A basic relation between invariants of matrices under the action of the special orthogonal group

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ENRICO ROGORA

RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO Serie II, Tomo LV (2006), pp. 82-94

A BASIC RELATION BETWEEN INVARIANTS OF MATRICES UNDER THE ACTION OF THE SPECIAL ORTHOGONAL GROUP ENRICO ROGORA

The first fundamental theorem of invariant theory for the action of the special orthogonal group on m tuples of matrices by simultaneous conjugation is proved in [2]. In this paper, as a first step in the direction of establishing the second fundamental theorem, we study a basic identity between S O(n, K) invariants of m matrices.

1. Introduction. Let Gl(n, K) be the group of invertible n × n matrices with coefficients in a field K of characteristic zero, and let (K)m n be the space of m-tuples of n × n matrices Gl(n, K) (and hence any subgroup G ⊆ Gl(n, K)) acts over (K) m n by matrices simultaneous conjugation (1)

A · (B1 , . . . , Bm ) = (A B1 A−1 , . . . , A Bm A−1 ) A ∈ Gl(n, K), (B1 , . . . , Bm ) ∈ (K)m n

m Let Am n denote the ring of polynomial functions over (K) n . For any m G G ⊆ Gl(n, K) one can study the ring (A n ) of polynomials f ∈ Am n which are invariant under the action induced by (1), i.e. under the action

(2)

(A · f )(B1, . . . , Bm ) = f (A −1 B1 A, . . . , A−1 Bm A) m A ∈ Gl(n, K), (B1 , . . . , Bm ) ∈ (K)m n , f ∈ An Two classical problems about these actions are:

G 1. Can we find a list of generators for (A m n) ?

2. Can we find a complete list of relations between the generators?

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Solutions of these problems are called the first and the second fundamental theorem for G-invariants of m matrices respectively. In [4] the first and the second fundamental theorem are proved for the general linear group Gl(n, K), for the orthogonal group O(n, K) and for the symplectic group Sp(n, K) . In [2] is proved the first fundamental theorem for the special orthogonal group S O(n, K) (see also [6]). In this paper, as a first step in the direction of establishing the second fundamental theorem, we study a basic identity between S O(n, K) invariants of matrices. Let us recall first fundamental theorem for invariants of matrices for the action of O(n, K) (see [4], Theorem 7.1, p.327). THEOREM 1.1 Every orthogonal invariant of m matrices (A 1, . . . , Am ) ∈ Km is a polynomial in the invariants tr(U i1 Ui2 . . . Uik ) where Ui = Ai or n

Ui = Ati for each i = 1, . . . , m.

Remark 1.2. Note that in the statement of Theorem 1.1, Ui1 Ui2 . . . Uik run over all possible noncommutative monomials in A 1 , . . . , An , At1 , . . . , Atn . A classical way to express the content of Theorem 1.1 is that the trace is the only typical invariant for the action of O(n, K) over m-tuples of matrices. If we consider the action of the special orthogonal group, the structure of m S O(n,K) turns out to depend on the parity of n: For odd n, under the action ( An ) of S O(2n − 1, K), no further invariants arise, (see [2] or [6]), hence S O(2n−1,K) O(2n−1,K) = ( Am . ( Am 2n−1 ) 2n−1 )

For n even one needs to introduce, beyond the trace, a new typical invariant Q, defined by   An − Atn A1 − At1 ,..., Q(A1 , . . . , An ) = Pf L 2 2 where Pf L is the complete polarization of the Pfaffian of an antisymmetric 2n × 2n matrix (see Definition 2.4). In order to describe more precisely S O(2n, K) invariants of matrices it is convenient to introduce the notions of even and odd invariants. Let W be any O(n, K)-module and let f be a polynomial function over W which is S O(n, K)-invariant: If f is also O(n, K)-invariant we say that f is even;

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If f (g · (w)) = det(g) f (w) for all g ∈ O(n, K) we say that f is odd. LEMMA 1.3. Any S O(n, K)-invariant f can be written as the sum of an odd and an even invariant. In fact, let f  be obtained by f by applying any orthogonal transformation f + f f − f whose determinant is −1 . Then is even, is odd and f = 2 2 f − f f + f + . 2 2 We are ready to state the first fundamental theorem for S O(2n, K)invariants ([2] or [6]). Note that, as we said above, S O(2n − 1, K) and O(2n − 1, K) invariants agree. THEOREM 1.4. Let F be a polynomial S O(n, K)-invariant of m matrices (A1 , . . . , Am ) ∈ Km 2n . If F is even it has the form already described in Theorem 1.1. If F is odd, then n must be even and F is a sum of terms of type Q(M1 , . . . , Mn ) f ∗ (A1 , . . . , Am ) where each Mi is a non commutative monomial in A 1 , . . . , Am , At1 , . . . , Atm and f ∗ is an even invariant. Since the product of two odd invariants is even we have in particular that for any pair of n-tuples of matrices (X 1 , . . . , X n ), (Y1 , . . . , Yn ) ∈ Kn2n the expression Q(X 1 , . . . , X n )Q(Y1 , . . . , Yn ) can be written as a polynomial in the basic even invariants tr(U i1 Ui2 . . . Uik ) described in Theorem 1.1. The goal of this paper is to give an explicit form for this polynomial.

2. The invariant Q. We shall identify the tensor product K n ⊗ Kn of the standard S O(n, K)module Kn with itself with the space of 2n × 2n matrices. Let e 1 , . . . e2n be the standard orthonormal basis of K n . We identify the reducible tensor (a1 e1 + . . . + an en ) ⊗ (b1 e1 + . . . + bn en ) with the matrix {ci j } = {ai b j }. This identification can be extended to all tensors by linearity. Under this identification the decomposition K n ⊗ Kn = S 2 (Kn ) ⊕ ∧2 (Kn ) corresponds to the

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decomposition of the space of square matrices in symmetric and antisymmetric ones. The map K n ⊗ K n → ∧2 ( K n ) A − At A → 2 sends each rank one matrix u ⊗ v onto the antisymmetric matrix u⊗v−v⊗u u∧v = 2 As anticipated in the Introduction, in order to describe all S O(2n, K) invariants of matrices we need to add a further typical invariant to the trace. This new invariant, which we denote by Q, must be multilinear and such that Q(u 1 ⊗ v1 , . . . , u n ⊗ vn ) coincides with det[u 1 , v1 , . . . , u n , vn ] (u i , vi ∈ K2n ). Before defining Q we recall the definition of the Pfaffian of an antisymmetric, even dimensional matrix. ´ 2.1. Let A = {ai j } be an antisymmetric 2n × 2n matrix. The DEFINITION Pfaffian of A is defined by

(3)

ωnA = n!Pf(A)e1 ∧ . . . ∧ e2n

where (4)

ωA =



2ai j ei ∧ e j

i< j

 Remark 2.2 If A = {ai j } is any matrix, we can write A = ai j ei ⊗ e j . If A = (ai j ) is antisymmetric A can be obviously written as alinear combination 2ai j ei ∧ e j which of the matrices ei ∧ e j . Note however the factor 2 in A = explain the factor 2 in (4). LEMMA 2.3. Let u 1 , v1 , . . . , u n , vn ∈ K2n and let X = u 1 ⊗ v1 + . . . + u n ⊗ vn . Then   X − Xt . det[u 1 , v1 , . . . , u n , vn ] = Pf 2 Proof. Let e1 ∧ . . . ∧ e2n be the volume form in V (an element of the 1 dimensional exterior power dim(V ) V ). Recall that u 1 ∧ v1 ∧ . . . ∧ u n ∧ vn (5) det[u 1 , v1 , . . . , u n , vn ] = e1 ∧ . . . ∧ e2n

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Then by (3)

 Pf

(6)

X − Xt 2

 = Pf(u 1 ∧ v1 + . . . + u n ∧ vn ) =

(u 1 ∧ v1 + . . . + u n ∧ vn )n n!(e1 ∧ . . . ∧ e2n ) 

hence the result.

We recall now the definition of full polarization. Let U 1 , . . . , Um be vector spaces. Recall that a polynomial f : U1 × . . . × Um → K is said to be multihomogeneous of multidegree h 1 , . . . , h m , if (7)

f (λ1 u 1 , ..., λm u m ) = λh1 1 · · · λhmm f (u 1, ..., u m ) ∀λ1 , ..., λm ∈ K, ∀(u 1 , ..., u m ) ∈ U1 × · · · × Um

Given a multihomogeneous polynomial f (u 1, . . . , u m ) having multidegree h 1 , h 2 , . . . , h m , it is possible to ”linearize” f in each variable, i.e, to get a polynomial g(u 11, . . . , u 1h 1 ; . . . ; u m1 , . . . , u mh m ), the full polarization of f , which is linear in each vector variable u i j ∈ Ui . We show how to proceed in the case m = 1; in the general case we need only to extend the procedure to each slot and it is only notationally more complicated. Let V be a finite dimensional vector space over K, let K[V ] be the ring of polynomial functions over V and let f ∈ K[V ] be a homogeneous polynomial of degree d . We can write  s s t11 · · · tdd f s1 ...sd (v1 , . . . , vd ) (8) f (t1 v1 + . . . + td vd ) = s1 +...+sd

where the polynomials f s1 ...sd ∈ K[V d ] are multihomogeneous of degree (s1 , . . . , sd ). ´ 2.4. (Full polarization) The multilinear polynomial f 11...1 ∈ DEFINITION d K[V ] in the decomposition (8) is called the full polarization of f . It will be denoted by P f .

We are finally ready to define Q by taking the complete polarization of the Pfaffian, (see Definition 2.4). ´ 2.5. Let Pf L = P(Pf) be the complete polarization of the DEFINITION Pfaffian. Let (A 1 , . . . , An ) ∈ (Kn2n ). We define Q(A 1 , . . . , An ) by   An − Atn A1 − At1 ,..., . (9) Q(A 1 , . . . , An ) = Pf L 2 2

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Q is obviously multilinear, let us show that it coincides with the determinant when A1 , . . . , An have rank one, as a consequence of Lemma 2.3. THEOREM 2.6. Let u 1 , v1 , . . . , u n , vn ∈ K2n . Then Q(u 1 ⊗ v1 , . . . , u n ⊗ vn ) = det[u 1 , v1 , . . . , u n , vn ] Proof. By Lemma 2.3,   λ1 u 1 ⊗ v1 + ... + λn u n ⊗ vn − (λ1 u 1 ⊗ v1 + ... + λn u n ⊗ vn )t (10) Pf 2 coincides with (11)

λ1 · . . . · λn det[u 1 , v1 , . . . , u n , vn ]

On the other hand we can expand (10) as a polynomial in λ 1 , . . . , λn as (12)    s u 1 ⊗ v1 − v1 ⊗ u 1 u n ⊗ vn − vn ⊗ u n sn 1 λ1 · . . . · λn Pfs1 ,...,sn ,..., . 2 2 The expressions (11) and (12) coincide as polynomials in λ 1 , . . . , λn , hence Pfs1 ,...,sn = 0 unless (s1 , . . . , sn ) = (1, . . . , 1) and therefore det[u 1 , v1 , . . . , u n , vn ] =   u 1 ⊗ v1 − v1 ⊗ u 1 u n ⊗ vn − vn ⊗ u n Pf L ,..., 2 2  hence the claim (13)

To conclude this section we shall list some easy well known facts which will be used in the next section. LEMMA 2.7. 1. u ⊗ v · u  ⊗ v  = u ⊗ #v, u  $v  2. tr(u ⊗ v) = #u, v$ 3. (v ⊗ w)t = w ⊗ v COROLLARY 2.8. (14) #x 1 , x2 $#x3 , x4 $ . . . #x2n−1 , x2n $ = tr(x 2n ⊗ x1 · x2 ⊗ x3 · · · x2n−2 ⊗ x2n−1 )

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3. The basic relation. Let X 1 , . . . , X n and Y1 , . . . , Yn be two n-tuples of 2n × 2n matrices. By Theorem 1.4 the even invariant (15)

Q(X 1, . . . , X n ) · Q(Y1 , . . . , Yn )

can be written as a polynomial in tr(Ui1 Ui2 . . . Ui j ) (notations as in Theorem 1.1). In this Section we want to be explicit about the form of this relation. Note that it is enough to find the form of this relation for rank one matrices X 1 , . . . , X n , Y1 , . . . , Yn since (15) is multilinear in the variables X 1 , . . . , X n , Y1 , . . . , Yn . We begin with an example to illustrate the procedure. Let n = 2 and let X 1 = x1 ⊗ x2 , X 2 = x3 ⊗ x4 , Y1 = y1 ⊗ y2 , Y2 = y3 ⊗ y4 . Then by Theorem 2.6 Q(X 1 , X 2 ) · Q(Y1 , Y2 ) = det[x 1 , x2 , x3 , x4 ] · det[y1 , y2 , y3 , y4 ]. This equals

⎡

#x1 , y1 $ ⎢ #x2 , y1 $ det ⎢ ⎣ #x3 , y1 $ #x4 , y1 $

#x1 , y2 $ #x2 , y2 $ #x3 , y2 $ #x4 , y2 $

#x1 , y3 $ #x2 , y3 $ #x3 , y3 $ #x4 , y3 $

⎤ #x1 , y4 $ #x2 , y4 $ ⎥ ⎥ #x3 , y4 $ ⎦ #x4 , y4 $

and by expanding this determinant we get  sign(ρ)#x 1, yρ(1) $#x2 , yρ(2) $#x3 , yρ(3) $#x4 , yρ(4) $ ρ∈S4

We need to write each product #x 1 , yρ(1) $#x2 , yρ(2) $#x3 , yρ(3) $#x4 , yρ(4) $ as a monomial in the basic invariants of Theorem 1.1. This is achieved by using Corollary 2.8. For example, the factor corresponding to ρ = (3, 2, 1, 4), i.e. (16)

#x 1 , y3 $#x2 , y2 $#x3 , y1 $#x4 , y4 $

can be rearranged as (17)

#x 1 , y3 $#y4 , x4 $#x3 , y1 $#y2 , x2 $

hence by Corollary 2.8 it can be rewritten in the form (18)

tr(x 2 ⊗ x 1 · y3 ⊗ y4 · x 4 ⊗ x 3 · y1 ⊗ y2 ) = tr(X 1t · Y2 · X 2t · Y1 )

By applying the same procedure to all terms we get the desired expression. We now describe the way to transform (15) in general. Let X i = x2i−1 ⊗ x2i and Yi = y2i−1 ⊗ y2i . Then

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Q(X 1, . . . , X n ) · Q(Y1 , . . . , Yn ) = det[(#x i , y j $)]   sign(ρ) #xi , yρ(i) $ = ρ∈S2n

i=1,...,2n

It is convenient to introduce some simple definitions and constructions related to permutations. We say that a pair of integers (i, j) is straight adjacent if i is odd and j = i + 1; is reverse adjacent if i is even and j = i − 1. A pair which is either reverse or straight adjacent is said to be adjacent. An adjacency cycle is an arrangement of an even number if integers   i 1 . . . i 2k (19) j1 . . . j2k such that: 1. the integers in each row are distinct; 2. (i 1 , i 2k ), (i 2 , i 3 ), (i 4 , i 5 ), . . . , (i 2k−2 , i 2k−1 ) are adjacent pairs; 3. ( j1 , j2 ), ( j3 , j4 ), . . . , ( jk−1, jk ) are adjacent pairs. For example (20)



1 6 5 9 4 3 5 6

is an adjacency cycle. If

10 2 7 8





 i 1 . . . i 2n j1 . . . j2n is a permutation of en even number of integers we can rearrange it by displaying its adjacency cycles as in the following example (the two adjacency cycles are separated by a vertical line).   1 2 3 4 5 6 7 8 9 10 = 10 3 7 1 4 9 5 8 6 2    (21) 3 8 7 9 10 4  1 6 5 2 7 8 5 6 2 1  10 9 4 3 ρ=

In general we can describe a simple iterative algorithm to rewrite any permutation of an even number of integers in a form in which its adjacency cycles are displayed side by side as in (21). The result of the application of this algorithm will be called the adjacency form of the permutation.

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Assume we have already found h adjacency cycles, i.e. assume we have written ρ in the form    i 1 . . . i k1  i k1 +1 . . . i k1 +k2  . . . ρ= j1 . . . jk1  jk1 +1 . . . jk1 +k2  . . . (22)    . . .  i k1 +k2 +...+kh−1 +1 . . . i k1 +k2 +...+ks  il1 . . . ilp . . .  jk1 +k2 +...+kh−1 +1 . . . jk1 +k2 +...+ks  jl1 . . . jlp where the first s blocks are in adjacency form. We rearrange the columns in the remaining part   il1 . . . ilp (23) jl1 . . . jlp according to the following inductive rule. The first column will be the one containing the lowest index in the first row. The i -th column will be chosen as follows: if i is even, choose the column containing in the second row the index adjacent to that on the second row of the (i − 1)-th column; if i is odd, choose the column containing in the first row the index adjacent to that on the first row of the (i − 1)-th column. We keep on going with this rule until we get a column whose first row contains the index adjacent to the one in the first row of the first column (necessarily after an even number of steps). If there remain indexes outside the new adjacency cycle we repeat the procedure with the remaining indexes until we find all adjacency cycles. Let k i be the number of columns in the i -th block. We finally rearrange the order of the blocks in such a way that k1 ≥ k2 ≥ . . . ≥ kh . If two blocks have the same length we consider first the one having the lowest index in the first row of the first column. LEMMA 3.1. Let (24)

ρ=



i1 . . . j1 . . .

i 2n j2n



be a permutation of an even number of integers in adjacency form. Let a be the number of reverse adjacent pairs in the first row and let b be the number of reverse adjacent pairs in the second row. Then (25)

sign(ρ) = (−1)a+b

Proof Let (26)

σ =



1 . . . 2n i 1 . . . i 2n



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and let (27)

τ=



1... j1 . . .

2n j2n

91



Then τ −1 ◦ ρ ◦ σ is equal to the identity, hence sign(ρ) = sign(τ ) · sign(σ ). The indexes τ (1), . . . τ (2n) can be rearranged in increasing order by first transposing the b reverse adjacent pairs which gives the contribution (−1) b to the sign of τ and then permuting adjacent pairs, which do not change the  sign. The same for σ We discuss now how to write each element (28)

#x 1 , yρ(1) $ . . . #x2n , yρ(2n) $

as a product of traces of products of the matrices X i and Y j . First, write the permutation ρ in adjacency form. Let   i1 ... ik (29) ρ(i 1 ) . . . ρ(2k ) be any of its adjacency cycles. Let (30)

#x i1 , yρ(i1 ) $#xi2 , yρ(i2 ) $ . . . #xik , yρ(ik ) $

be the corresponding sub product of (28). We can write it as (31)

#x i1 , yρ(i1 ) $#yρ(i2 ) , xi2 $ . . . #yρ(ik ) , xik $

by reversing each product #x i2s , yρ(i2s ) $ at even position. In this way the indexes of the elements in position 2i , 2i + 1 are adjacent. Note that the indexes of the first and the last element are adjacent. By Corollary 2.8, the expression (31) equals (32)

tr(x ik ⊗ x i1 · yρ(i1 ) ⊗ yρ(i2 ) · x i2 ⊗ x i3 · · · yρ(ik−1 ) ⊗ yρ(ik ) )

We introduce the last piece of notation. Let γ = (γ 1 , γ2 ) be an adjacent pair of integers. We define  γ2 /2 if the pair γ is straight adjacent I(γ ) = γ1 /2 if the pair γ is reverse adjacent and S(γ ) =

1 if the pair γ is straight adjacent t (for transposition) if the pair γ is reverse adjacent

Hence (32) can be written   S(ρ(ik−1 ),ρ(ik )) S(i ,i ) S(ρ(i ),ρ(i )) tr X I(ikk ,i11 ) · YI(ρ(i11),ρ(i22)) · · · YI(ρ(ik−1 (33) ),ρ(ik ))

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We denote the expression (32) corresponding to an adjacency cycle c of ρ by trc . We have therefore   sign(ρ) trci (34) Q(X 1, . . . , X n )Q(Y1, . . . , Yn ) = ρ∈S2n

ci ∈C(ρ)

It is possible to have a more explicit form for (34) but we need to introduce some further notation. Let P(n) denote the set of all partitions of the integer n i.e. the set of all integer sequences k = (k1 , . . . , kh )  such that k1 ≥ k2 ≥ . . . ≥ kh ≥ 1 and hi=1 ki = n and let Q n be the set of functions from {1, . . . , n} to the symbols {1, t} (t stands for ”transposition”). For each partition k ∈ Pn , let Sn (k) be the set of permutations σ ∈ Sn such that σ (1) = min{σ (1), . . . , σ (k1 )} σ (k1 + 1) = min{σ (k1 + 1), . . . , σ (k1 + k2 )} ... σ (k1 + . . . + kh−1 + 1) = min{σ (k1 + . . . + kh−1 + 1), . . . , σ (n)}, and such that if k i = ki+1 then σ (k1 + . . . + ki−1 + 1) < σ (k1 + . . . + ki ) + 1. Let Q n (k) = { f ∈ Q n s.t. f (1) = f (k 1 + 1) = f (k1 + k2 + 1) = (35) . . . = f (k1 + . . . + kh−1 + 1) = t} We are now ready to state our result. THEOREM 3.2. The invariant Q(X 1 , . . . , X n )Q(Y1 , . . . , Yn ) is equal to 



β(1)

β(k )

α(k1 ) 1 (α, β)tr(X σα(1) (1) Yτ (1) . . . X σ (k1 ) Yτ (k1 ) )·

k∈P(n) σ ∈Sn (k),τ ∈Sn ,α∈Q n (k),β∈Q n

(36)

α(k +1)

β(k +1)

α(k +k )

β(k +k )

tr(X σ (k11 +1) Yτ (k11+1) . . . X σ (k11 +k22 ) Yτ (k11+k22) )·

··· α(k +...+k +1) β(k1 +...+kh−1 +1) ·tr(X σ (k11 +...+kh−1 Y h−1 +1) τ (k1 +...+kh−1 +1)

β(n)

. . . X σα(n) (n) Yτ (n) )

where (α, β) is defined as follows. Let h be the number of integers in the partition k, let a be the number of indexes j such that α( j) = t and let b be the number of indexes j such that β( j) = t ; then (α, β) = (−1) a+b−h .

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Proof Let ρ ∈ S2n be a permutation and let C(ρ) = {c1 , . . . , ch } be the set of adjacency cycles of ρ. Let k = (k1 , . . . , kh ) be the partition of n associated to C(ρ). Let   i1 . . . i2 p (37) c= j1 . . . j2 p be an adjacency cycle ρ ∈ S2n . Out of c we extract the following two ordered lists of adjacency pairs: from the first row (i 2 p , i 1 ), (i 2 , i 3 ), . . . , (i 2 p−2 , i 2 p−1 ) and from the second row ( j1 , j2 ), ( j3 , j4 ), . . . , ( j2 p−1 , j2 p ). Making this extraction cycle by cycle of the adjacency form of a permutation, we associate to a permutation two ordered lists of adjacency pairs: from the first row μ1 , . . . , μn and from the second row ν1 , . . . , νn Let us define two permutations σ, τ ∈ Sn by σ (i) = I(μi )

τ (i) = I(νi )

and two functions α, β : {1, . . . , n} → {1, t} by α(i) = C(μi )

τ (i) = C(νi ).

Note that if k is the partition associated to ρ, then σ ∈ S n (k) and α ∈ Q n (k). We have shown in this way how each factor  trci

ci ∈C(ρ)

in (34) corresponds to one and only one product of traces in (36). For showing that (34) is equivalent to (36) it remains only to check that also the signs match  and this follows from Lemma 3.1.

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Acknowledgments. I wish to express my gratitude to Claudio Procesi for his help and encouragement. REFERENCES [1] Boffi G., Procesi C., A primer of invariant theory, Brandeis lecture notes 1. [2] Aslaksen, Eng-Chye Tan, Chen Bo Zhu, Invariant theory of special orthogonal groups, Pacific Journal of Mathematics, 168 (1995), pp.207-215. [3] Kraft H., Procesi C. Classical invariant theory: A primer, http://www.math.unibas.ch/˜ Kraft/Papers/KP-Primer.pdf [4] Procesi C. The invariant theory of n × n matrices, Adv. in Math., 19 (1976), 306-381. [5] Procesi C. Lie groups: an approach through invariants and representations, Springer, to appear. [6] Rogora E. Invariant of matrices under the action of the special orthogonal group, Universit`a di Roma, “La Sapienza”, preprint 10/5, Available on line at: http./www.mat.uniroma1.it/people/rogora/pdf/lavoro.pdf. [7] Weyl H. The classical groups: Their invariants and representations, Fifteenth printing. Princeton Landmarks in Mathematics. Princeton Paperbacks. Princeton University Press, Princeton, NJ, (1946) Pervenuto il 22 ottobre 2005.

Dipartimento di Matematica, Universit`a di Roma “La Sapienza” e-mail: [email protected]