s h)u(x ¡ h) =u(x): (6) s=1
s=1
Proof 2.3. The lemma follows from the fact that equation (1) with respect to tj is obtained by the substituting tj in equation (2).
2.3 The sl2 reproduction By Lemma 2.2, each h-critical point de¯nes a di®erence equation together with a polynomial solution. This di®erence equation is a linear di®erence equation of the second order with polynomial coe±cients. In this section we show that the space of solutions of that di®erence equation contains a complex two-dimensional space of polynomials all of which (except for scalar multiples of ¯nitely many) represent h-critical points.
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Lemma 2.4. Let the roots of the polynomial u(x) be distinct and satisfy the Bethe equation (1). Then the corresponding di®erence equation (2) has two linearly independent polynomial solutions. Q Proof 2.5. Let u(x) = ni=1 (x ¡ ti ). We look for the second polynomial solution v(x) in the form v(x) = c(x)u(x). After multiplication by a non-zero number, we have the equation v(x + h)u(x) ¡
v(x)u(x + h) =
n Y ¤s Y
(x ¡
zs + ih):
(7)
s=1 i=1
Therefore c(x) is a solution of the equation Qn Q¤s (x ¡ zs + ih) c(x + h) ¡ c(x) = s=1 i=1 : u(x)u(x + h)
(8)
The right hand side of this equation is of the form f (x) +
l µ X j=1
aj ¡ x ¡ tj + h
bj x¡
tj
¶
;
where f (x) is a polynomial and aj ; bj are some numbers. Moreover, equation (1) implies that aj = bj for all j. The equation c(x + h) ¡ c(x) = f (x) has a polynomial solution. Denote it c~(x). Then P c(x) = c~(x) + lj=1 aj =(x ¡ tj ) is a solution to (8). Then the function v(x) = c(x)u(x) is a polynomial solution of equation (2). Thus, starting from a solution of the Bethe equation (1), we obtain a di®erence equation of order two which has a two-dimensional space of polynomial solutions. We call the projectivisation of this space the population of h-critical points related to and denote it P ( ). Generic polynomials with respect to ; ¤ form a Zariski open subset of the population. Roots of a generic polynomial form an h-critical point by Lemma 2.2. Let @ : f (x) ! f (x + h) be the shift operator acting on functions of one variable. The di®erence operator @ 2 + B(x)=A(x)@ + C (x)=A(x) obtained from an h-critical point ¹ 2 P ( ) does not depend on the choice of ¹. We call this operator the fundamental operator associated to the population P ( ) and denote it by D P ( ) .
2.4 Fertile polynomials We give another equivalent condition for a polynomial to de¯ne an h-critical point. Q Lemma 2.6. Let a polynomial u(x) = li=1 (x¡ ti ) be generic with respect to ; ¤ . Then the roots of u(x) form a solution of equation (1) if and only if there exists a polynomial solution v(x) to equation (7).
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Proof 2.7. If the roots of u(x) form a solution of equation (5), then equation (2) with the polynomial B(x) given by (6) has two linearly independent polynomial solutions by Lemma 2.4, and hence equation (7) has a polynomial solution. Now assume that v(x) is a polynomial solution to (7). Then the rational function c(x) = v(x)=u(x) is a solution of equation (8). Then Qn Q¤s Qn Q¤s ¡ (x z + ih) (x ¡ zs + ih) s Resx=tj s=1 i=1 + Resx=tj ¡h s=1 i=1 =0 u(x) u(x + h) u(x)u(x + h) for j = 1; : : : ; l. This system of equations is equivalent to equation (1). A polynomial u(x) is called fertile if there exist a polynomial v(x) such that the discrete Wronskian of u and v is T (x), W (u; v)(x) = T (x). All polynomials in a population of h-critical points are fertile. Moreover, the h-critical points correspond to fertile generic polynomials. We get the following immediate corollary. Lemma 2.8. If equation (2) has two polynomial solutions of degree l and l 0 , l 6= l0 , then l + l 0 ¡ 1 = ¤1 + ¢ ¢ ¢ + ¤n .
3
The case of slN +1
Let ¬ 1 ; : : : ; ¬ N be the simple roots of slN+1 . We have (¬ i ; ¬ i ) = 2, (¬ i ; ¬ all other scalar products equal to zero.
i§1 )
= ¡ 1 with
3.1 De¯nition of h-critical points In the case of slN+1 we ¯x the following parameters: rami¯cation points = (z1 ; : : : ; zn ) 2 Cn ; non-zero dominant integral slN+1 weights ¤ = (¤1 ; : : : ; ¤n ), relative shifts (i) = (i) (i) (b1 ; : : : ; bn ) 2 Cn , i = 1; : : : ; N , and the step h 2 C, h 6= 0. We call the set of parameters ; ¤ ; (i) , i = 1; : : : ; N , the initial data. We denote (i) ¤s = (¤s ; ¬ i ) 2 Z¸0 . Let = (l1 ; : : : ; lN ) 2 ZN ¸0 . The slN +1 -weight ¤1 =
n X
N X
¤s ¡
s=1
li ¬ i ;
(9)
i=1
is called the weight at in¯nity. The following system of algebraic equations for variables j = 1; : : : ; li , (i) n Y tj ¡ s=1
li¡ (i) (i) 1 (i) zs + bs h + ¤s h Y tj ¡
(i)
tj ¡
(i)
(i)
zs + bs h
Y t(i) j £ (i) k6=j tj
¡
¡
(i) tk (i) tk
(i¡1)
tk
¡
k=1 li+1
hY
+h
tj ¡
(i¡1)
tk
(i) tj ¡
(i) k=1 tj
¡
+h
(i)
= (tj ), i = 1; : : : ; N ,
£
(i+1)
tk
(i+1)
tk
¡
h
= 1;
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245
is called the slN+1 Bethe equation associated with the initial data and the weight at in¯nity. (j) (j) Note that by a shift of variables ~ti = ti + jh=2 the system can be written in a slightly more symmetrical way. Note that in the quasiclassical limit h ! 0, system (10) becomes system (2.2) of [MV1] specialized to the case of slN +1 . The product of symmetric groups S = Sl1 £ ¢ ¢ ¢ £ SlN acts on the set of solutions of (10) permuting the coordinates with the same upper index. An S orbit of solutions of the Bethe equation such that (i) 6 t(i) tj = k ;
(i) 6 t(i) tj = k + h;
(i) 6 t(i+1) tj = ; m
(i) 6 zs ¡ tj =
b(i) s ¡
(i) 6 t(i+1) tj = + h; m
rh;
(i)
6 j and r = 1; : : : ; ¤s is called an h-critical point associated with for all j; i; k; m; s, k = the initial data and the weight at in¯nity. Let L¤ be the irreducible slN +1 module with highest weight ¤. Let the initial data Pj (j) (i) satisfy bs = ¡ i=1 ¤s . We have the following conjecture.
Conjecture 3.1. If ¤1 is integral dominant, then for generic the number of h-critical points associated with the initial data and the weight at in¯nity equals to the multiplicity of L¤1 in L¤1 « ¢ ¢ ¢ « L¤n .
3.2 Di®erence equations of the second order In this section, given an h-critical point we obtain N di®erence operators of order two. The ith operator is the fundamental operator with respect to the sl2 2 slN+1 in the direction ¬ i . Let be an h-critical point. For i = 1; : : : ; N , introduce polynomials
yi (x) =
li Y
(x ¡
(i)
tj ):
(10)
j=1
We also set y0 (x) = 1, yN+1(x) = 1. The N -tuple uniquely determines the h-critical point and we say that represents . We consider the tuple up to multiplication of each coordinate by a non-zero number, since we are interested only in the roots of polynomials y1 ; : : : ; yN . Thus the tuple de¯nes a point in the direct product ( C[x])N of N copies of the projective space associated with the vector space of polynomials of x. In this paper we view all N -tuples of polynomials as elements of ( C[x])N .
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Introduce the polynomials (i)
T i (x) =
¤s n Y Y
(x ¡
zs + b(i) s h + jh);
(11)
s=1 j=1
A i (x) =
n Y
(x ¡
li¡ 1
zs +
b(i) s h)
s=1
Ci (x) =
n Y
Y
(x ¡
(i¡1) tk )
k=1
(x ¡
zs +
s=1
b(i) s h
+
¤(i) s h)
li+1 Y
(x ¡
(i+1)
tk
¡
+ h)
li+1 Y
h);
k=1 li¡ 1
Y
(x ¡
(i¡1) tk
k=1
(x ¡
(i+1)
tk
):
k=1
We say that a tuple of polynomials 2 ( C[x])N is generic with respect to the initial data if for all i the polynomial yi(x) has no multiple roots and no common roots with polynomials yi (x + h), yi¡1 (x + h), yi+1(x), T i (x). For i = 1; : : : ; N , consider di®erence equations of the second order of the form Ai (x)u(x + h) + Bi (x)u(x) + Ci (x)u(x ¡
h) = 0;
(12)
where Bi (x) are any functions. Lemma 3.2. Assume that for all i, yi is a polynomial solution of i-th equation in (12) for some polynomial Bi (x). If the tuple = (y1 ; : : : ; yN ) is generic with respect to the initial data then represents an h-critical point. Conversely, if a tuple of polynomials = (y1 ; : : : ; yN ) represents an h-critical point, then for all i the polynomial yi satis¯es the i-th equation in (12), where Bi (x) is a polynomial given by Bi (x) = ¡ (Ai (x)yi (x + h) + Ci (x)yi (x ¡ h))=yi(x). Proof 3.3. The lemma follows directly from the sl2 counterpart, Lemma 2.2.
3.3 Fertile tuples In this section we discuss yet another criteria for a tuple of polynomials to represent an h-critical point. We say that a tuple of polynomials 2 ( C[x])N is fertile with respect to the initial data if for every i there exists a polynomial y~i such that the discrete Wronskian is W (yi ; y~i)(x) = T i (x)yi¡1 (x + h)yi+1 (x): Lemma 3.4. A generic tuple
(13)
is fertile if and only if it represents an h-critical point.
Proof 3.5. The lemma follows from the sl2 considerations, see Lemma 2.6. Let be fertile and let yi ; y~i satisfy (13). Then polynomials of the form c1 yi + c2 y~i with c1 ; c2 2 C span the two dimensional space of polynomial solutions of equation (12). The tuples (y1 ; : : : ; c1 yi + c2 y~i ; : : : ; yN ) 2 (C[x])N are called immediate descendents of
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247
in the i-th direction. Note that if is generic, then all but ¯nitely many immediate descendents are generic. Let Cd [x] be the space of polynomials of degree not greater than d. The set of fertile tuples is closed in ( Cd [x])N : Lemma 3.6. Assume that a sequence of fertile tuples of polynomials k , k = 1; 2; : : : , has a limit 1 in ( Cd [x])N as k tends to in¯nity. Then the limiting tuple 1 is fertile. Assume in addition that for some i, all immediate descendents of all k in the i(i) th direction are in ( Cd [x])N . If 1 is an immediate descendant of 1 in the i-th (i) (i) direction, then 1 2 ( Cd [x])N and there exist immediate descendants k of k in the (i) (i) i-th direction such that 1 is the limit of k . Proof 3.7. Let us prove 1 is fertile in direction i. For each k we have a di®erence equation (12), which we denote Ek , and a plane of polynomial solutions Pk . By the assumptions of the lemma there is a limiting equation E1 as k ! 1. The space P k de¯nes a point in the projective Grassmanian variety of planes in Cd [x]N . Let P1 be a limiting plane, then all polynomials in P 1 are solutions of equation E1 and the lemma is proved.
3.4 Reproduction procedure Given an h-critical point, we describe a procedure of obtaining a family of h-critical points. Theorem 3.8. Let represent an h-critical point and let yi ; y~i satisfy (13). Assume that (i) = (y1 ; : : : ; y~i ; : : : ; yN ) is generic. Then (i) represents an h-critical point. (i) Proof 3.9. Denote t¹j the roots of y¹i = c1 yi + c2 y~i . Let z be a root of Ti (x)yi¡1 (x + h)yi+1 (x). Then we have yi(z + h) y¹i (z + h) = : yi(z) y¹i (z) (i¡1)
In particular we choose z = tk Y j
Y j
(i+1)
¡
(i¡1)
h and z = tk (i)
¡
tk
(i¡1)
tk
tj (i) tj ¡
¡
(i+1) ¡ tk (i+1) tk
¡
(i) tj
h +h
(i)
tj
=
Y j
=
and obtain (i¡1)
(i¡1)
¹t(i) j ¡ ¡
tk
Y t(i+1) ¡ k
(i+1)
j
(i) t¹j
¡
tk
tk
h
;
¡
¹t(i) j +h ¹t(i) j
for all k and i. In addition Lemma 2.2 implies that for all k, Y t(i) k ¡
(i) j; j6=k tk
¡
(i) tj ¡ (i)
h
tj + h
=
Y t¹(i) k ¡
¹(i) j; j6=k tk
¡
(i) t¹j ¡
h
¹t(i) j +h
:
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That proves the theorem. Thus, starting with a tuple , representing an h-critical point and an index i 2 f1; : : : ; N g, we construct a closed subvariety of immediate descendants (i) in ( C[x])N which is isomorphic to 1 . All but ¯nitely many points in this subvariety correspond to N -tuples of polynomials which represent h-critical points. All constructed N -tuples of polynomials are fertile. We call this construction the simple reproduction procedure in the i-th direction. Now we can repeat the simple reproduction procedure in some other j-th direction applied to all points obtained in the previous step and obtain more critical points. We continue the process until no simple reproduction in any direction applied to any polynomial obtained in the previous steps produces a new polynomial. The result is called the population originated at the h-critical point and is denoted P ( ). More formally, the population of is the set of all N -tuples of polynomials N ¹ 2 ( C[x]) such that there exist N -tuples 1 = , 2 ; : : : ; k = ¹, such that i is an immediate descendant of i¡1 for all i = 2; : : : ; k. Obviously, if two populations intersect, then they coincide.
4
Fundamental space of an slN +1 population
Given an slN+1 population of critical points we construct a space of polynomials called the fundamental space. Elements of the population are in natural correspondence with full °ags in the fundamental space. Then we show that the problem of counting h-critical points is equivalent to a problem of Schubert calculus.
4.1 Degrees of polynomials in a population and the slN +1 Weyl group Let be the Cartan subalgebra of slN +1 and let ( ; ) be the standard scalar product on ¤ . Denote » 2 ¤ the half sum of positive roots. 2 End( ¤ ) is generated by re°ections si , i = 1; : : : ; N , The Weyl group si (¶ ) = ¶ ¡
¶ 2
(¶ ; ¬ i )¬ i ;
¤
:
We use the notation w ¢ ¶ = w(¶ + » ) ¡
» ;
w2
; ¶ 2
¤
;
(14)
for the shifted action of the Weyl group. Let some initial data be given. Let = (y1 ; : : : ; yN ) 2 ( C[x])N , and let li be the degree of the polynomial yi for i = 1; : : : N . Let ¤1 be the weight at in¯nity de¯ned by (9). Assume the tuple is fertile. Let the tuple (i) = (y1 ; : : : ; y~i ; : : : ; yr ) be an immediate (i) descendant of in the direction i and let ¤1 be the slN+1 weight at in¯nity of (i) . The following lemma follows from Lemma 2.8.
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249
Lemma 4.1. If the degree of y~i is not equal to the degree of yi , then ¤(i) 1 = si ¢ ¤ 1 ; where si ¢ is the shifted action of the i-th generating re°ection of the Weyl group. Therefore we obtain the following Proposition 4.2. Let a tuple 0 represent an h-critical point associated with a given initial data and weight at in¯nity ¤1 . Let P be the population of h-critical points originated at 0 . Then ° For any tuple 2 P , there is an element w of the slN +1 Weyl group , such that the weight at in¯nity of is w ¢ ¤1 . ° For any element w 2 , there is a tuple 2 P whose weight at in¯nity is w ¢ ¤1 . Proposition 4.2 gives su±cient conditions for the absence of h-critical points. Corollary 4.3. There is no h-critical point in either of the two cases Pn (1) if there is an element w of the Weyl group such that w ¢ ¤1 does not s=1 ¤s ¡ belong to the cone Z¸0 ¬ 1 © ¢ ¢ ¢ © Z¸0 ¬ N ; (2) if ¤1 belongs to one of the re°ection hyperplanes of the shifted action of the Weyl group. The next corollary says that under certain conditions on weights there is exactly one population of h-critical points. The tuple (1; : : : ; 1) 2 ( C[x])N is the unique N -tuple of non-zero polynomials of P degree 0. The weight at in¯nity of (1; : : : ; 1) is ¤1;(1;:::;1) = ns=1 ¤s . Let P(1;:::;1) be the population associated to the initial data and originated at (1; : : : ; 1). Corollary 4.4. Let some w 2 . Then
be an h-critical point such that ¤1 has the form w ¢ ¤1;(1;:::;1) for belongs to the population P (1;:::;1) .
4.2 The di®erence operator of a population In this section we describe a linear di®erence operator of order N + 1 related to a population of h-critical points. Let an initial data ; ¤ ; (i) be given. Let the polynomials Ti , i = 1; : : : ; N , be de¯ned by formula (11). Let = (y1 ; : : : ; yN ) be an N -tuple of non-zero polynomials. Set y0 = yN+1 = 1.
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De¯ne a linear di®erence operator of order N + 1 with meromorphic coe±cients N yN (x) Y Ts (x + (N ¡ s + 1)h) )£ yN (x + h) s=1 T s (x + (N ¡ s)h)
D( ) = (@ ¡
N ¡1 yN (x + h) yN¡1 (x) Y Ts (x + (N ¡ s)h) ) £ ¢¢¢£ yN (x) yN¡1 (x + h) s=1 T s (x + (N ¡ 1 ¡ s)h)
(@ ¡
y2 (x + h) y1 (x) T 1 (x + h) y1 (x + h) ) (@ ¡ )= y2 (x) y1 (x + h) T 1 (x) y1 (x) Ã ! 0!N N ¡i Y yN+1¡i (x + h) yN¡i(x) Y Ts (x + (N ¡ i ¡ s + 1)h) = @¡ : (15) yN+1¡i (x) yN ¡i (x + h) s=1 Ts (x + (N ¡ i ¡ s)h) i (@ ¡
Notice that the ¯rst coordinate y1 of the N -tuple belongs to the kernel of the operator D( ). Theorem 4.5. Let P be a population of h-critical points originated at some tuple Then the operator D( ) does not depend on the choice of in P .
0
.
Proof 4.6. We have to prove that if ; ~ 2 P then D( ) = D(~). It is enough to show the case when ~ is an immediate descendant of in some direction i and both ; ~ represent critical points. Therefore, we assume that yj = y~j , for all j; j 6= i, and W (yi; y~i )(x) = T i (x) yi¡1 (x + h)yi+1(x). In this case, all factors of D( ) and D(~) are the same except the two factors which involve yi or y~i . So it is left to show that for any function u(x) we have (@ ¡
i yi+1(x + h) yi (x) Y T s (x + (i ¡ s + 1)h) )£ yi+1(x) yi (x + h) s=1 T s (x + (i ¡ s)h) i¡1
(@ ¡
yi (x + h) yi¡1 (x) Y T s (x + (i ¡ s)h) )u(x) = yi (x) yi¡1 (x + h) s=1 Ts (x + (i ¡ s ¡ 1)h)
(@ ¡ (@ ¡
i yi+1(x + h) y~i (x) Y T s (x + (i ¡ s + 1)h) )£ yi+1(x) y~i (x + h) s=1 T s (x + (i ¡ s)h)
i¡1 y~i(x + h) yi¡1 (x) Y Ts (x + (i ¡ s)h) ) u(x) : y~i (x) yi¡1 (x + h) s=1 T s (x + (i ¡ s ¡ 1)h)
After the change of variables v(x) = u(x)
Qi¡1
+ (i ¡ s ¡ yi¡1 (x)
s=1 T s (x
1)h)
;
we have to prove that (@ ¡ (@ ¡
yi+1 (x + h) yi (x) yi¡1 (x + 2h) T i (x + h) yi (x + h) )(@ ¡ ) v(x) = yi+1 (x) yi (x + h) yi¡1 (x + h) T i (x) yi (x) yi+1(x + h) y~i (x) yi¡1 (x + 2h) Ti (x + h) y~i (x + h) )(@ ¡ ) v(x) : yi+1(x) y~i (x + h) yi¡1 (x + h) Ti (x) y~i (x)
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251
This identity is easily checked directly using the equation connecting yi and y~i . The di®erence operator (15) is called the fundamental operator associated to the population P and is denoted DP . Corollary 4.7. Let be a member of a population P . Then the ¯rst coordinate y1 of is in the kernel of the operator D P . Proposition 4.8. The space of polynomial solutions to the di®erence equation DP u = 0
(16)
has dimension N + 1. Proof 4.9. Let be a member of the population P . Assume that represents an h-critical point. We construct polynomials u1 ; : : : ; uN +1 , satisfying equation (16). Set u1 = y1 . The polynomial u1 is a solution of (16). Let u2 be a polynomial such that W (u1 ; u2 )(x) = T1 (x)y2 (x). The polynomial u2 is a solution of (16). Let y~2 be a polynomial such that W (y2 ; y~2 )(x) = T 2 (x)y1 (x + h)y3 (x). Such a polynomial can be chosen so that (y1 ; y~2 ; : : : ; yN ) is generic, and therefore represents an h-critical point. Choose a polynomial u3 to satisfy equation W (u3 ; y1 )(x) = T 1 (x)~ y2 (x). The polynomial u3 is a solution of (16). In general, to construct a polynomial ui+1 we ¯nd ~ i = (y1 ; : : : ; y~i ; : : : ; yN ), such that ~ i is generic and W (yi ; y~i )(x) = Ti (x)yi¡1 (x + h)yi+1 (x) and then repeat the construction for ui using ~ i instead of . Let V be the complex vector space spanned by polynomials u1 ; : : : ; uN+1. We show that the space V has dimension N + 1. Let W (g 1 ; : : : ; g s ) be the discrete Wronskian of functions g 1 ; : : : ; gs de¯ned by W (g1 ; : : : ; gs )(x) = det(gi (x + (j ¡
1)h))si;j=1
For s = 0, we de¯ne the corresponding discrete Wronskian to be 1. The proposition is proved with the following lemma: Lemma 4.10. For i = 1; : : : ; N + 1, we have W (u1 ; : : : ; ui)(x) = yi (x)
i¡1 Y
j=1
T 1 (x + (j ¡
1)h)
i¡2 Y
T 2 (x + (j ¡
1)h) : : : Ti¡1 (x):
j=1
Proof 4.11. The lemma is proved in the same way as Lemma 5.5 in [MV1]; by making use of the Wronskian identities in Lemmas 9.4 and 9.7. A linear di®erence equation of order N + 1 cannot have more than N + 1 polynomial solutions linearly independent over C. The complex (N + 1)-dimensional vector space of
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polynomial solutions of equation (16) is called the fundamental space of the population P and is denoted VP .
4.3 Frames We describe an h-analogue of rami¯cation properties of a space of polynomials. A space of polynomials V is called a space without base points if for any z 2 C there exists v 2 V such that v(z) 6= 0. Let V be an (N + 1)-dimensional vector space of polynomials without base points. Let U i (x) be the monic polynomial which is the greatest common divisor of the family of polynomials fW (u1 ; : : : ; ui) j u1 ; : : : ; ui 2 V g. Lemma 4.12. There exist a unique sequence of monic polynomials T1 (x); : : : ; T N (x) such that i¡1 i¡2 Y Y U i (x) = T1 (x + (j ¡ 1)h) T2 (x + (j ¡ 1)h) : : : T i¡1 (x); j=1
j=1
for i = 1; : : : ; N + 1.
We call the sequence of monic polynomials T1 (x); : : : ; T N (x) the frame of V . Proof 4.13. We construct the polynomials T i by induction on i. For i = 0, we have U 1 = 1. For i = 1 we just set T1 = U 2 . Suppose the lemma is proved for all i = 1; : : : ; i0 ¡ 1. Then we set iY 0¡2 iY 0 ¡i S(x) = T i (x + (j ¡ 1)h); T i0¡1 (x) = U i0 (x)=S(x): i=1 j=1
We only have to show that T i0¡1 is a polynomial. In other words, we have to show that a Wronskian of any i0 dimensional subspace in V is divisible by S(x). Consider the Grassmanian Gr(i0 ¡ 2; V ) of (i0 ¡ 2)-dimensional spaces in V . For any z 2 C, the set of points in Gr(i0 ¡ 2; V ), such that the corresponding Wronskian divided by U i0 ¡2 does not vanish at z, is a Zariski open algebraic set. Therefore, we have a Zariski open set of points in Gr(i0 ¡ 2; V ) such that the corresponding Wronskian divided by U i0¡2 does not vanish at roots of S(x ¡ h). We call such subspaces acceptable. Therefore we have a Zariski open set of points in Gr(i0 ; V ) such that the corresponding i0 dimensional space contains an acceptable i0 ¡ 2 dimensional subspace. Let u1 ; : : : ; ui0 2 V be such that u1 ; : : : ; ui0¡2 span an acceptable space. It is enough to show that W (u1 ; : : : ; ui0 ) is divisible by S(x). Then using Wronskian identities in Lemmas 9.4 and 9.7, we have for suitable polynomials f1 ; f2 ; g: W (W (u1 ; : : : ; ui0¡1 ); W (u1 ; : : : ; ui0¡2 ; ui0 )) W (U i0 ¡1 f1 ; U i0¡1 f2 ) W (u1 ; : : : ; ui0 ) = = W (u1 ; : : : ; ui0¡2 )(x + h) U i0¡2 (x + h)g(x + h) U i ¡1 (x)U i0 ¡1 (x + h) W (f1 ; f2 ) W (f1 ; f2 ) = 0 = S(x) : U i0¡2 (x + h) g(x + h) g(x + h)
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253
Since the space spanned by u1 ; : : : ; ui0 2 V is accepatble, the polynomial g(x + h) = W (u1 ; : : : ; ui0¡2 )(x + h)=U i0¡2 (x + h) and S(x) are relatively prime. Therefore the Wronskian W (u1 ; : : : ; ui0 ) is divisible by S(x).
4.4 Frames of the fundamental spaces Let = (y1 ; : : : ; yN ) represent an h-critical point associated to the initial data ; ¤ ; Let P be a population of h-critical points originated at .
(i)
.
Proposition 4.14. The fundamental space V P of the population P has no base points. The polynomials T 1 (x); : : : ; TN (x) given by (11) form a frame of VP . Proof 4.15. We construct polynomials u1 ; : : : ; uN +1 2 V P as in the proof of Proposition 4.8. We have W (u1 ; : : : ; ui ) = yi (x)
i¡j i¡1 Y Y
T j (x + (r ¡
1)h);
T j (x + (r ¡
1)h);
j=1 r=1
W (u1 ; : : : ; ui¡1 ; ui+1 ) = y~i (x)
i¡j i¡1 Y Y
j=1 r=1
where (y1 ; : : : ; y~i; : : : ; yN ) is a descendant of in the i direction. Qi¡1 Qi¡j In particular W (u1 ; : : : ; ui ) is divisible by j=1 r=1 Tj (x + (r ¡ 1)h). Therefore Wronskians of all i dimensional planes are divisible by this polynomial. Moreover, we have W (yi; y~i ) = T i (x)yi¡1 (x + h)yi+1(x). Since yi and Ti (x)yi¡1 (x + h)yi+1 (x) have no common roots, the polynomials yi (x) and y~i (x) have no common roots. It follows that the greatest common divisor of i-dimensional Wronskians is Qi¡1 Qi¡j 1)h). j=1 r=1 Tj (x + (r ¡ The absence of base points for V P follows from the case of i = 1. The converse statement is also true. Proposition 4.16. Let V be a space of polynomials of dimension N + 1 without base points and with the frame T i . Let ; ¤ ; (i) be the initial data related to polynomials T i by (11). Then V is the fundamental space of a population of h-critical points associated to the initial data ; ¤ ; (i) . Proof 4.17. We postpone the proof until after Section 4.6, where the generating morphism is described. Then the proof is similar to the proof of Proposition 5.17 in [MV1] combined with Lemma 5.20 in [MV1]. From Propositions 4.14 and 4.16 we obtain the following theorem. Theorem 4.18. There is a bijective correspondence between populations of h-critical
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points associated to a given initial data and the spaces of polynomials with framing Ti , where Ti are related to the initial data by (11).
4.5 Schubert calculus The problem of counting the number of populations for a special choice of relative shifts (i) can be approached via Schubert calculus. Let be a complex vector space of dimension d + 1 and = f0 » F1 » F2 » ¢ ¢ ¢ » Fd+1 = g;
dim F i = i;
a full °ag in . Let Gr(N +1; ) be the Grassmanian of all (N +1)-dimensional subspaces in . Let = (a1 ; : : : ; aN +1 ), d ¡ N ¶ a1 ¶ a2 ¶ ¢ ¢ ¢ ¶ aN +1 ¶ 0, be a non-increasing sequence of non-negative integers. De¯ne the Schubert cell G 0 ( ) associated to the °ag and sequence to be the set fV 2 Gr(N + 1; ) j dim(V \ Fd¡N+i¡ai ) = i; dim(V \ Fd¡N+i¡ai¡1 ) = i ¡
1; for i = 1; : : : ; N + 1g:
The closure G ( ) of the Schubert cell is called the Schubert cycle. For a ¯xed °ag F , the Schubert cells form a cell decomposition of the Grassmanian. The codimension of G0 ( (z)) in the Grassmanian is j j = a1 + ¢ ¢ ¢ + aN+1 . Let = Cd [x] be the space of polynomials of degree not greater than d, dim = d +1. For any z 2 C [ 1, de¯ne a full °ag in Cd [x], (z) = f0 » F1 (z) » F 2 (z) » ¢ ¢ ¢ » F d+1 (z) = g : For z 2 C and any i, we set F i (z) to be the subspace of all polynomials divisible by Qd+1¡i j=1 (x ¡ z ¡ (j ¡ 1)h). For any i, we set Fi (1) to be the subspace of all polynomials of degree less than i. Let V 2 Gr(N + 1; Cd [x]). For any z 2 C[ 1, let (z) be such a unique sequence that V belongs to the cell G 0 (z) ( (z)). We say that a point z 2 C [ 1 is an h-rami¯cation point for V , if (z) 6= (0; : : : ; 0). We call (z) the rami¯cation condition of V at z. If u1 ; : : : ; uN +1 is a basis of V then the rami¯cation points are zeroes of the discrete Wronskian W (x) = W (u1 ; : : : ; uN+1)(x). Indeed, given z 2 C, without loss of generality we can assume that ui 2 d+2¡i¡ai(z) (z), and therefore the matrix fui(x + (j ¡ 1))gN+1 i;j=1 is upper triangular. Moreover, this matrix has a zero diagonal element if and only if jaj > 0. Fix h-rami¯cation conditions at z1 ; : : : ; zn ; 1 so that n X
j (zs )j + j (1)j = dim Gr(N + 1; Cd [x]) :
s=1
Counting Problem. Compute the number of spaces of polynomials of dimension N + 1 with these rami¯cation properties.
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255
In other words we ask to count the number of points in the intersection of Schubert cycles. The intersection index of the Schubert cycles is well known. Namely, for a non-increasing sequence = (a1 ; : : : ; aN +1 ), a1 ¶ a2 ¶ : : : ; ¶ aN+1 ¶ 0, ~ the ¯nite dimensional irreducible glN +1 -module with of non-negative integers, denote L ~ . highest weight . Let L be the slN +1 module obtained by restriction of L Theorem 4.19. ([F]) The intersection index of Schubert cycles G (zs)(F (zs )), s = 1; : : : ; n, and G (1) (F (1)) equals the multiplicity of the trivial slN +1 -module in the tensor product of slN +1 -modules L
(z1 )
« ¢¢¢« L
(zn)
«L
(1)
:
Conjecturally, for almost all z1 ; : : : zn the number of spaces V with such rami¯cation conditions is equal to the above multiplicity. Fix an initial data ; ¤ ; (i) . Until the end of this section we assume that zs ¡ zr 62 hZ for all s 6= r and
b(j) s
=¡
j X
¤(i) s :
(17)
i=1
Fix an slN weight and write it in the form w ¢ ¤1 , where ¤1 is dominant integral and w is an element of Weyl group. Let be an h-critical point associated to the initial data and the weight at in¯nity w ¢ ¤1 . Let P be the population of h-critical points originated at and let V P be the corresponding fundamental space. Let d be large enough, so that VP » Cd [x]. Theorem 4.20. The points z1 ; : : : ; zn and 1 are rami¯cation points of V P . The ramPN +1¡i (j) i¯cation condition at zs is ai (zs ) = ¤s . The rami¯cation condition at 1 is j=1 Pi¡1 (j) ai (1) = d ¡ N ¡ l1 ¡ and ¤1 via (9). j=1 ¤1 , where l1 is de¯ned from ¤ Proof 4.21. Follows from Lemma 4.10, cf. proof of Lemmas 5.8, 5.10 in [MV1].
For an integral dominant slN +1 weight ¤, we denote L¤ the irreducible slN +1 module with highest weight ¤. Corollary 4.22. . The number of (discretely lying) populations associated to initial data i ; ¤ ; (i) such that (17) holds, is not greater than the multiplicity of L¤1 in the tensor product L¤1 « ¢ ¢ ¢ « L¤n . Proof 4.23. The corollary holds since the number of isolated points of the intersection of the corresponding Schubert cycles is not greater than the intersection index of the cycles.
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4.6 Generating morphism We identify a population with the variety of full °ags in the fundamental space. Let V = V P be the fundamental space of a population P . By Proposition 4.14, the polynomials Ti de¯ned in (11) form a frame of V P . Let F L(V ) be the variety of all full °ags = f0 » F1 » F 2 » ¢ ¢ ¢ » F N+1 = V g in V . For any 2 F L(V ) de¯ne an N -tuple of polynomials = (y1 ; : : : ; yN ) as follows. Let u1 ; : : : ; uN +1 be a basis in V such that for any i the polynomials u1 ; : : : ; ui form a basis in F i . We say that this basis is adjusted to the °ag and the °ag is generated by the basis u1 ; : : : ; uN +1. De¯ne the polynomials yi (x) = Qi¡1
j=1 T1 (x + (j ¡
The correspondence
7!
W (u1 ; : : : ; ui )(x) Q 1)h) i¡2 j=1 T 2 (x + (j ¡
1)h) : : : T i¡1 (x)
:
(18)
gives a morphism
: F L(V ) ! ( C[x])N ;
(19)
called the generating morphism of V . Theorem 4.24. Let P be a population of h-critical points with the fundamental space V . Then the generating morphism de¯nes an isomorphism of F L(V ) and the population P » ( C[x])N . Proof 4.25. Proof is the same as the proof of the ¯rst part of Theorem 5.12 in [MV1]. Remark. Recall that V is the (N + 1)-dimensional complex vector space of polynomials which is contained in the kernel of the fundamental operator DP , and D P is a linear difference operator of order N +1. Therefore the full °ags in V also label the decompositions of D P to N + 1 linear factors of the form (@ ¡ f (x)), where f (x) is a rational function of x. Thus, h-critical points are in bijective correspondence with such factorizations of the fundamental operator D P . 2 F L(V ) de¯ne a permutation w( ) in the Fix a °ag 0 2 F L(V ). For any N+1 symmetric group S as follows. De¯ne w1 ( ) as the minimum of i such that F1 » F i0 . Fix a basis vector u1 2 F1 . De¯ne w2 ( ) as the minimum of i such that there is a basis in F 2 of the form u1 ; u2 with u2 2 F i0 . Assume that w1 ( ); : : : ; wj ( ) and u1 ; : : : ; uj are determined. De¯ne wj+1( ) as the minimum of i such that there is a basis in F j+1 of the form u1 ; : : : ; uj ; uj+1 with uj+1 2 F i0 . As a result of this procedure we de¯ne w( ) = (w1 ( ); : : : ; wN+1( )) 2 S N +1 and a basis u1 ; : : : ; uN+1 which generates and such that ui 2 F w0i( ) .
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257
For w 2 S N +1 , de¯ne 0
Gw = f 2 F L(V ); w( ) = wg: 0
The algebraic variety Gw is called the Bruhat cell associated with The set of all Bruhat cells form a cell decomposition of F L(V ):
0
and w 2 S N+1 .
0
F L(V ) = tw2S N+1 G w : Recall that the symmetric group S N +1 is identi¯ed with the slN +1 Weyl group in such a way that the simple transposition (i; i + 1) corresponds to the simple re°ection with respect to ¬ i . Now we are ready to describe the set of (N + 1)-tuples in a population of a ¯xed degree. Let represent an h-critical point associated with the initial data and the weight at in¯nity w ¢ ¤1 , where ¤1 is dominant integral. Let P = P ( ) be the corresponding population and : F L(V ) ! P the generating isomorphism. Theorem 4.26. The set of N -tuples ¹ in P ( ) associated with a weight at in¯nity w ¢¤1 , (1) where ¤1 is integral dominant, coincides with the image of the Bruhat cell (Gw ). Proof 4.27. The theorem is proved similar to Corollary 5.23 in [MV1]. In particular, each population contains at most one tuple which represents an hcritical point associated to a integral dominant weight at in¯nity. Thus the number of critical points associated to an initial data and an integral dominant weight is bounded from above by the number of the populations. We conjecture that for generic values of zi this bound is exact.
5
h-selfdual vector spaces of polynomials
5.1 Dual spaces Let V be a space of polynomials of dimension N + 1 with frame T1 (x); : : : ; T N (x). For u1 ; : : : ; ui 2 V , the polynomial W y(u1 ; : : : ; ui )(x) := Qi¡1
j=1
T1 (x + (j ¡
W (u1 ; : : : ; ui )(x) Qi¡2 1)h) j=1 T 2 (x + (j ¡
1)h) : : : Ti¡1 (x)
is called the divided Wronskian. Note that if polynomials u1 ; : : : ; uN+1 form a basis of V , then the divided Wronskian y W (u1 ; : : : ; uN +1) is a non-zero constant. Let V y be the set of polynomials W y (u1 ; : : : ; uN ), where ui 2 V . Clearly V y is a vector space of dimension N + 1. We call V y the h-dual space of V. We have a non-degenerate pairing V « V y ! C;
u « W y (u1 ; : : : ; uN ) 7! W y(u; u1 ; : : : ; uN )(x):
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For i = 1; : : : ; N; set T iy(x) = TN +1¡i (x + (i ¡
1)h):
(20)
The following two lemmas are obtained from Wronskian identities in Lemmas 9.7 and 9.4. y
y
Lemma 5.1. The polynomials T1 (x); : : : ; TN (x) form a frame of V y . For a space of polynomials V and a 2 C, we denote V (x + a) the space of polynomials spanned by u(x + a), u(x) 2 V . Lemma 5.2. We have V yy = V (x + (N ¡
1)h).
5.2 h-selfdual spaces and canonical bilinear form We say that V is h-selfdual, if V y = V (x ¡ (N ¡ 1)h=2). A space V is h-selfdual if and only if V y is h-selfdual. If V is h-selfdual, then Ti (x) = TN +1¡i (x + (i ¡
1)h ¡
(N ¡
1)h=2):
(21)
In particular, if Ti are of the form (11), then (N+1¡i) ¤(i) ; s = ¤s
(N +1¡i) b(i) + (i ¡ s = bs
1)h ¡
(N ¡
1)h=2:
For instance, the space of polynomials of degree not greater than N is h-selfdual. In this case all polynomials T i are equal to 1. Let V be h-selfdual. De¯ne a non-degenerate pairing ( ; ) : V « V ! C. If u; v 2 V , then we write v(x) = W y (u1 ; : : : ; uN )(x ¡ (N ¡ 1)h=2) with ui 2 V and set (u; v) = W y(u; u1 ; : : : ; uN ). This pairing is called the canonical bilinear form. A basis u1 ; : : : ; uN+1 in a space of polynomials V is called a Witt basis if for i = 1; : : : ; N + 1 we have bN+2¡i ; : : : ; uN +1)(x ¡ ui (x) = W y (u1 ; : : : ; u
(N ¡
1)h=2):
(22)
Clearly, if V has a Witt basis, then V is h-selfdual.
Theorem 5.3. Let V be h-selfdual. Then V has a Witt basis. In particular, the form ( ; ) : V « V ! C is symmetric if the dimension of V is odd, and skew-symmetric if the dimension of V is even. Proof 5.4. The theorem is proved similarly to Theorem 6.4 in [MV1].
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5.3 Isotropic °ags Let u1 ; : : : ; uN+1 be a basis in a vector space V of polynomials. bi ; : : : ; uN +1 )(x ¡ (N ¡ 1)h=2). W y(u1 ; : : : ; u
Denote W i (x) =
Lemma 5.5. If for i = 1; : : : ; N
N¡ 1 h); (23) 2 where c1 ; : : : ; cN are some non-zero complex numbers, then for every i, the polynomial ui (x) is a linear combination of W N+1 (x); W N (x); : : : ; W N+2¡i (x). W y(u1 ; : : : ; ui )(x) = ci W y (u1 ; : : : ; uN +1¡i )(x + (i ¡
1)h ¡
Proof 5.6. The proof of the lemma is similar to the proof of Theorem 6.8 in [MV1]. Corollary 5.7. If V has a basis satisfying (23), then V is h-selfdual. Let V be an h-selfdual space of polynomials. For a subspace U » V denote U ? its orthogonal complement. A full °ag = f0 » F 1 » ¢ ¢ ¢ » FN +1 = V g is called isotropic if F i? = FN +1¡i for i = 1; : : : ; N . Proposition 5.8. Let be a °ag in an h-selfdual space of polynomials V and u1 ; : : : ; uN +1 a basis in V adjusted to . Then is isotropic if and only if (23) holds. Proof 5.9. If F i? = FN +1¡i , then we have two bases in F N+1¡i : the basis u1 ; : : : ; uN +1¡i and the basis W N +1 ; W N ; : : : ; W i+1. Hence W y(u1 ; : : : ; uN+1¡i )(x) = constW y(W N +1 ; W N ; : : : ; W i+1)(x) = = constW y(u1 ; : : : ; ui )(x + (N ¡
i)h ¡
(N ¡
1)h=2):
The forward part of the lemma is proved. Now, let (23) hold. We prove that the spaces spanned by u1 (x); : : : ; ui (x) and by W N+1 (x ¡ (N ¡ 1)h=2); : : : ; W N¡i (x ¡ (N ¡ 1 + 2)h=2) are the same by induction on i. For i = 1 it is just equation (23). Assume that the statement is proved for i = 1; : : : ; i0 ¡ 1. Then W y (u1 ; : : : ; ui0 )(x) = constW y(u1 ; : : : ; uN +1¡i0 )(x + (i0 ¡
1)h ¡
constW y(W N +1 ; W N ; : : : ; W N ¡i0+2)(x ¡ y
(N ¡
(N ¡
constW (u1 (x); : : : ; ui0¡1 (x); W N¡i0+2 (x ¡
1)h=2)
1)h=2)
(N ¡
1)h=2));
and the step of induction follows.
6
The case of B N
Consider the root system of type BN corresponding to the Lie algebra so2N+1 . Let ¬ 1 ; : : : ; ¬ N¡1 be long simple roots and ¬ N the short one. We have (¬
N; ¬ N)
= 2;
(¬ i ; ¬ i ) = 4;
(¬ i ; ¬
i+1 )
= ¡ 2;
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for i = 1; : : : ; N ¡
1, and all other scalar products are zero.
6.1 De¯nition of critical points of B N type We ¯x rami¯cation points = (z1 ; : : : ; zn ) 2 Cn , non-zero dominant integral BN weights (i) (i) ¤ = (¤1 ; : : : ; ¤n ) and relative shifts (i) = (b1 ; : : : ; bn ) 2 Cn , i = 1; : : : ; N . We call this (i) data a BN initial data. We also set ¤s = (¤s ; ¬ _i ). Given = (l1 ; : : : ; lN ) 2 ZN ¸0 , we de¯ne the B N weight at in¯nity ¤1 by formula (9). For a BN initial data ; ¤ ; (i) and a weight at in¯nity ¤1 , we de¯ne an sl2N initial data, which consists of rami¯cation points zi , non-zero dominant integral sl2N weights (i);A ¤A , and we also de¯ne an sl2N weight at in¯nity ¤A 1 . Namely, i , and relative shifts A given a BN weight ¤, the sl2N weight ¤ is de¯ned by (¤A ; ¬ A i
where i = 1; : : : ; N and ¬
A i )
= (¤A ; ¬
A 2N¡i )
are roots of sl2N . The shifts ¡i);A b(i);A = b(2N + (i ¡ s s
_ i );
= (¤; ¬
(i);A
are de¯ned by
N )h = b(i) s :
(i)
Given a set of complex numbers tj , i = 1; : : : ; N , j = 1; : : : li , we represent it by the Qli (i) N -tuple of polynomials = (y1 ; : : : ; yN ), where yi (x) = j=1 (x ¡ tj ). We de¯ne the corresponding (2N ¡ 1)-tuple A by the formula yiA (x) = yi (x);
A yN (x) = yN (x);
A yi+N (x) = yN¡i(x + ih);
(24)
where i = 1; : : : ; N ¡ 1. We propose the following de¯nition of the h-critical points of type BN , cf. Lemma (i) 7.1 in [MV1]. The set of complex numbers tj , i = 1; : : : ; N , j = 1; : : : ; li , is called an h-critical point of BN type associated to the initial data , ¤ , (i) and the weight at in¯nity ¤1 if the corresponding (2N ¡ 1)-tuple A represents an sl2N h-critical point associated to the initial data , ¤ A , (i);A and the weight at in¯nity ¤A 1. In this case we say that the N -tuple represents an h-critical point of the BN type. (i) Equivalently, the set of numbers tj ; i = 1; : : : ; N , j = 1; : : : ; li , is an h-critical point of BN type if it satis¯es the following system of algebraic equations: (i) n Y tj ¡ s=1
li¡ (i) (i) 1 (i) zs + bs h + ¤s h Y tj ¡
(i) tj ¡
(i)
zs + b s h
£
Y t(i) j ¡ (i) tj ¡
k6=j
lN¡ 1
£
Y
k=1
Ã
(N)
tj
(i)
tk
(N) n Y tj
¡
(N )
tj
(i)
tk
¡
(i¡1)
tk
+h
(i) (i¡1) tj ¡ tk k=1 li+1 (i) (i+1) ¡ hY tj ¡ tk (i) (i+1) ¡ + h k=1 tj ¡ tk (N ) (N) ¡ zs + b s h + ¤ s h (N )
(N)
tj ¡ zs + bs h s=1 !2 (N¡1) ) Y t(N) ¡ t(N ¡ tk +h j k (N¡1)
tk
(N )
k6=j
tj
¡
(N)
tk
£
h
= 1;
£ h
+h
= 1;
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261
where i = 1; : : : ; N ¡ 1. Note that in the quasiclassical limit h ! 0, this system becomes system (2.2) of [MV1] specialized to the case of BN .
6.2 Reproduction procedure of B N type We describe the concepts of reproduction and population. All of that follows from the de¯nitions in the case of sl2N . An N -tuple is called fertile in the BN sense if there exist polynomials y~i , i = 1; : : : ; N ¡ 1 and y~N such that W (yi ; y~i)(x) = yi+1(x)yi¡1 (x + h)Ti (x); 2 W (yN ; y~N )(x) = yN ¡1 (x + h)T N (x):
The N -tuple (i) = (y1 ; : : : ; y~i ; : : : ; yN ) is called an immediate descendant of in the direction i. Immediate descendants (i) are also fertile in the BN sense. From Lemma 3.4 we obtain that an N -tuple represents a critical point if and only if it is fertile and A is generic. In this case if ( (i) )A is generic, then (i) also represents an h-critical point of type BN . The BN population originated at a critical point is the minimal set P ( ) of fertile N tuples such that 2 P ( ) and if ¹ 2 P ( ) then the immediate descendants ¹ (i) 2 P ( ). Obviously, the BN population is contained in the corresponding sl2N population: if ¹ 2 P ( ), then ¹ A 2 P A ( A ), where we denote P A ( A ) to be the sl2N population of critical points originated at A . The fundamental space of the sl2N population P A ( A ) contains a °ag with the property (22), and therefore is h-selfdual. We call this space the fundamental space of the BN population P ( ). The corresponding fundamental operator is the di®erence operator of order 2N obtained from (15) by substituting (24) and (21): Ã ! N!1 i¡1 Y Y yN ¡i (x + (i + 1)h) yN +1¡i (x + (i ¡ 1)h) T N¡s (x + ih) £ @¡ T (x + ih) y (x + ih) y (x + ih) T 1)h) ¡s (x + (i ¡ N¡i N+1¡i N i s=1 Ã ! ¡i 1!N N Y Y yN +1¡i (x + h) yN ¡i (x) T s (x + (N ¡ i ¡ s + 1)h) @¡ ; ¡ ¡ y (x) y T (x + (N i s)h) ¡i (x + h) N+1¡i N s i s=1 Q where T (x) = N s)h)=T s (x + (N ¡ s ¡ 1)h). s=1 T s (x + (N ¡ Clearly, two populations of h-critical points of type BN either do not intersect or coincide.
6.3 A B N population and the C N °ag variety Let be an h-critical point of BN type and let V be the fundamental space of the population P A ( A ). Then V is an h-selfdual space of dimension 2N and therefore has a non-degenerate skew-symmetric canonical bilinear form.
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The special symplectic Lie algebra of V consists of all traceless endomorphisms x of V such that (xv; v 0 ) + (v; xv 0 ) = 0 for all v; v 0 2 V . Let = (u1 ; : : : ; u2N ) be a Witt basis in V . We have (ui; u2N+1¡i ) = (¡ 1)i+1 , i = 1; : : : ; N , and (ui ; uj ) = 0 if i + j 6= 2N + 1. This choice of basis identi¯es the special symplectic Lie algebra with a Lie subalgebra of sl2N , which is denoted sp2N . The Lie algebra sp2N has the root system of type CN . Denote Ei;j the matrix with zero entries except 1 at the intersection of the i-th row and the j-th column. The lower triangular part of sp2N is spanned by matrices Ei;i+1 +E2N¡i;2N +1¡i for i = 1; : : : ; N ¡ 1 and EN;N +1 . Denote these matrices X1 ; : : : ; XN , respectively. Now we describe the action of one-parametric subgroups in the special symplectic group in the direction of Xi on the basis = (u1 ; : : : ; u2N ). We have ecXi e
cXi
= (u1 ; : : : ; ui¡1 ; ui + cui+1; ui+1; : : : ; u2N¡i ; u2N +1¡i + cu2N+2¡i ; u2N +2¡i ; : : : ; u2N ); = (u1 ; : : : ; uN¡1 ; uN + cuN +1 ; uN+1; : : : ; u2N );
where i = 1; : : : ; N ¡ We also set e1Xi e1Xi
1.
= (u1 ; : : : ; ui¡1 ; ui+1 ; ui; : : : ; u2N ¡i ; u2N +2¡i ; u2N+1¡i ; : : : ; u2N ); = (u1 ; : : : ; uN¡1 ; uN +1 ; uN ; : : : ; u2N ):
Note ecXi is a Witt basis for all i; c and the corresponding °ag is isotropic. Given a basis = (u1 ; : : : ; u2N ) of V , we denote ( ) to be the full °ag generated by this basis. Let F L(V ) be the variety of all full °ags of V . Denote F L? (V ) » F L(V ) the subvariety of all isotropic °ags and de¯ne the BN generating morphism : F L? (V ) ! ( C[x])N , ( ) 7! (y1 ; : : : ; yN ), where yi = W y(u1 ; : : : ; ui ). Lemma 6.1. If F ( ) is an isotropic °ag, then ( ( )) is fertile. Moreover the set of all immediate descendants of ( ( )) in the direction i coincides with the set ( (ecXi )), ¹ c 2 C. Proof 6.2. Follows from the Wronskian identities W (yi ; W y(u1 ; : : : ; ui¡1 ; ui+1))(x) = Ti (x)yi¡1 (x + h)yi+1 (x); W (yN ; W y(u1 ; : : : ; uN ¡1 ; uN +1 ))(x) = TN (x)(yN ¡1 (x + h))2 ; where i = 1; : : : ; N ¡
1.
Theorem 6.3. The generating morphism of type BN gives rise to the isomorphism F L? (V ) ! P ( ) » ( C[x])N . Proof 6.4. The generating morphism is an isomorphism of F L? (V ) to the image by Theorem 4.24. Clearly, the image of contains the population P ( ). The image of coincides with P ( ) by Lemma 6.1.
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263
Note that F L? is isomorphic to the °ag variety of the special symplectic group which corresponds to the CN root system. This °ag variety has a Bruhat cell decomposition (1);C F L? (V ) = tw2 G w , where is the CN Weyl group. Note that the Weyl groups of BN and CN are canonically identi¯ed. For details, see section 7.3 in [MV1]. As before, the set of all N -tuples in a BN population of the same degree is the Bruhat cell in the CN °ag variety. Theorem 6.5. The set of N -tuples ¹ in P ( ) associated with a weight at in¯nity w ¢ ¤1 , (1);C where ¤1 is integral dominant, coincides with the image of the Bruhat cell (Gw ). Proof 6.6. Completely parallel to the proof of Corollary 7.12 in [MV1].
7
The case of C N
Consider the root system of type CN corresponding to the Lie algebra sp2N . Let ¬ 1 ; : : : ; ¬ N¡1 be short simple roots and ¬ N the long one. We have (¬
N; ¬ N)
for i = 1; : : : ; N ¡
= 4;
(¬ i ; ¬ i ) = 2;
(¬
N¡1 ; ¬ N )
=¡ 2
(¬
i¡1 ; ¬ i )
= ¡ 1;
1 and all other scalar products are zero.
7.1 De¯nition of critical points of C N type We ¯x the rami¯cation points = (z1 ; : : : ; zn ) 2 C, the dominant integral CN with (i) n weights ¤ = (¤1 ; : : : ; ¤n ) and the relative shifts (i) = (b1 ; : : : ; (i) n ) 2 C , i = 1; : : : ; N . (i) We call this data a CN initial data. We also set ¤s = (¤s ; ¬ _i ). Given = (l1 ; : : : ; lN ) 2 ZN ¸0 , we de¯ne the C N weight at in¯nity ¤1 by the formula (9). For a CN initial data ; ¤ ; (i) and a weight at in¯nity ¤1 , we de¯ne the sl2N +1 initial data, which consists of rami¯cation points zi , the dominant integral ¤A i , and relative shifts (i);A A and we also de¯ne an sl2N+1 weight at in¯nity ¤1 . Namely, given a CN weight ¤, the sl2N+1 weight ¤A is de¯ned by (¤A ; ¬ where i = 1; : : : ; N and ¬
A i
A i )
= (¤A ; ¬
A 2N +1¡i )
= (¤; ¬
are roots of sl2N +1 . The shifts
+1¡i);A b(i);A = b(2N + (i ¡ s s
_ i ); (i);A
are de¯ned by
N + 1=2)h = b(i) s :
(i) (N ) Given a set of complex numbers tj , tk where i = 1; : : : ; N ¡ 1, j = 1; : : : ; li , k = 1; : : : ; 2lN , we represent it by the N -tuple of polynomials = (y1 ; : : : ; yN ), where Qi Q N (i) (N ) A yi (x) = lj=1 (x ¡ tj ), yN = 2l j=1 (x ¡ tj ). We de¯ne the corresponding (2N )-tuple by the formula
yiA (x) = yi (x);
A yi+N (x) = yN+1¡i(x + (i ¡
1=2)h);
(25)
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where i = 1; : : : ; N . We propose the following de¯nition of the h-critical points of type CN , cf. Section (i) (N ) 7.2 in [MV1]. The set of complex numbers tj , tk , where i = 1; : : : ; N ¡ 1, j = 1; : : : li , k = 1; : : : ; 2lN , is called an h-critical point of CN type associated to the initial data , ¤ , (i) and the weight at in¯nity ¤1 if the corresponding (2N )-tuple A represents an sl2N +1 h-critical point associated to the initial data , ¤ A , (i);A , and the weight at in¯nity ¤A 1. In this case we say that the N -tuple represents an h-critical point of the CN type. (i) (N ) Equivalently, the set of numbers tj , tk , where i = 1; : : : ; N ¡ 1, j = 1; : : : ; li , k = 1; : : : ; 2lN , is an h-critical point of CN type if it satis¯es the following system of algebraic equations: (i) n Y tj ¡ s=1
(i) (i) (i) zs + bs h + ¤s h Y tj ¡
(i) tj ¡
£
(i)
zs + bs h (i) Y t(i) tk ¡ j ¡ (i)
k6=j
(N) n Y tj s=1
tj ¡
(i)
(i) tj ¡
k
hY
tk + h
(i¡1)
tk
k
(i¡1)
tk
(i) tj ¡
(i)
tj ¡
+h
£
(i+1)
tk
(i+1)
tk
¡
h
lN¡ (N) (N ¡1) 1 (N) ¡ zs + + ¤s h Y tj ¡ tk +h £ (N ) (N) (N) (N ¡1) tj ¡ zs + bs h tj ¡ tk k=1 (N ) (N) 2lN (N ) Y t(N) ¡ tk ¡ h Y tj ¡ tk + h=2 j £ (N ) (N) (N ) ¡ t(N) + h k=1 tj ¡ tk ¡ h=2 k6=j tj k (N ) bs h
= 1;
= 1:
Note that in the quasiclassical limit h ! 0, this system becomes system (2.2) of [MV1] specialized to the case of CN .
7.2 Reproduction procedure of C 1 type Fix a polynomial T (x). Suppose that we have a polynomial y(x), such that (y(x); y(x + h=2)) is a critical point of sl3 with weights and shifts given by (T (x); T (x + h=2)) via (11). Then there exists a polynomial y~ such that W (y; y~)(x) = y(x + h=2)T (x): The fundamental space V of sl3 population P A originated at (y(x); y(x + h=2) is 3dimensional. It is spanned by u1 = y(x), u2 = y~(x) and u3 satisfying the identities W (u1 ; u2 )(x) = u1 (x + h=2)T (x); W (u1 ; u3 )(x) = u2 (x + h=2)T (x);
(26)
W (u2 ; u3 )(x) = u3 (x + h=2)T (x): These three equations constitute the fact that u1 ; u2 ; u3 form a Witt basis of V . Equations (26) imply the following lemma. Lemma 7.1. Let v be a polynomial in V . The pair (v(x); v(x + h=2)) is in the population PA if and only if v(x) is a scalar multiple of u1 + ¬ u2 + ¬ 2 u3 =2 for some ¬ 2 C or of u3 (i.e., ¬ = 1).
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265
We say that the polynomials v(x) described in the lemma and considered as elements of C[x] form a C1 population of h-critical points with weight T (x). Note that unlike the case of sl2 , a C1 population is not a linear space.
7.3 Reproduction procedure of C N type Here, we describe the concepts of reproduction and population. All such follows from the de¯nitions in the cases of sl2N+1 and C1 . An N -tuple is called fertile in the CN sense if there exist polynomials y~i, i = 1; : : : ; N ¡ 1 and y¹N such that W (yi; y~i )(x) = yi+1 (x)yi¡1 (x + h)T i (x); W (yN ; y¹N )(x) = yN¡1 (x + h)yN (x + 1=2h)TN (x): For i = 1; : : : ; N ¡ 1, the N -tuple (i) = (y1 ; : : : ; y~i ; : : : ; yN ) is called an immediate descendant of in the direction i. It follows from the N -th equation that the polynomial yN belongs to a C1 population with weight yN ¡1 (x + h)TN (x). Let y~N be any element of that C1 population. Then (N ) = (y1 ; : : : ; yN¡1 ; y~N ) is called an immediate descendant of in the direction N . For i = 1; : : : ; N , immediate descendants (i) are fertile in the CN sense. From Lemma 3.4 we obtain that an N -tuple represents a critical point of CN type if and only if it is fertile, and A is generic in the sl2N +1 sense. If represents a critical point of CN type, and if ( (i) )A is generic in the sl2N+1 sense, then (i) also represents an h-critical point of type CN . The CN population originated at a critical point is the minimal set P ( ) of fertile N tuples such that 2 P ( ), and if ¹ 2 P ( ), then the immediate descendants ¹ (i) 2 P ( ). Obviously, the CN population is contained in the corresponding sl2N +1 population: if ¹ 2 P ( ), then ¹ A 2 P A ( A ), where we denote P A ( A ) to be the sl2N+1 population of critical points originated at A . Moreover, the fundamental space of the sl2N +1 population PA ( A ) contains a °ag with the property (22), and therefore is h-selfdual. We call this space the fundamental space of the CN population P ( ). The corresponding fundamental operator is the di®erence operator of order 2N + 1 obtained from (15) by substituting (25) and (21): N!1 Y µ
yN ¡i (x + (i + 3=2)h) yN+1¡i (x + (i ¡ 1=2)h) T (x + ih)£ y ¡i (x + (i + 1=2)h) yN +1¡i (x + (i + 1=2)h) N i !µ ¶ i¡1 Y TN ¡s (x + (i ¡ 1=2)h) yN (x + 3=2)h) yN (x) @¡ T (x) £ ¡ T (x + (i 3=2)h) y (x + 1=2h) y (x + h) ¡s N N N s=1 Ã ! 1!N N¡i Y yN +1¡i (x + h) yN ¡i (x) Y T s (x + (N ¡ i ¡ s + 1)h) @¡ ; yN +1¡i (x) yN¡i (x + h) s=1 T s (x + (N ¡ i ¡ s)h) i @¡
where T (x) =
QN
s=1
T s (x + (N ¡
s + 1)h)=T s (x + (N ¡
s)h).
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Clearly two populations of h-critical points of type CN either do not intersect or coincide.
7.4 A C N population and the B N °ag variety Let be an h-critical point of CN type and let V be the fundamental space of the population P A ( A ). Then V is an h-selfdual space of dimension 2N + 1, and therefore has a non-degenerate symmetric canonical bilinear form. The special orthogonal Lie algebra of V consists of all traceless endomorphisms x of V such that (xv; v0 ) + (v; xv 0 ) = 0 for all v; v 0 2 V . Let = (u1 ; : : : ; u2N +1 ) be a i+1 Witt basis in V . We have (ui; u2N+2¡i ) = (¡ 1) , i = 1; : : : ; N + 1 , and (ui ; uj ) = 0 if i + j 6= 2N + 2. The choice of the basis identi¯es the special orthogonal Lie algebra with a Lie subalgebra of sl2N +1 , which is denoted so2N +1 . The Lie algebra so2N+1 has the root system of type Bk . The lower triangular part of so2N+1 is spanned by matrices Ei;i+1 + E2N +1¡i;2N+2¡i for i = 1; : : : ; N . Denote these matrices X 1 ; : : : ; XN , respectively. Now we describe the action of one-parametric subgroups in the special orthogonal group in the directions of Xi on the basis = (u1 ; : : : ; u2N +1 ). We have ecXi e
= (u1 ; : : : ; ui¡1 ; ui + cui+1 ; ui+1 ; : : : ; u2N +1¡i ; u2N+2¡i + cu2N +3¡i ; : : : ; u2N +1 );
cXi
= (u1 ; : : : ; uN ¡1 ; uN + cuN+1 + c2 uN +2 =2; uN+1 + cuN+2; uN +2 ; : : : ; u2N +1 );
where i = 1; : : : ; N ¡ We also set e1Xi e
1Xi
1.
= (u1 ; : : : ; ui¡1 ; ui+1; ui ; : : : ; u2N +1¡i ; u2N +3¡i ; u2N+2¡i ; : : : ; u2N+1); = (u1 ; : : : ; uN¡1 ; uN +2 ; uN+1; uN : : : ; u2N +1 ):
Note ecXi is a Witt basis for all i; c, and the corresponding °ag is isotropic. Denote F L? (V ) » F L(V ) to be the subvariety of all isotropic °ags, and de¯ne the CN generating morphism : F L? (V ) ! ( C[x])N ; ( ) 7! (y1 ; : : : ; yN ), where yi = W y (u1 ; : : : ; ui ). ¹ coincides with the set of all immediate descenLemma 7.2. The set ( (ecXi )), c 2 C dants of ( ( )) in the direction i. Proof 7.3. Follows from the Wronskian identities W (yi ; W y(u1 ; : : : ; ui¡1 ; ui+1 ))(x) = T i (x)yi¡1 (x + h)yi+1(x); W (yN ; W y(u1 ; : : : ; uN¡1 ; uN +1 ))(x) = T N (x)yN¡1 (x + h)yN (x + h=2); W (yN ; W y(u1 ; : : : ; uN¡1 ; uN +2 ))(x) = T N (x)yN¡1 (x + h) W y (u1 ; : : : ; uN¡1 ; uN+1)(x + h=2); where i = 1; : : : ; N ¡
1.
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267
Theorem 7.4. The generating morphism of type CN gives rise to the isomorphism F L? (V ) ! P ( ) » ( C[x])N . Proof 7.5. The generating morphism is an isomorphism of F L? (V ) to the image by Theorem 4.24. Clearly, the image of contains the population P ( ). The image of coincides with P ( ) by Lemma 7.2. Note that F L? is isomorphic to the °ag variety of the special symplectic group which corresponds to the BN root system. This °ag variety has a Bruhat cell decomposition (1);B F L? (V ) = tw2 G w , where is the BN Weyl group. The Weyl groups of BN and CN are canonically identi¯ed. For details, see section 7.4 in [MV1]. As before, the set of all N -tuples in a CN population of the same degree is the Bruhat cell in the BN °ag variety. Theorem 7.6. The set of N -tuples ¹ in a CN population P ( ) associated with a weight at in¯nity w ¢ ¤1 , where ¤1 is integral dominant, coincides with the image of the Bruhat (1);B cell (Gw ). Proof 7.7. Completely parallel to the proof of Corollary 7.14 in [MV1].
8
Appendix A: an example of an sl3 population
Consider the population of h-critical points associated to N = 2 and n = 0 and originated at 0 = (1; 1). The pair (1; 1) represents the h-critical point with no variables. We claim that this population consists of pairs of non-zero polynomials = (y1 ; y2 ), where yi = a2;i x2 + a1;i x + a0;i ;
i = 1; 2 ;
(27)
and (a1;1 + a2;1 h)(a1;2 ¡
a2;2 h) = 2a0;1 a2;2 + 2a2;1 a0;2 :
For any generic pair = (y1 ; y2 ) of this form, the roots of the polynomials y1 ; y2 form an h-critical point with the initial data where n = 0. In other words, the roots of y1 ; y2 satisfy the equations Y t(1) j ¡ (1) k6=j tj
(1) tk ¡
¡
(2) l1 Y tj ¡
k=1
(1)
l2 h Y
(1) k=1 tj
(2)
tk
¡
tk ¡
(2)
h
(1) (2) tk + h Y tj ¡
(2) tk ¡
h
tk + h
(2) tj ¡
(1) tj ¡
(1)
tk
k6=j
(2) tj ¡
(2)
tk + h
= 1;
j = 1; : : : ; l1 ;
= 1;
j = 1; : : : ; l2 ;
where l1 = deg y1 and l2 = deg y2 . Equations (13) take the form W (y1 ; y~1 )(x) = y2 (x);
W (y2 ; y~2 )(x) = y1 (x + h);
(28)
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and the reproduction procedure works as follows. We start with 0 = (1; 1). Equations (28) have the form W (1; y~1 )(x) = 1, W (1; y~2 )(x) = 1. Using the second of them, we get pairs = (1; x + a) for all numbers a. Equations (28) now are W (1; y~1 ) = x + a, W (x + a; y~2 ) = 1. Using the ¯rst equation we get pairs = (x2 + (2a ¡ h)x + b; x + a) for all a; b. Equations (28) take the form W (x2 + (2a ¡ h)x + b; y~1 ) = x + a, W (x + a; y~2 ) = (x + h)2 + (2a ¡ h)(x + h) + b. Using the second of them we get = (x2 + (2a ¡ h)x + b; x2 + cx + ac ¡ ah ¡ b) for all a; b; c. It is easy to see that the union of all those pairs is our population, and nothing else can be constructed starting from 0 = (1; 1). Now, it is easy to see that the family of pairs (27) (where each pair is considered up to multiplication of its coordinates by non-zero numbers) is isomorphic as an algebraic variety to the variety of all full °ags in the three dimensional vector space V = C2 [x] of the ¯rst coordinates of the pairs. Namely, y1 generates a line in V and y2 de¯nes a plane in V containing the line generated by y1 . The space V is the fundamental space of the population. In the example above the possible degrees of polynomials y1 ; y2 are (0,0), (1,0), (0,1), (1,2), (2,1), (2,2). The corresponding parts of the family are isomorphic to open Bruhat cells of dimensions 0, 1, 1, 2, 2, 3, respectively. The fundamental space of the population V is the space of polynomials of degree at most 2. This is an h-selfdual space. The canonical form in the basis (1; x; x(x ¡ 1)=2) is described by the matrix: 0 1 1 C B0 0 B C B0 ¡ 1 1=2 C : B C @ A 1 1=2 ¡ 1=8
It is a symmetric non-degenerate form. The basis (1; x ¡ 1=2; (x2 ¡ x ¡ 1=8)=2) is a Witt basis. The set of isotropic vectors is the C1 population. In this example, this is the set of polynomials of the form (x(x ¡
1) ¡
1=8)=2 + ¬ (x ¡
1=2) + ¬
2
=2 = (x + ¬ ¡
1=2)2 ¡
1=8;
where ¬ 2 C and also 1 2 , which corresponds to ¬ = 1.
9
Appendix B: the Wronskian identities
In this appendix we collect identities involving discrete Wronskians. All functions in this section are functions of one variable x. Set ¢f (x) = f (x + h) ¡ f (x): Set ¢(0) f (x) = f (x) and ¢(n+1) f (x) = ¢(¢(n) f )(x) :
E. Mukhin, A. Varchenko / Central European Journal of Mathematics 2 (2003) 238{271
269
The discrete Wronskian of functions g1 ; : : : ; g s is de¯ned by W (g 1 ; : : : ; gs )(x) = det(g i (x + (j ¡
1)h))si;j=1 = det(¢ (j¡1) g i (x))si;j=1 :
We follow the convention that for s = 0 the corresponding discrete Wronskian equals 1. In this section we write W s (g 1 ; : : : ; gs ) instead of W (g 1 ; : : : ; gs ) to stress the order of the Wronskian. Lemma 9.1. We have W s+1(1; g1 ; : : : ; gs )(x) = W s (¢g 1 ; : : : ; ¢g s )(x). Now we describe the Wronskian W s (¢g 1 ; : : : ; ¢g s )(x) in more detail. Apriori, this Wronskian consists of 2s s! terms. However, there are cancellations. We describe the surviving terms. Lemma 9.2. The Wronskian W s (¢g 1 ; : : : ; ¢g s )(x) is equal to the alternating sum of Q (s + 1)! terms of the form si=1 gi (x + !i h), where (!1 ; : : : ; !s ) is a permutation ! of the set f0; : : : ; ^j; : : : ; sg for some j. The sign of this term is (¡ 1)j sgn(w). Proof 9.3. This lemma is straightforward. Now we proceed to other identities. Lemma 9.4. We have W s (f g1 ; : : : ; f g s )(x) = W s (g 1 ; : : : ; gs )(x)
Qs¡1
j=0
f (x + jh) .
For given functions g 1 ; : : : ; gs+1 and an integer k, 0 µ k µ s, denote Vs¡k+1(i) = W s¡k+1(g1 ; : : : ; g s¡k ; gi ). Lemma 9.5. We have W k+1(V s¡k+1(s ¡
k + 1); : : : ; Vs¡k+1(s + 1))(x) = k Y W s+1(g 1 ; : : : ; gs+1 )(x) W s¡k (g1 ; : : : ; g s¡k )(x + jh) : j=1
Proof 9.6. This lemma is proved by induction on s. The case of k = s is trivial. Suppose that the lemma is proved for s = k; k + 1; : : : ; s0 ¡ 1. Divide both sides of the identity Q 0¡k Qk for s = s0 by sj=0 i=0 g 1 (x + (i + j)h) and use Lemma 9.4 to carry g 1 inside all Wronskians. Then one of the functions in each Wronskian is 1 and we can reduce the order by Lemma 9.1. Then the identity for s = s0 follows from the induction hypothesis applied to fi = ¢(g i+1 =g 1 ), i = 1; : : : ; s0 . For given functions g 1 ; : : : ; gs+1, denote W s (i) = W (g 1 ; : : : ; gbi ; : : : ; g s+1) the Wronskian of all functions except gi .
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Lemma 9.7. We have W k+1(W s (s + 1); W s (s); : : : ; W s (s ¡ W s¡k (g1 ; : : : ; g s¡k )(x + kh)
k + 1))(x) = k Y
W s+1 (g1 ; : : : ; g s+1)(x + (j ¡
1)h) :
j=1
Proof 9.8. First we prove the case of s = k by induction on k. The case of k = 0 is trivial. Suppose the case of k < k0 is proved. Divide both sides of our identity in the case Q 0¡1 Qk0 of s = k = k 0 by ki=0 j=0 g1 (x + (i + j)h): By Lemmas 9.1 and 9.4 we are reduced to the identity W k0+1 (W k¢h [k 0 ]; : : : ; W k¢h [1]; W k0 (h1 ; : : : ; hk0 )) 0¡1 0¡1
=
kY 0¡1
W (¢h1 ; : : : ; ¢hk0 )(x + ih);
i=0
di ; : : : ; ¢hk ). where hi = g i+1 =g 1 and W k¢h [i] = W k0¡1 (¢h1 ; : : : ; ¢h 0 0¡1 The left hand side of the last identity is a determinant of size k 0 + 1. Add to the last row the row number i with coe±cient (¡ 1)k0 ¡i+1hk0¡i+1 (x + k0 h), i = 1; : : : ; k 0 . Then using Lemma 9.2 we observe that the last row becomes (0; : : : ; 0; W k0 (¢h1 ; : : : ; ¢hk0 )(x + (k0 ¡
1)h))
and the lemma for k = k0 follows from the induction hypothesis applied to functions f1 = ¢h1 ; : : : ; fk0 = ¢hk0 . Now we continue by induction on s. Suppose that the lemma is proved for s = Q 0¡1 Q k k; : : : ; s0 ¡ 1. Divide both sides of the identity for s = s0 by si=0 j=0 g 1 (x + (i + j)h). Then the identity for s = s0 follows from the induction hypothesis applied to fi = ¢(g i+1 =g 1 ), i = 0; : : : ; s0 ¡ 1.
Acknowledgments Research of E.M. is supported in part by NSF grant DMS-0140460. Research of A.V. is supported in part by NSF grant DMS-9801582.
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