arXiv:math/0506614v2 [math.RA] 21 Feb 2006

COMPUTING WITH MATRIX INVARIANTS VESSELIN DRENSKY Abstract. We present an introduction to the theory of the invariants under the action of GLn (C) by simultaneous conjugation of d matrices of size n × n. Then we survey some results, old or recent, obtained by a dozen of mathematicians, on minimal sets of generators, the defining relations of the algebras of invariants and on the multiplicities of the Hilbert series of these algebras. The picture is completely understood only in the case n = 2. Besides, explicit minimal sets of generators are known for n = 3 and any d and for n = 4, d = 2. The multiplicities of the Hilbert series are obtained only for n = 3, 4 and d = 2. For n > 2 most of the concrete results are obtained with essential use of computers.

1. Introduction to invariant theory All considerations in this paper are over an arbitrary field K of characteristic 0. If not explicitly stated, all vector spaces and algebras are over K, and the algebras are unitary and commutative. To get some idea about invariant theory, we start with the following well known result from the undergraduate algebra course. Let n ≥ 2 be an integer and let A = K[X] = K[x1 , . . . , xn ] be the algebra of polynomials in n variables. The algebra of symmetric polynomials ASn = K[X]Sn consists of all polynomials f (x1 , . . . , xn ) ∈ A such that f (xσ(1) , . . . , xσ(n) ) = f (x1 , . . . , xn ) for all permutations σ in the symmetric group Sn . Theorem 1.1. (i) The algebra ASn is generated by the elementary symmetric functions X e1 = x1 + · · · + xn = xi , X xi xj , e2 = x1 x2 + x1 x3 + · · · + xn−1 xn = i n. Much later it was discovered that this theorem was first established in 1943 by Dubnov and Ivanov [DI] but their paper was overlooked by the mathematical community. The class of nilpotency N (n) in the Nagata-Higman theorem is related in the following nice way with invariant theory of matrices. Theorem 2.7. (Formanek [F2], Procesi [P2, P3], Razmyslov [R2]) Let N (n) be the class of nilpotency in the Nagata-Higman theorem. Then the algebra of invariants n ΩGL is generated by the traces tr(Xi1 · · · Xim ) of degree ≤ N (n). For d sufficiently nd large this bound is sharp. It is important to know the exact value of the class of nilpotency N (n) in the Nagata-Higman theorem. The upper bound given in the proof of Higman [Hi] is N (n) ≤ 2n − 1. The best known upper bound is due to Razmyslov [R2]. Applying trace polynomial identities of matrices, he obtained the bound N (n) ≤ n2 . The proof of the theorem of Razmyslov is given also in his book [R3] or in the book by Formanek [F4]. For a lower bound, Kuzmin [Ku] showed that N (n) ≥ 21 n(n + 1). A proof of the result of Kuzmin may be found also in the books by Drensky and Formanek [DF] or by Kanel-Belov and Rowen [KBR]. Hence n(n + 1) ≤ N (n) ≤ n2 . 2 Problem 2.8. Find the exact value N (n) of the class of nilpotency of nil algebras of index n over a field of characteristic 0. Conjecture 2.9. (Kuzmin [Ku]) The exact value N (n) of the class of nilpotency of nil algebras of index n over a field of characteristic 0 is N (n) =

n(n + 1) . 2

The only values of N (n) are known for n ≤ 4: Dubnov [Du] obtained in 1935 N (1) = 1,

N (2) = 3,

In 1993, Vaughan-Lee [VL] proved that N (4) = 10.

N (3) = 6.

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In this way the conjecture of Kuzmin is confirmed for n ≤ 4. Recently, Shestakov and Zhukavets [SZ] have proved that the class of nilpotency of the two-generated algebras satisfying the identity x5 = 0 is equal to 15, which agrees with the conjecture of Kuzmin for n = 5. They have obtained the same result also in the more general setup of 2-generated superalgebras. Their proof is based on computer calculations with the GAP package. 2 Since N (2) = 3, the algebra ΩGL is generated by products of traces of degree 2d ≤ 3. The following result was established by Sibirskii [Si]. Theorem 2.10. The elements tr(Xi ), tr(Xi2 ),

i = 1, . . . , d,

tr(Xi Xj Xk ),

tr(Xi Xj ),

1 ≤ i < j ≤ d,

1 ≤ i < j < k ≤ d,

constitute a minimal set of generators of the algebra of 2 × 2 matrix invariants 2 ΩGL 2d . Now we give an idea about the Razmyslov-Procesi theory which is related with the second fundamental theorem of the matrix invariants, see Razmyslov [R2] and Procesi [P2], as well as the book by Razmyslov [R3] for other applications of his method. For simplicity we consider the case n = 2 only. The Cayley-Hamilton theorem for 2 × 2 matrices implies that X 2 − tr(X)X + det(X) = 0. The Newton formulas give that 1 2 (tr (X) − tr(X 2 )). 2 This can be seen also directly. If ξ1 , ξ2 are the eigenvalues of X, then det(X) =

tr(X) = ξ1 + ξ2 ,

tr(X 2 ) = ξ12 + ξ22 ,

1 1 ((ξ1 + ξ2 )2 − (ξ12 + ξ22 )) = (tr2 (X) − tr(X 2 )). 2 2 In this way we obtain the mixed trace identity det(X) = ξ1 ξ2 =

1 c(X) = X 2 − tr(X)X + (tr2 (X) − tr(X 2 )) = 0. 2 Now we consider the identity c(X1 + X2 ) − c(X1 ) − c(X2 ) = 0, i.e. we linearize the identity c(X) = 0. In this way we obtain the mixed Cayley-Hamilton identity Ψ2 (X1 , X2 ) = X1 X2 +X2 X1 −tr(X1 )X2 −tr(X2 )X1 +tr(X1 )tr(X2 )−tr(X1 X2 ) = 0. Since the trace is a nondegenerate bilinear form on M2 (K), the vanishing of the polynomial Ψ2 (X1 , X2 ) on M2 (K) is equivalent to the vanishing of the pure CayleyHamilton identity Φ2 (X1 , X2 , X3 ) = tr(Ψ2 (X1 , X2 )X3 ) = 0 on all 2 × 2 matrices. Direct calculations show that 0 = Φ2 (X1 , X2 , X3 ) = tr(Ψ2 (X1 , X2 )X3 ) = tr(X1 X2 X3 ) + tr(X2 X1 X3 ) −tr(X1 )tr(X2 X3 ) − tr(X2 )tr(X1 X3 ) + tr(X1 )tr(X2 )tr(X3 ) − tr(X1 X2 )tr(X3 ).

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If we delete the symbols of traces and the X’s in the above expression, we shall obtain the following linear combination of permutations X (123) + (213) − (1)(23) − (2)(13) + (1)(2)(3) − (12)(3) = sign(σ). σ∈S3

This suggests the following construction. We write the permutations in the symmetric group Sm as products of disjoint cycles, including the cycles of length 1, σ = (i1 . . . ip )(j1 . . . jq ) · · · (k1 . . . kr ). We define the associated trace function trσ (x1 , . . . , xm ) = tr(xi1 · · · xip )tr(xj1 · · · xjq ) · · · tr(xk1 · · · xkr ). For every element X

ασ σ ∈ KSm ,

ασ ∈ K, σ ∈ Sm ,

σ∈Sm

where KSm is the group algebra of Sm , we define the trace polynomial X f (x1 , . . . , xm ) = ασ trσ (x1 , . . . , xm ). σ∈Sm

We also assume that for m ≤ k the symmetric group Sm acts on 1, . . . , m and leaves invariant m + 1, . . . , k, i.e. Sm is canonically embedded into Sk . Theorem 2.11. (The Second Fundamental Theorem of Matrix Invariants, Razmyslov [R2], Procesi [P2]) Let X f (x1 , . . . , xm ) = ασ trσ (x1 , . . . , xm ), ασ ∈ K, σ∈Sm

be a multilinear trace polynomial of degree m. Then f = 0 is a trace identity for the n × n matrix algebra, i. e. f (a1 , . . . , am ) = 0 for all a1 , . . . , am ∈ Mn (K), if and only if X ασ σ σ∈Sm

belongs to the two-sided ideal J(n, m) of the group algebra KSm generated by the element X sign(σ)σ. σ∈Sn+1

As in the case of 2 × 2 matrices, the fundamental trace identity X (sign σ)trσ (x1 , . . . , xn+1 ) = 0 σ∈Sn+1

is actually the linearization of the Cayley-Hamilton polynomial. There are several important objects related with invariant theory of matrices. As above, n, d ≥ 2 are fixed integers and X1 , . . . , Xd are d generic n × n matrices: The algebra Rnd generated by X1 , . . . , Xd ; n The pure (or commutative) trace algebra Cnd = ΩGL generated by the traces of nd all products tr(Xi1 · · · Xik ); The mixed (or noncommutative) trace algebra Tnd generated by Rnd and Cnd regarding the elements of Cnd as scalar matrices; The field of fractions Q(Cnd ) of the algebra Cnd ; The algebra Q(Cnd )Rnd .

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All these algebras have no zero divisors and play important roles in mathematics. See the books [DF, F4, J, P1] for different aspects of the theory of algebras of matrix invariants and their applications to combinatorial and structure theory of PI-algebras, central division algebras, etc. The algebra Rnd is a well known object in the theory of PI-algebras, or algebras with polynomial identities. Let Khx1 , . . . , xd i be the free associative algebra (i.e. the algebra of polynomials in noncommuting variables). A polynomial f (x1 , . . . , xd ) ∈ Khx1 , . . . , xd i is a polynomial identity in d variables for Mn (K) if f (a1 , . . . , ad ) = 0 for all a1 , . . . , ad ∈ Mn (K). The set I(Mn (K)) of all polynomial identities in d variables is a two-sided ideal of Khx1 , . . . , xd i and Rnd is isomorphic to the factor algebra Khx1 , . . . , xd i/I(Mn (K)). Clearly, the algebra Cnd is the algebra of matrix invariants. The algebra Tnd is also the algebra of invariant polynomial functions under a suitable action of GLn . It is called the algebra of matrix concominants. It is a finitely generated Cnd -module and, as a Cnd -module, has a generating set consisting of products Xj1 · · · Xjk , where k < N (n), the class of nilpotency in the NagataHigman theorem. The field of fractions Q(Cnd ) appears naturally in field theory. One of the main problems related with Q(Cnd ) is whether it is a purely transcendent extension of K. Finally, Q(Cnd )Rnd is a central division algebra of dimension n2 over its centre Q(Cnd ) and serves as a source of counterexamples to the theory of central division algebras. General invariant theory gives that Cnd and Tnd have nice algebraic properties. Theorem 2.12. (Van den Bergh, [VB1]) The algebra Cnd is a Cohen-Macaulay and even Gorenstein unique factorization domain. The algebra Tnd is a CohenMacaulay module over Cnd . Recall that the Noether normalization theorem gives that Cnd contains a homogeneous set of algebraically independent elements {a1 , . . . , ak }, where k = (d−1)n2 +1 is the transcendence degree of the quotient field Q(Cnd ) (k is also equal to the Krull dimension of Cnd ), such that Cnd is integral over the polynomial algebra K[a1 , . . . , ak ]. Such a set {a1 , . . . , ak } is called a homogeneous system of parameters for Cnd . By a result of Stanley [St1], a graded Cnd -module is Cohen-Macaulay if and only if it is a free module with respect to some homogeneous system of parameters {a1 , . . . , ak } of Cnd . The algebras Rnd , Cnd , and Tnd are multigraded and their homogeneous components of degree (k1 , . . . , kd ) consist of all polynomials which are homogeneous of degree ki in the generic matrix Xi . Hence we may consider their Hilbert series in d variables. For example, X (k ,...,kd ) H(Cnd , t1 , . . . , td ) = dim Cnd1 tk11 · · · tkdd , (k ,...,k )

d where Cnd1 is the homogeneous component of degree (k1 , . . . , kd ). Le Bruyn [LB1] for the case n = 2, and Formanek [F3] and Teranishi [T1, T3] in the general case proved:

Theorem 2.13. Let d ≥ 2 for n ≥ 3 and d > 2 for n = 2. Then H(Cnd , t1 , . . . , td ) and H(Tnd , t1 , . . . , td ) satisfy the functional equation 2

−1 k n H(Cnd , t−1 1 , . . . , td ) = (−1) (t1 · · · td ) H(t1 , . . . , td ),

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where k = (d − 1)n2 + 1 is the Krull dimension of Cnd , and similarly for the Hilbert series of Tnd . The proofs of this theorem given by Formanek and Teranishi are quite different and use, respectively, representation theory of general linear groups and the MolienWeyl integral formula. Later, Van den Bergh paid attention that the proof can be considerably simplified using results of Stanley on Hilbert series of Cohen-Macaulay algebras. We need some background on symmetric polynomials and representation theory of the general linear group, see e.g. the books by Weyl [W2] and Macdonald [Mc]. As in the case of polynomial algebras, the general linear group GLd acts canonically on the vector space with basis {X1 , . . . , Xd }. If g = (gij ), gij ∈ K, then the action of g on Xj is defined by g(Xj ) = g1j X1 + · · · + gdj Xd . This action is extended diagonally on Rnd , Cnd , Tnd . If f (X1 , . . . , Xd ) is any polynomial expression depending on X1 , . . . , Xd (maybe including also traces), then g(f (X1 , . . . , Xd )) = f (g(X1 ), . . . , g(Xd )),

g ∈ GLd .

Representation theory of GLd says that every submodule of the GLd -modules Rnd , Cnd , Tnd is a direct sum of irreducible (or simple) submodules. The irreducible GLd -submodules which appear in the decomposition are polynomial modules and are described in terms of partitions of integers. If λ = (λ1 , . . . , λd ),

λ1 ≥ · · · ≥ λd ≥ 0,

is a partition of k (notation λ ⊢ k) in not more than d parts, then we denote the related GLd -module by W (λ) = W (λ1 , . . . , λd ). To be explicit, below we consider the case of Cnd only. If M Cnd = m(λ)W (λ), m(λ) ≥ 0, i.e. there are m(λ) direct summands isomorphic to W (λ), then we say that W (λ) appears with multiplicity m(λ). The multiplicities m(λ) for Rnd , Cnd , and Tnd play important role in the quantitative study of polynomial identities of matrices. (See the survey by Regev [Re] and the book of the author [D2] for applications of representation theory of Sn and GLd to the theory of PI-algebras.) The Hilbert series of Cnd has the form X H(Cnd , t1 , . . . , td ) = m(λ)Sλ (t1 , . . . , td ), where Sλ (t1 , . . . , td ) is the Schur function associated to λ. Schur functions are important combinatorial objects and appear in many places in mathematics. For example, they form a basis of the vector space of all symmetric polynomials in d variables. One of the possible ways to define Schur functions is via Vandermondelike determinants. For a partition µ = (µ1 , . . . , µd ), define the determinant µ1 µ1 µ1 t1 µ2 tµ2 2 · · · tµd 2 t1 · · · td t2 V (µ1 , . . . , µd ) = . . .. . .. .. .. . . µ t d tµd · · · tµd 1 2 d

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Then the Schur function is V (λ1 + d − 1, λ2 + d − 2, . . . , λd−1 + 1, λd ) . V (d − 1, d − 2, . . . , 1, 0) The Schur functions play the role of characters of the corresponding representation of GLd . If we know the Hilbert series H(Cnd , t1 , . . . , td ), then we can uniquely determine the GLd -module structure of Cnd . Sλ (t1 , . . . , td ) =

3. Concrete computations We give a survey of some concrete results about generators, defining relations and Hilbert series of the algebras of matrix invariants. We start with 2×2 matrices. The first important reduction is the following. We take the generic 2 × 2 matrix X and present it in the form x11 +x22 x11 −x22 x11 x12 0 x12 2 2 X= = + x11 +x22 22 x21 x22 0 x21 − x11 −x 2 2 tr(X) tr(X) y y12 E + 11 = E + Y, = y21 −y11 2 2 where E is the identity matrix and y11 y12 Y = y21 −y11 is a generic 2 × 2 traceless matrix. This reduction implies that C2d is generated by tr(Xi ), i = 1, . . . , d, and tr(Yi1 · · · Yik ), where Y1 , . . . , Yd are generic traceless matrices and k ≥ 2. Since the class of nilpotency in the Nagata-Higman theorem is N (2) = 3 for n = 2, we obtain that it is sufficient to consider the cases k = 2, 3 only. Similarly, T2d is generated by C2d and Y1 , . . . , Yd . It is also easy to see: Proposition 3.1. The algebra C2d has the presentation C2d = (K[tr(X1 ), . . . , tr(Xd )]) ⊗K C0 , where C0 = K[tr(Yi Yj ), tr(Yp Yq Yr ) | 1 ≤ i ≤ j ≤ d, 1 ≤ p < q < r ≤ d]/I is the algebra generated by products of generic traceless 2 × 2 matrices. Hence we may choose all defining relations as linear combinations of the traces of products of traceless matrices. The description of T2d is easier than that of C2d . Theorem 3.2. (i) (Procesi [P4]) The noncommutative trace algebra T2d is isomorphic to the tensor product of K-algebras K[tr(X1 ), . . . , tr(Xm )] ⊗K Wd , where Wd is the associative algebra generated by the generic traceless matrices Y1 , . . . , Ym . (ii) (Razmyslov [R1]) The algebra of the generic traceless matrices Wd is isomorphic to the factor algebra Khy1 , . . . , yd i/I(M2 (K), sl2 (K)) of the free associative algebra Khy1 , . . . , yd i, where sl2 (K) is the Lie algebra of all traceless 2 × 2 matrices and I(M2 (K), sl2 (K)) is the ideal of all polynomials in Khy1 , . . . , yd i which vanish under the substitutions of yi by elements in sl2 (K). (Such polynomials are called weak polynomial identities for the pair (M2 (K), sl2 (K)).) The ideal

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I(M2 (K), sl2 (K)) is generated as a weak T-ideal by the weak polynomial identity [x21 , x2 ] = 0, i.e. I(M2 (K), sl2 (K)) is the minimal ideal of weak polynomial identities containing the element [x21 , x2 ]. An equivalent description is that as an ideal of the free algebra, I(M2 (K), sl2 (K)) is generated by all elements [uv + vu, w], where u, v, w are all possible commutators [yj1 , . . . , yjk ] in the variables y1 , . . . , yd . (iii) (Drensky and Koshlukov [DK]) The ideal I(M2 (K), sl2 (K)) is the minimal ideal of the free associative algebra Khy1 , . . . , yd i which is invariant under the diagonal action of GLd and contains the elements [y12 , y2 ] and X sign(σ)yσ(1) yσ(2) yσ(3) yσ(4) , s4 (y1 , y2 , y3 , y4 ) = σ∈S4

the second polynomial appears for d ≥ 4 only. Hence the algebra of 2 × 2 generic traceless matrices has a uniform set of defining relations for any d ≥ 2. (iv) (Procesi [P4]) As a GLd -module Wd has the description M W (λ1 , λ2 , λ3 ), Wd ∼ =

where the sum is on all partitions λ in at most three parts. (v) (Formanek [F1]) The GLd -module T2d has the decomposition M (λ1 − λ2 + 1)(λ2 − λ3 + 1)(λ3 − λ4 + 1)W (λ1 , λ2 , λ3 , λ4 ). T2d ∼ =

(vi) (See [D1, D2, F1, P4, LB1, LB2] for other descriptions of the Hilbert series.) The Hilbert series of T2d is H(T2d , t1 , . . . , td ) =

d Y

1 X S(λ1 ,λ2 ,λ3 ) (t1 , . . . , td ). 1 − ti i=1

Now we give some results on C2d . We follow the way we used in the previous theorem. For more details, especially for the Hilbert series and the GLd -module decomposition of C2d , see [F1, LB2, P4] or [DF]. Theorem 3.3. (i) The commutative trace algebra C2d is isomorphic to the tensor product of K-algebras K[tr(X1 ), . . . , tr(Xm )] ⊗K C(Wd ), where C(Wd ) is the centre of the algebra Wd defined above. (ii) As a subalgebra of Wd , its centre C(Wd ) is generated by Yi2 ,

i = 1, . . . , d, Yi Yj + Yj Yi , 1 ≤ i < j ≤ d, X sign(σ)Yσ(1) Yσ(2) Yσ(3) , 1 ≤ i < j < k ≤ d, s3 (Yi , Yj , Yk ) = σ∈S3

where the symmetric group S3 acts on {i, j, k}. (iii) As a GLd -module Wd has the decomposition M W (λ1 , λ2 , λ3 ), Wd ∼ =

where the sum is on all partitions λ in at most three parts such that both λ1 − λ2 and λ2 − λ3 are even. Concerning the defining relations of C2d , the case d = 2 is trivial. Formanek, Halpin and Li [FHL] showed that C22 is generated by the algebraically independent elements tr(X1 ), tr(X2 ), det(X1 ), det(X2 ), tr(X1 X2 ).

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For d = 3 Sibirskii [Si] found one relation between the generators of C23 and, using the Hilbert series of C23 , Formanek [F1] proved that there are no more relations. In the general case, the description of the defining relations of C2d is reduced to a similar description of the defining relations of the subalgebra of C2d generated by tr(Yi2 ),

i = 1, . . . , d, tr(Yi Yj Yk ),

tr(Yi Yj ),

1 ≤ i < j ≤ d,

1 ≤ i < j < k ≤ d,

where Y1 , . . . , Yd are generic traceless 2 × 2 matrices. Since GL2 acts on the generic matrices by conjugation, we may replace its action with the action of SL2 and even with the action of P SL2 . Since P SL2 (C) ∼ = SO3 (C), the special orthogonal group, we may apply invariant theory of special linear groups. (The restriction K = C is not essential in the final version of the result.) We consider the action of the special orthogonal group SO3 = SO3 (K), i.e. the group of orthogonal 3 × 3 matrices with determinant 1, on the polynomial algebra in 3d variables (3) (2) (1) K[ui , ui , ui | i = 1, . . . , d], (1)

(2)

(3)

induced by the action of SO3 on the three-dimensional vectors ui = (ui , ui , ui ). (j) It is a classical result that the algebra of invariants K[ui ]SO3 is generated by all scalar products (3) (3)

(2) (2)

(1) (1)

hui , uj i = ui uj + ui uj + ui uj , and all 3 × 3 determinants of the coordinates (1) (1) (1) u uk uj i (2) (2) ∆(ui , uj , uk ) = u(2) uk , uj i (3) (3) (3) u u u i

j

1 ≤ i ≤ j ≤ d,

1 ≤ i < j < k ≤ d.

k

The defining relations express the fact that the underlying vector space is threedimensional and every four vectors are linearly dependent. In particular, they use the properties of the Gram determinant: hui , up i hui , uq i hui , ur i hui , us i huj , up i huj , uq i huj , ur i huj , us i = 0, Γ4 (ui , uj , uk , ul ; up , uq , ur , us ) = huk , up i huk , uq i huk , ur i huk , us i hul , up i hul , uq i hul , ur i hul , us i 1 ≤ i < j < k < l ≤ d,

1 ≤ p < q < r < s ≤ d,

∆(ui , uj , uk )∆(up , uq , ur ) − Γ3 (ui , uj , uk ; up , uq , ur ) = 0, hup , ui i∆(uj , uk , ul ) − hup , uj i∆(ui , uk , ul ) +hup , uk i∆(ui , uj , ul ) − hup , ul i∆(ui , uj , uk ) = 0. In order to apply invariant theory of SO3 we need a scalar product (i.e. nondegenerate symmetric bilinear form) on sl2 (K). We use the trace and define hu, vi = tr(uv),

u, v ∈ sl2 (K).

The following result gives the generators and the defining relations of C2d for d ≥ 2. It is a translation of the description of the invariants of SO3 .

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Theorem 3.4. (i) The algebra C2d is generated by tr(Xi ),

tr(Yi2 ),

tr(Yi Yj ),

tr(s3 (Yi , Yj , Yk )),

where i, j, k = 1, . . . , d, and in the traces involving two or three traceless matrices we require i < j or i < j < k, respectively. (ii) Drensky [D3]) The defining relations of C2d with respect to the above generators are tr(Yi Yp ) tr(Yi Yq ) tr(Yi Yr ) tr(s3 (Yi , Yj , Yk ))tr(s3 (Yp , Yq , Yr )) + 18 tr(Yj Yp ) tr(Yj Yq ) tr(Yj Yr ) = 0, tr(Yk Yp ) tr(Yk Yq ) tr(Yk Yr )

tr(Yp Yi )tr(s3 (Yj , Yk , Yl )) − tr(Yp Yj )tr(s3 (Yi , Yk , Yl )) +tr(Yp Yk )tr(s3 (Yi , Yj , Yl )) − tr(Yp Yl )tr(s3 (Yi , Yj , Yk )) = 0, where, again, i, j, k, p, q, r = 1, . . . , d, and, where necessary, we require i < j < k < l and p < q < r.

In order to work efficiently with an algebra R = K[x1 , . . . , xp ]/I, it is not sufficient to know the generators of the ideal I. For computational purposes one needs also the Gr¨ obner basis of I with respect to some ordering on the monomials of K[x1 , . . . , xp ], see e.g. the book by Adams and Loustaunau [AL]. The Gr¨obner basis of C2d is given by Domokos and Drensky [DD], see their paper for more details. Now we consider two generic 3 × 3 matrices. Using the Molien-Weyl formula, Teranishi [T1] calculated the Hilbert series of C32 , namely, H(C32 , t1 , t2 ) =

1 + t31 t32 , (1 − t1 )(1 − t2 )q2 (t1 , t2 )q3 (t1 , t2 )(1 − t21 t22 )

where q2 (t1 , t2 ) = (1 − t21 )(1 − t1 t2 )(1 − t22 ), q3 (t1 , t2 ) = (1 − t31 )(1 − t21 t2 )(1 − t1 t22 )(1 − t32 ). He also found the following system of generators of C32 : tr(X1 ), tr(X2 ), tr(X12 ), tr(X1 X2 ), tr(X22 ), tr(X13 ), tr(X12 X2 ), tr(X1 X22 ), tr(X23 ), tr(X12 X22 ), tr(X12 X22 X1 X2 ), where X1 , X2 are generic 3 × 3 matrices. He showed that the first ten of these generators form a homogeneous system of parameters of C32 and C32 is a free module with generators 1 and tr(X12 X22 X1 X2 ) over the polynomial algebra on these ten elements. Abeasis and Pittaluga [AP] found a system of generators of C3d in terms of representation theory of the symmetric and general linear groups, in the spirit of its use in theory of PI-algebras. They showed that C3d has a minimal system of generators which spans a GLd -module isomorphic to G = W (1) ⊕ W (2) ⊕ W (3) ⊕ W (13 ) ⊕ W (22 ) ⊕ W (2, 12 ) ⊕W (3, 12 ) ⊕ W (22 , 1) ⊕ W (15 ) ⊕ W (32 ) ⊕ W (3, 13 ). (The partitions in [AP] are given in “Francophone” way, i.e., transposed to ours.) It follows from the description of the generators of C32 given by Teranishi [T1], that tr(X12 X22 X1 X2 ) satisfies a quadratic equation with coefficients depending on the other ten generators. The explicit (but very complicated) form of the equation

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was found by Nakamoto [N], over Z, with respect to a slightly different system of generators. A much simpler description of C32 was obtained by Aslaksen, Drensky, and Sadikova [ADS]. The following generators are in the spirit of the ideas of [AP]. Proposition 3.5. Let X1 , X2 and Y1 , Y2 be, respectively, two generic and two generic traceless 3 × 3 matrices. The algebra C32 is generated by tr(X1 ), tr(X2 ), tr(Y12 ), tr(Y1 Y2 ), tr(Y22 ), tr(Y13 ), tr(Y12 Y2 ), tr(Y1 Y22 ), tr(Y23 ), V = tr(Y12 Y22 ) − tr(Y1 Y2 Y1 Y2 ),

W = tr(Y12 Y22 Y1 Y2 ) − tr(Y22 Y12 Y2 Y1 ).

Now we define the following elements of C32 : tr(Y12 ) tr(Y1 Y2 ) U = tr(Y1 Y2 ) tr(Y22 ) W1 = U 3 ,

W6 =

,

W2 = U 2 V, W4 = U V 2 , W7 = V 3 , tr(Y12 ) tr(Y1 Y2 ) tr(Y22 ) tr(Y12 Y2 ) tr(Y1 Y22 ) , W5 = V tr(Y13 ) tr(Y12 Y2 ) tr(Y1 Y22 ) tr(Y23 ) 2 tr(Y23 ) tr(Y1 Y22 ) tr(Y13 ) tr(Y13 ) tr(Y1 Y22 ) −4 tr(Y1 Y22 ) tr(Y12 Y2 ) tr(Y12 Y2 ) tr(Y12 Y2 ) tr(Y23 ) tr(Y12 ) tr(Y1 Y2 ) tr(Y22 ) tr(Y12 Y2 ) tr(Y1 Y22 ) , W3′ = U tr(Y13 ) tr(Y12 Y2 ) tr(Y1 Y22 ) tr(Y23 )

tr(Y12 Y2 ) tr(Y1 Y22 )

,

where U, V are defined above. Finally, we define one more element W3′′ as follows. Recall that a linear mapping δ of an algebra R is a derivation if δ(rs) = δ(r)s+rδ(s) for all r, s ∈ R. We consider the derivation δ of C32 which commutes with the trace and satisfies the conditions δ(X1 ) = 0, Then

δ(X2 ) = X1 ,

δ(Y1 ) = 0,

δ(Y2 ) = Y1 .

6

W3′′ =

1 X (−1)i δ i (tr3 (Y22 ))δ 6−i (tr2 (Y23 )). 144 i=0

The following theorem gives the defining relation of C32 . It uses representation theory of GL2 , combinatorics, computations by hand and easy computer calculations with standard functions of Maple. Theorem 3.6. (Aslaksen, Drensky, Sadikova [ADS]) The algebra of invariants C32 of two 3×3 matrices is generated by the elements from the previous theorem, subject to the defining relation 2 4 1 1 2 1 4 1 W1 − W2 + W3′ + W3′′ + W4 − W5 − W6 − W7 = 0. W2 − 27 9 15 90 3 3 3 27 The calculation of the Hilbert series of Cnd and Tnd based on the Molien-Weyl formula is quite complicated because requires evaluations of multiple integrals. Van den Bergh [VB2] sujected a way which involves graph theory. As a consequence, he established important properties of H(C2d , t1 , . . . , td ) and H(T2d , t1 , . . . , td ).

18

VESSELIN DRENSKY

Berele and Stembridge [BS] applied the method of van den Bergh [VB2] and calculated the Hilbert series of T32 . Using the above results of Aslaksen, Drensky, and Sadikova on C32 and the explicit form of the Hilbert series of T32 , Benanti and Drensky [BD] found a polynomial subalgebra S of C32 and a finite set of generators of the free S-module T32 . They gave also a set of defining relations of T32 as an algebra and a Gr¨obner basis of the corresponding ideal. (See the survey article by Ufnarovski [U] for a background on Gr¨obner bases in the noncommutative case, as well as the paper by Mikhalev and Zolotykh [MZ] which is closer to the situation in [BD].) For two generic 4 × 4 matrices, the Hilbert series of C42 was calculated (with some typos) by Teranishi [T1, T2] and corrected by Berele and Stembridge [BS]. Teranishi found also a homogeneous system of parameters and a system of generators, in the spirit of the 3 × 3 case. Recently, Drensky and Sadikova [DS] have found another system of generators of C42 which is minimal and seems to be more convenient for concrete calculations. The Hilbert series of a graded vector space with GLd -module structure determines uniquely its decomposition into irreducible submodules. Hence, in principle, one may calculate the multiplicities m(λ) if one knows the concrete form of the Hilbert series. Berele [B1] used the Hilbert series of C32 found by Teranishi [T1] and described the asymptotics of m(λ1 , λ2 ). (Due to a technical error (an omitted summand) some of the coefficients of the polynomials in the asymptotics of Berele are slightly different from the real ones.) Another approach to the problem was suggested by Drensky and Genov [DG1]. Let X f (t1 , t2 ) = aij ti1 tj2 , i,j≥0

aij ∈ K, aij = aji , be a symmetric function in two variables which is a formal power series from K[[t1 , t2 ]]. We present it in the form X m(λ1 , λ2 )S(λ1 ,λ2 ) (t1 , t2 ) f (t1 , t2 ) = λ1 ≥λ2

and want to find the multiplicities m(λ1 , λ2 ). In most of the cases which we consider, f (t1 , t2 ) is given explicitly as a rational function. So, it is natural to express m(λ1 , λ2 ) not in terms of the coefficients aij but in a more direct way. We introduce the generating function of the multiplicities X m(λ1 , λ2 )tλ1 uλ2 ∈ K[[t, u]] M (f, t, u) = λ1 ≥λ2

and call it the multiplicity series of f (t1 , t2 ). It is more convenient to introduce a new variable v = tu and to consider the series X m(λ1 , λ2 )tλ1 −λ2 v λ2 ∈ K[[t, v]], M ′ (f, t, v) = λ1 ≥λ2

because the mapping M ′ : K[[t1 , t2 ]]S2 → K[[t, v]] is a bijective linear mapping which is continuous with respect to the formal power series topology. It is easy to see that f (t1 , t2 ) and M ′ (f, t, v) are related by f (t1 , t2 ) =

t1 M ′ (f, t1 , t1 t2 ) − t2 M ′ (f, t2 , t1 t2 ) . t1 − t2

COMPUTING WITH MATRIX INVARIANTS

19

Hence, if we have a potential candidate h(t, v) for M ′ (f, t, v), it is easy to verify whether h(t, v) = M ′ (f, t, v). Also, the elementary symmetric function e2 = t1 t2 behaves like a constant, M ′ (g(t1 t2 )f (t1 , t2 ), t, v) = g(v)M ′ (f, t, v), and this simplifies the calculations. Applying quite complicated (also technically) arguments, Drensky and Genov [DG1] found the multiplicity series of the Hilbert series of C32 . They also corrected the technical errors in [B1]. Theorem 3.7. (i) [DG1] The multiplicity series of the Hilbert series of the algebra C32 of invariants of two 3 × 3 matrices is 1 × M ′ (H(C32 , t1 , t2 ), t, v) = 2 (1 − v )(1 − v 3 )2 (1 + v 2 + v 4 )((1 + v 2 )(1 − t2 v) + 2tv(1 − v)) + × 3(1 − v)(1 − v 2 )3 (1 − t)2 (1 − t2 ) (1 − v 2 )(1 − tv) (1 − v)(1 + tv) + + 3(1 − v 2 )(1 − t)(1 − t2 ) 3(1 − v 3 )(1 − t3 ) v 3 ((1 − v + v 2 )(1 − t2 v 2 ) + tv(1 − v 2 )) . − (1 − v)(1 − v 2 )2 (1 − v 4 )(1 − t)(1 − t2 )(1 − tv) (ii) [B1, DG1] Let λ = (p, q) and let m(p, q) be the multiplicity of S(p,q) (t1 , t2 ) in H(C32 , t1 , t2 ). Then for p > 2q ≥ 0 q7 (p − q)q 6 (p − q)2 q 5 + + + O((p + q)6 ) 5 2 4 2 7!2 .3 6!2 .3 2!5!23 32 p2 q 5 11pq 6 71q 7 = − + + O((p + q)6 ); 17280 103680 1451520

m(p, q) =

for 2q ≥ p ≥ q ≥ 0 q7 (p − q)q 6 (p − q)2 q 5 (2q − p)7 + + − + O((p + q)6 ) 5 2 4 2 3 2 7!2 .3 6!2 .3 2!5!2 3 7!25 .32 p7 p6 q p5 q 2 p4 q 3 p3 q 4 7p2 q 5 7pq 6 19q 7 = − + − + − + − +O((p+q)6 ). 1451520 103680 17280 5184 2592 17280 34560 483840 Later the methods for calculating the multiplicity series of symmetric functions of special kinds were significantly improved [DG2]. The Hilbert series of C42 calculated by Teranishi [T1, T2] (with some typos corrected in [BS]) and the Hilbert series of T32 and T42 calculated by Berele and Stembridge [BS] allowed to express their multiplicity series and to determine the asympotics of the multiplicities. We shall state simplified versions of the results: m(p, q) =

Theorem 3.8. (i) (Drensky, Genov, Valenti [DGV]) The multiplicities m(λ1 ,λ2 ) (C32 ) and m(λ1 ,λ2 ) (T32 ) of the Hilbert series of C32 and T32 , respectively, are related by m(λ1 ,λ2 ) (T32 ) ≈ 9m(λ1 ,λ2 ) (C32 ). (ii)( Drensky and Genov [DGV]) Let λ = (λ1 , λ2 ). The multiplicities mλ (C42 ) of the Hilbert series of C42 satisfy the condition 13 if λ1 > 3λ2 , m1 + O((λ1 + λ2 ) ), mλ (C42 ) = m1 + m2 + O((λ1 + λ2 )13 ), if 3λ2 ≥ λ1 > 2λ2 , 13 m1 + m2 + m3 + O((λ1 + λ2 ) ), if 2λ2 ≥ λ1 ,

20

VESSELIN DRENSKY

where m1 =

(λ1 − λ2 )2 λ12 127(λ1 − λ2 )λ13 305λ14 (λ1 − λ2 )3 λ11 2 2 2 2 − + − , 8 2 8 3 10 4 11!3!2 3 12!2!2 3 13!2 3 14!29 35

(3λ2 − λ1 )14 , 14!210 35 52 7(2λ2 − λ1 )14 (λ1 − λ2 )(2λ2 − λ1 )13 . − m3 = − 13!210 32 5 14!293 · 52 The multiplicities mλ (T42 ) satisfy m2 =

mλ (T42 ) = 16mλ (C42 ) + O((λ1 + λ2 )13 ). We want to mention that Berele and Stembridge [BS] computed also the Hilbert series of C33 and T33 but the methods of [DG1, DG2, DG3, DGV] do not work successfully for symmetric functions in three variables. One can introduce the multiplicity series of a symmetric function in any number of variables, generalizing in an obvious way the case of symmetric functions in two variables. A recent theorem of Berele [B2] gives the rationality of the multiplicity series of a class of rational symmetric functions in any number of variables, including the Hilbert series of Cnd and Tnd . Unfortunately, it is not clear how to perform the concrete calculations, even for the Hilbert series of C33 and T33 . References [AP] S. Abeasis, M. Pittaluga, On a minimal set of generators for the invariants of 3 × 3 matrices, Commun. Algebra 17 (1989), 487-499. [AL] W.W. Adams, P. Loustaunau, An Introduction to Gr¨ obner Bases, Graduate Studies in Math. 3, AMS, Providence, R.I., 1994. [ADS] H. Aslaksen, V. Drensky, L. Sadikova, Defining relations of invariants of two 3×3 matrices, J. Algebra (to appear). Announcement: C.R. Acad. Bulg. Sci. 58 (2005), No. 6, 617-622. [BD] F. Benanti, V. Drensky, Defining relations of noncommutative trace algebra of two 3 × 3 matrices, Adv. Appl. Math. (to appear). Preprint, http://xxx.lanl.gov/abs/math.RA/0501219. [B1] A. Berele, Approximate multiplicities in the trace cocharacter sequence of two three-by-three matrices, Commun. Algebra 25 (1997), 1975-1983. [B2] A. Berele, Applications of Belov’s theorem to the cocharacter sequence of P.I. algebras, J. Algebra (to appear). [BS] A. Berele, J.R. Stembridge, Denominators for the Poincar´ e series of invariants of small matrices, Israel J. Math. 114 (1999), 157-175. [Ch] C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 67 (1955), 778-782. [DC] J.A. Dieudonn´ e, J.B. Carrell, Invariant Theory, Old and New, Academic Press, New YorkLondon, 1971. [DD] M. Domokos, V. Drensky, Gr¨ obner bases for the rings of invariants of special orthogonal ans 2 × 2 matrix invariants, J. Algebra 243 (2001), 706-716. [D1] V. Drensky, Codimensions of T-ideals and Hilbert series of relatively free algebras, J. Algebra 91 (1984), 1-17. [D2] V. Drensky, Free Algebras and PI-Algebras, Springer-Verlag, Singapore, 1999. [D3] V. Drensky, Defining relations for the algebra of invariants of 2 × 2 matrices, Algebras and Representation Theory 6 (2003), 193-214. [DF] V. Drensky, E. Formanek, Polynomial Identity Rings, Advanced Courses in Mathematics, CRM Barcelona, Birkh¨ auser, Basel-Boston, 2004. [DG1] V. Drensky, G.K. Genov, Multiplicities of Schur functions in invariants of two 3×3 matrices, J. Algebra 264 (2003), No. 2, 496-519. [DG2] V. Drensky, G.K. Genov, Multiplicities of Schur functions with applications to invariant theory and PI-algebras, C.R. Acad. Bulg. Sci. 57 (2004), No. 3, 5-10.

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[DG3] V. Drensky, G.K. Genov, Multiplicities in the trace cocharacter sequence of two 4 × 4 matrices, Mediterr. J. Math. 2 (2005), 231-241. [DGV] V. Drensky, G.K. Genov, A. Valenti, Multiplicities in the mixed trace cocharacter sequence of two 3 × 3 matrices, International J. Algebra and Computations (to appear). Preprint, http://xxx.lanl.gov/abs/math.RA/0411082. [DK] V. Drensky, P. Koshlukov, Weak polynomial identities for a vector space with a symmetric bilinear form, in “Math. and Education in Math. Proc. of 16-th Spring Conf. of the Union of Bulg. Mathematicians”, Sofia, Publ. House of the Bulg. Acad. of Sci., 1987, 213-219. [DS] V. Drensky, L. Sadikova, Generators of invariants of two 4 × 4 matrices, preprint, http://xxx.lanl.gov/abs/math.RA/0503146. [Du] J. Dubnov, Sur une g´ en´ eralisation de l’´ equation de Hamilton-Cayley et sur les invariants simultan´ es de plusieurs affineurs, Proc. Seminar on Vector and Tensor Analysis, Mechanics Research Inst., Moscow State Univ. 2/3 (1935), 351-367 (see also Zbl. f¨ ur Math. 12 (1935), p. 176). [DI] J. Dubnov, V. Ivanov, Sur l’abaissement du degr´ e des polynˆ omes en affineurs, C.R. (Doklady) Acad. Sci. USSR 41 (1943), 96-98. (See also MR 6 (1945), p. 113, Zbl. f¨ ur Math. 60 (1957), p. 33). [F1] E. Formanek, Invariants and the ring of generic matrices, J. Algebra 89 (1984), 178-223. [F2] E. Formanek, Generating the ring of matrix invariants, Lect. Notes in Math. 1195, Springer Verlag, Berlin-Heidelberg-New York, 1986, 73-82. [F3] E. Formanek, Functional equations for character series associated with n × n matrices, Trans. Amer. Math. Soc. 294 (1986), 647-663. [F4] E. Formanek, The Polynomial Identities and Invariants of n × n Matrices, CBMS Regional Conf. Series in Math. 78, Published for the Confer. Board of the Math. Sci. Washington DC, AMS, Providence RI, 1991. [F5] E. Formanek, The ring of generic matrices, J. Algebra 258 (2002), 310-320. [FHL] E. Formanek, P. Halpin, W.-C.W. Li, The Poincar´ e series of the ring of 2 × 2 generic matrices, J. Algebra 69 (1981), 105-112. [Fr] G. Freudenburg, A survey of counterexamples to Hilbert’s fourteenth problem, Serdica Math. J. 27 (2001), 171-192. [Gu] B. G. Gurevich, Foundations of the Theory of Algebraic Invariants, Noordhoff, Groningen, 1964. [Hi] G. Higman, On a conjecture of Nagata, Proc. Camb. Philos. Soc. 52 (1956), 1-4. ¨ [H1] D. Hilbert, Uber die Theorie der algebraischen Formen, Math. Ann. 36 (1890), 473-534. Reprinted in “Gesammelte Abhandlungen, Band II, Algebra, Invariantentheorie, Geometrie”, Zweite Auflage, Springer-Verlag, Berlin-Heidelberg-New York, 1970, 199-257. [H2] D. Hilbert, Mathematische Probleme, Archiv f. Math. u. Phys. 1 (1901), 44-63, 213-237. Reprinted in “Gesammelte Abhandlungen, Band III, Analysis, Grundlagen der Mathematik, Physik, Verschiedenes, Lebensgeschichte”, Zweite Auflage, Springer-Verlag, BerlinHeidelberg-New York, 1970, 290-329. [J] N. Jacobson, PI-Algebras: An Introduction, Lecture Notes in Math. 441, Springer-Verlag, Berlin-New York, 1975. [KBR] A. Kanel-Belov, L.H. Rowen, Computational Aspects of Polynomial Identities, Research Notes in Mathematics 9, A.K. Peters, Wellesley, MA, 2005. [Ki] A. A. Kirillov, Certain division algebras over a field of rational functions (Russian), Funkts. Anal. Prilozh. 1 (1967), No. 1, 101-102. Translation: Funct. Anal. Appl. 1 (1967), 87-88. [Ku] E.N. Kuzmin, On the Nagata-Higman theorem (Russian), in “Mathematical Structures, Computational Mathematics, Mathematical Modelling. Proc. Dedicated to the 60th Birthday of Acad. L. Iliev”, Sofia, 1975, 101-107. [LB1] L. Le Bruyn, The functional equation for Poincar´ e series of trace rings of generic 2 × 2 matrices, Israel J. Math. 52 (1985), 355-360. [LB2] L. Le Bruyn, Trace Rings of Generic 2 by 2 Matrices, Memoirs of AMS, 66, No. 363, Providence, R.I., 1987. [Mc] I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Univ. Press (Clarendon), Oxford, 1979. Second Edition, 1995. [MZ] A.A. Mikhalev, A.A. Zolotykh, Standard Gr¨ obner-Shirshov bases of free algebras over rings. I: Free associative algebras, Int. J. Algebra Comput. 8 (1998), No. 6, 689-726.

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¨ [Mo] T. Molien, Uber die Invarianten der linearen Substitutionsgruppen, Sitz. K¨ onig Preuss. Akad. Wiss. (1897), N 52, 1152-1156. [Na1] M. Nagata, On the nilpotency of nil algebras, J. Math. Soc. Japan 4 (1953), 296-301. [Na2] M. Nagata, On the 14th problem of Hilbert, Amer. J. Math. 81 (1959), 766-772. [N] K. Nakamoto, The structure of the invariant ring of two matrices of degree 3, J. Pure Appl. Algebra 166 (2002), No. 1-2, 125-148. [No] E. Noether, Der Endlichkeitssatz der Invarianten endlicher Gruppen, Math. Ann. 77 (1916), 89-92. Reprinted in “Gesammelte Abhandlungen. Collected Papers”, Springer-Verlag, BerlinHeidelberg-New York-Tokyo, 1983, 181-184. [P1] C. Procesi, Rings with Polynomial Identities, Marcel Dekker, New York, 1973. [P2] C. Procesi, The invariant theory of n × n matrices, Adv. Math. 19 (1976), 306-381. [P3] C. Procesi, Trace indentities and standard diagrams, in “Ring Theory”, Lect. Notes in Math. 51, Marcel Dekker, New York, 1979, 191-218. [P4] C. Procesi, Computing with 2 × 2 matrices, J. Algebra 87 (1984), 342-359. [R1] Yu.P. Razmyslov, Finite basing of the identities of a matrix algebra of second order over a field of characteristic 0 (Russian), Algebra i Logika 12 (1973), 83-113. Translation: Algebra and Logic 12 (1973), 43-63. [R2] Yu.P. Razmyslov, Trace identities of full matrix algebras over a field of characteristic zero (Russian), Izv. Akad. Nauk SSSR, Ser. Mat. 38 (1974), 723-756. Translation: Math. USSR, Izv. 8 (1974), 727-760. [R3] Yu. P. Razmyslov, Identities of Algebras and Their Representations (Russian), “Sovremennaya Algebra”, “Nauka”, Moscow, 1989. Translation: Translations of Math. Monographs 138, AMS, Providence, R.I., 1994. [Re] A. Regev, On the codimensions of matrix algebras, in “Algebra – Some Current Trends (Varna, 1986)”, Lect. Notes in Math. 1352, Springer-Verlag, Berlin-New York, 1988, 162172. [ST] G.C. Shephard, J.A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274304. [SZ] I.P. Shestakov, N. Zhukavets, On associative algebras satisfying the identity x5 = 0, Algebra Discrete Math. (2004) No. 1, 112-120. [Si] K. S. Sibirskii, Algebraic invariants for a set of matrices (Russian), Sib. Mat. Zh. 9 (1968), No. 1, 152-164. Translation: Siber. Math. J. 9 (1968), 115-124. [Sp] T.A. Springer, Invariant Theory, Lect. Notes in Math. 585, Springer-Verlag, BerlinHeidelberg-New York, 1977. [St1] R. P. Stanley, Combinatorics and invariant theory, in “Relations between Combinatorics and Other Parts of Mathematics” Ed. D. K. Ray-Chaudhuri, Proc. Sympos. Pure Math., 34, AMS, Providence, RI, 1979, 345-355. [St2] R.P. Stanley, Combinatorics and Commutative Algebra, Progress in Math. 41, Birkh¨ auser Boston, Boston, MA, 1983, Second Edition, 1996. [St] B. Sturmfels, Algorithms in Invariant Theory, Texts and Monographs in Symbolic Computation, Springer-Verlag, Wien, 1993. [T1] Y. Teranishi, The ring of invariants of matrices, Nagoya Math. J. 104 (1986), 149-161. [T2] Y. Teranishi, Linear diophantine equations and invariant theory of matrices, “Commut. Algebra and Combinatorics (Kyoto, 1985)”, Adv. Stud. Pure Math. 11, North-Holland, Amsterdam-New York, 1987, 259-275. [T3] Y. Teranishi, The Hilbert series of matrix concominants, Nagoya Math. J. 111 (1988), 143156. [U] V.A. Ufnarovski, Combinatorial and asymptotic methods in algebra, in A.I. Kostrikin, I.R. Shafarevich (Eds.), “Algebra VI”, Encyclopedia of Math. Sciences 57, Springer-Verlag, 1995, 1-196. [VB1] M. Van den Bergh, Trace rings are Cohen-Macaulay, J. Amer. Math. Soc. 2 (1989), 775-799. [VB2] M. Van den Bergh, Explicit rational forms for the Poincar´ e series of the trace rings of generic matrices, Isr. J. Math. 73 (1991), 17-31. [VL] M. R. Vaughan-Lee, An algorithm for computing graded algebras, J. Symbolic Comput. 16 (1993), 345-354. [W1] H. Weyl, Zur Darstellungstheorie und Invariantenabz¨ ahlung der projektiven, der Komplex- und der Drehungsgruppe, Acta Math. 48 (1926), 255-278. Reprinted in “Gesammelte Abhandlungen”, Band III, Springer-Verlag, Berlin-Heidelberg-New York, 1968, 1-25.

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[W2] H. Weyl, The Classical Groups, Their Invariants and Representations, Princeton Univ. Press, Princeton, N.J., 1946, New Edition, 1997. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria E-mail address: [email protected]

COMPUTING WITH MATRIX INVARIANTS VESSELIN DRENSKY Abstract. We present an introduction to the theory of the invariants under the action of GLn (C) by simultaneous conjugation of d matrices of size n × n. Then we survey some results, old or recent, obtained by a dozen of mathematicians, on minimal sets of generators, the defining relations of the algebras of invariants and on the multiplicities of the Hilbert series of these algebras. The picture is completely understood only in the case n = 2. Besides, explicit minimal sets of generators are known for n = 3 and any d and for n = 4, d = 2. The multiplicities of the Hilbert series are obtained only for n = 3, 4 and d = 2. For n > 2 most of the concrete results are obtained with essential use of computers.

1. Introduction to invariant theory All considerations in this paper are over an arbitrary field K of characteristic 0. If not explicitly stated, all vector spaces and algebras are over K, and the algebras are unitary and commutative. To get some idea about invariant theory, we start with the following well known result from the undergraduate algebra course. Let n ≥ 2 be an integer and let A = K[X] = K[x1 , . . . , xn ] be the algebra of polynomials in n variables. The algebra of symmetric polynomials ASn = K[X]Sn consists of all polynomials f (x1 , . . . , xn ) ∈ A such that f (xσ(1) , . . . , xσ(n) ) = f (x1 , . . . , xn ) for all permutations σ in the symmetric group Sn . Theorem 1.1. (i) The algebra ASn is generated by the elementary symmetric functions X e1 = x1 + · · · + xn = xi , X xi xj , e2 = x1 x2 + x1 x3 + · · · + xn−1 xn = i n. Much later it was discovered that this theorem was first established in 1943 by Dubnov and Ivanov [DI] but their paper was overlooked by the mathematical community. The class of nilpotency N (n) in the Nagata-Higman theorem is related in the following nice way with invariant theory of matrices. Theorem 2.7. (Formanek [F2], Procesi [P2, P3], Razmyslov [R2]) Let N (n) be the class of nilpotency in the Nagata-Higman theorem. Then the algebra of invariants n ΩGL is generated by the traces tr(Xi1 · · · Xim ) of degree ≤ N (n). For d sufficiently nd large this bound is sharp. It is important to know the exact value of the class of nilpotency N (n) in the Nagata-Higman theorem. The upper bound given in the proof of Higman [Hi] is N (n) ≤ 2n − 1. The best known upper bound is due to Razmyslov [R2]. Applying trace polynomial identities of matrices, he obtained the bound N (n) ≤ n2 . The proof of the theorem of Razmyslov is given also in his book [R3] or in the book by Formanek [F4]. For a lower bound, Kuzmin [Ku] showed that N (n) ≥ 21 n(n + 1). A proof of the result of Kuzmin may be found also in the books by Drensky and Formanek [DF] or by Kanel-Belov and Rowen [KBR]. Hence n(n + 1) ≤ N (n) ≤ n2 . 2 Problem 2.8. Find the exact value N (n) of the class of nilpotency of nil algebras of index n over a field of characteristic 0. Conjecture 2.9. (Kuzmin [Ku]) The exact value N (n) of the class of nilpotency of nil algebras of index n over a field of characteristic 0 is N (n) =

n(n + 1) . 2

The only values of N (n) are known for n ≤ 4: Dubnov [Du] obtained in 1935 N (1) = 1,

N (2) = 3,

In 1993, Vaughan-Lee [VL] proved that N (4) = 10.

N (3) = 6.

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In this way the conjecture of Kuzmin is confirmed for n ≤ 4. Recently, Shestakov and Zhukavets [SZ] have proved that the class of nilpotency of the two-generated algebras satisfying the identity x5 = 0 is equal to 15, which agrees with the conjecture of Kuzmin for n = 5. They have obtained the same result also in the more general setup of 2-generated superalgebras. Their proof is based on computer calculations with the GAP package. 2 Since N (2) = 3, the algebra ΩGL is generated by products of traces of degree 2d ≤ 3. The following result was established by Sibirskii [Si]. Theorem 2.10. The elements tr(Xi ), tr(Xi2 ),

i = 1, . . . , d,

tr(Xi Xj Xk ),

tr(Xi Xj ),

1 ≤ i < j ≤ d,

1 ≤ i < j < k ≤ d,

constitute a minimal set of generators of the algebra of 2 × 2 matrix invariants 2 ΩGL 2d . Now we give an idea about the Razmyslov-Procesi theory which is related with the second fundamental theorem of the matrix invariants, see Razmyslov [R2] and Procesi [P2], as well as the book by Razmyslov [R3] for other applications of his method. For simplicity we consider the case n = 2 only. The Cayley-Hamilton theorem for 2 × 2 matrices implies that X 2 − tr(X)X + det(X) = 0. The Newton formulas give that 1 2 (tr (X) − tr(X 2 )). 2 This can be seen also directly. If ξ1 , ξ2 are the eigenvalues of X, then det(X) =

tr(X) = ξ1 + ξ2 ,

tr(X 2 ) = ξ12 + ξ22 ,

1 1 ((ξ1 + ξ2 )2 − (ξ12 + ξ22 )) = (tr2 (X) − tr(X 2 )). 2 2 In this way we obtain the mixed trace identity det(X) = ξ1 ξ2 =

1 c(X) = X 2 − tr(X)X + (tr2 (X) − tr(X 2 )) = 0. 2 Now we consider the identity c(X1 + X2 ) − c(X1 ) − c(X2 ) = 0, i.e. we linearize the identity c(X) = 0. In this way we obtain the mixed Cayley-Hamilton identity Ψ2 (X1 , X2 ) = X1 X2 +X2 X1 −tr(X1 )X2 −tr(X2 )X1 +tr(X1 )tr(X2 )−tr(X1 X2 ) = 0. Since the trace is a nondegenerate bilinear form on M2 (K), the vanishing of the polynomial Ψ2 (X1 , X2 ) on M2 (K) is equivalent to the vanishing of the pure CayleyHamilton identity Φ2 (X1 , X2 , X3 ) = tr(Ψ2 (X1 , X2 )X3 ) = 0 on all 2 × 2 matrices. Direct calculations show that 0 = Φ2 (X1 , X2 , X3 ) = tr(Ψ2 (X1 , X2 )X3 ) = tr(X1 X2 X3 ) + tr(X2 X1 X3 ) −tr(X1 )tr(X2 X3 ) − tr(X2 )tr(X1 X3 ) + tr(X1 )tr(X2 )tr(X3 ) − tr(X1 X2 )tr(X3 ).

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If we delete the symbols of traces and the X’s in the above expression, we shall obtain the following linear combination of permutations X (123) + (213) − (1)(23) − (2)(13) + (1)(2)(3) − (12)(3) = sign(σ). σ∈S3

This suggests the following construction. We write the permutations in the symmetric group Sm as products of disjoint cycles, including the cycles of length 1, σ = (i1 . . . ip )(j1 . . . jq ) · · · (k1 . . . kr ). We define the associated trace function trσ (x1 , . . . , xm ) = tr(xi1 · · · xip )tr(xj1 · · · xjq ) · · · tr(xk1 · · · xkr ). For every element X

ασ σ ∈ KSm ,

ασ ∈ K, σ ∈ Sm ,

σ∈Sm

where KSm is the group algebra of Sm , we define the trace polynomial X f (x1 , . . . , xm ) = ασ trσ (x1 , . . . , xm ). σ∈Sm

We also assume that for m ≤ k the symmetric group Sm acts on 1, . . . , m and leaves invariant m + 1, . . . , k, i.e. Sm is canonically embedded into Sk . Theorem 2.11. (The Second Fundamental Theorem of Matrix Invariants, Razmyslov [R2], Procesi [P2]) Let X f (x1 , . . . , xm ) = ασ trσ (x1 , . . . , xm ), ασ ∈ K, σ∈Sm

be a multilinear trace polynomial of degree m. Then f = 0 is a trace identity for the n × n matrix algebra, i. e. f (a1 , . . . , am ) = 0 for all a1 , . . . , am ∈ Mn (K), if and only if X ασ σ σ∈Sm

belongs to the two-sided ideal J(n, m) of the group algebra KSm generated by the element X sign(σ)σ. σ∈Sn+1

As in the case of 2 × 2 matrices, the fundamental trace identity X (sign σ)trσ (x1 , . . . , xn+1 ) = 0 σ∈Sn+1

is actually the linearization of the Cayley-Hamilton polynomial. There are several important objects related with invariant theory of matrices. As above, n, d ≥ 2 are fixed integers and X1 , . . . , Xd are d generic n × n matrices: The algebra Rnd generated by X1 , . . . , Xd ; n The pure (or commutative) trace algebra Cnd = ΩGL generated by the traces of nd all products tr(Xi1 · · · Xik ); The mixed (or noncommutative) trace algebra Tnd generated by Rnd and Cnd regarding the elements of Cnd as scalar matrices; The field of fractions Q(Cnd ) of the algebra Cnd ; The algebra Q(Cnd )Rnd .

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All these algebras have no zero divisors and play important roles in mathematics. See the books [DF, F4, J, P1] for different aspects of the theory of algebras of matrix invariants and their applications to combinatorial and structure theory of PI-algebras, central division algebras, etc. The algebra Rnd is a well known object in the theory of PI-algebras, or algebras with polynomial identities. Let Khx1 , . . . , xd i be the free associative algebra (i.e. the algebra of polynomials in noncommuting variables). A polynomial f (x1 , . . . , xd ) ∈ Khx1 , . . . , xd i is a polynomial identity in d variables for Mn (K) if f (a1 , . . . , ad ) = 0 for all a1 , . . . , ad ∈ Mn (K). The set I(Mn (K)) of all polynomial identities in d variables is a two-sided ideal of Khx1 , . . . , xd i and Rnd is isomorphic to the factor algebra Khx1 , . . . , xd i/I(Mn (K)). Clearly, the algebra Cnd is the algebra of matrix invariants. The algebra Tnd is also the algebra of invariant polynomial functions under a suitable action of GLn . It is called the algebra of matrix concominants. It is a finitely generated Cnd -module and, as a Cnd -module, has a generating set consisting of products Xj1 · · · Xjk , where k < N (n), the class of nilpotency in the NagataHigman theorem. The field of fractions Q(Cnd ) appears naturally in field theory. One of the main problems related with Q(Cnd ) is whether it is a purely transcendent extension of K. Finally, Q(Cnd )Rnd is a central division algebra of dimension n2 over its centre Q(Cnd ) and serves as a source of counterexamples to the theory of central division algebras. General invariant theory gives that Cnd and Tnd have nice algebraic properties. Theorem 2.12. (Van den Bergh, [VB1]) The algebra Cnd is a Cohen-Macaulay and even Gorenstein unique factorization domain. The algebra Tnd is a CohenMacaulay module over Cnd . Recall that the Noether normalization theorem gives that Cnd contains a homogeneous set of algebraically independent elements {a1 , . . . , ak }, where k = (d−1)n2 +1 is the transcendence degree of the quotient field Q(Cnd ) (k is also equal to the Krull dimension of Cnd ), such that Cnd is integral over the polynomial algebra K[a1 , . . . , ak ]. Such a set {a1 , . . . , ak } is called a homogeneous system of parameters for Cnd . By a result of Stanley [St1], a graded Cnd -module is Cohen-Macaulay if and only if it is a free module with respect to some homogeneous system of parameters {a1 , . . . , ak } of Cnd . The algebras Rnd , Cnd , and Tnd are multigraded and their homogeneous components of degree (k1 , . . . , kd ) consist of all polynomials which are homogeneous of degree ki in the generic matrix Xi . Hence we may consider their Hilbert series in d variables. For example, X (k ,...,kd ) H(Cnd , t1 , . . . , td ) = dim Cnd1 tk11 · · · tkdd , (k ,...,k )

d where Cnd1 is the homogeneous component of degree (k1 , . . . , kd ). Le Bruyn [LB1] for the case n = 2, and Formanek [F3] and Teranishi [T1, T3] in the general case proved:

Theorem 2.13. Let d ≥ 2 for n ≥ 3 and d > 2 for n = 2. Then H(Cnd , t1 , . . . , td ) and H(Tnd , t1 , . . . , td ) satisfy the functional equation 2

−1 k n H(Cnd , t−1 1 , . . . , td ) = (−1) (t1 · · · td ) H(t1 , . . . , td ),

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VESSELIN DRENSKY

where k = (d − 1)n2 + 1 is the Krull dimension of Cnd , and similarly for the Hilbert series of Tnd . The proofs of this theorem given by Formanek and Teranishi are quite different and use, respectively, representation theory of general linear groups and the MolienWeyl integral formula. Later, Van den Bergh paid attention that the proof can be considerably simplified using results of Stanley on Hilbert series of Cohen-Macaulay algebras. We need some background on symmetric polynomials and representation theory of the general linear group, see e.g. the books by Weyl [W2] and Macdonald [Mc]. As in the case of polynomial algebras, the general linear group GLd acts canonically on the vector space with basis {X1 , . . . , Xd }. If g = (gij ), gij ∈ K, then the action of g on Xj is defined by g(Xj ) = g1j X1 + · · · + gdj Xd . This action is extended diagonally on Rnd , Cnd , Tnd . If f (X1 , . . . , Xd ) is any polynomial expression depending on X1 , . . . , Xd (maybe including also traces), then g(f (X1 , . . . , Xd )) = f (g(X1 ), . . . , g(Xd )),

g ∈ GLd .

Representation theory of GLd says that every submodule of the GLd -modules Rnd , Cnd , Tnd is a direct sum of irreducible (or simple) submodules. The irreducible GLd -submodules which appear in the decomposition are polynomial modules and are described in terms of partitions of integers. If λ = (λ1 , . . . , λd ),

λ1 ≥ · · · ≥ λd ≥ 0,

is a partition of k (notation λ ⊢ k) in not more than d parts, then we denote the related GLd -module by W (λ) = W (λ1 , . . . , λd ). To be explicit, below we consider the case of Cnd only. If M Cnd = m(λ)W (λ), m(λ) ≥ 0, i.e. there are m(λ) direct summands isomorphic to W (λ), then we say that W (λ) appears with multiplicity m(λ). The multiplicities m(λ) for Rnd , Cnd , and Tnd play important role in the quantitative study of polynomial identities of matrices. (See the survey by Regev [Re] and the book of the author [D2] for applications of representation theory of Sn and GLd to the theory of PI-algebras.) The Hilbert series of Cnd has the form X H(Cnd , t1 , . . . , td ) = m(λ)Sλ (t1 , . . . , td ), where Sλ (t1 , . . . , td ) is the Schur function associated to λ. Schur functions are important combinatorial objects and appear in many places in mathematics. For example, they form a basis of the vector space of all symmetric polynomials in d variables. One of the possible ways to define Schur functions is via Vandermondelike determinants. For a partition µ = (µ1 , . . . , µd ), define the determinant µ1 µ1 µ1 t1 µ2 tµ2 2 · · · tµd 2 t1 · · · td t2 V (µ1 , . . . , µd ) = . . .. . .. .. .. . . µ t d tµd · · · tµd 1 2 d

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Then the Schur function is V (λ1 + d − 1, λ2 + d − 2, . . . , λd−1 + 1, λd ) . V (d − 1, d − 2, . . . , 1, 0) The Schur functions play the role of characters of the corresponding representation of GLd . If we know the Hilbert series H(Cnd , t1 , . . . , td ), then we can uniquely determine the GLd -module structure of Cnd . Sλ (t1 , . . . , td ) =

3. Concrete computations We give a survey of some concrete results about generators, defining relations and Hilbert series of the algebras of matrix invariants. We start with 2×2 matrices. The first important reduction is the following. We take the generic 2 × 2 matrix X and present it in the form x11 +x22 x11 −x22 x11 x12 0 x12 2 2 X= = + x11 +x22 22 x21 x22 0 x21 − x11 −x 2 2 tr(X) tr(X) y y12 E + 11 = E + Y, = y21 −y11 2 2 where E is the identity matrix and y11 y12 Y = y21 −y11 is a generic 2 × 2 traceless matrix. This reduction implies that C2d is generated by tr(Xi ), i = 1, . . . , d, and tr(Yi1 · · · Yik ), where Y1 , . . . , Yd are generic traceless matrices and k ≥ 2. Since the class of nilpotency in the Nagata-Higman theorem is N (2) = 3 for n = 2, we obtain that it is sufficient to consider the cases k = 2, 3 only. Similarly, T2d is generated by C2d and Y1 , . . . , Yd . It is also easy to see: Proposition 3.1. The algebra C2d has the presentation C2d = (K[tr(X1 ), . . . , tr(Xd )]) ⊗K C0 , where C0 = K[tr(Yi Yj ), tr(Yp Yq Yr ) | 1 ≤ i ≤ j ≤ d, 1 ≤ p < q < r ≤ d]/I is the algebra generated by products of generic traceless 2 × 2 matrices. Hence we may choose all defining relations as linear combinations of the traces of products of traceless matrices. The description of T2d is easier than that of C2d . Theorem 3.2. (i) (Procesi [P4]) The noncommutative trace algebra T2d is isomorphic to the tensor product of K-algebras K[tr(X1 ), . . . , tr(Xm )] ⊗K Wd , where Wd is the associative algebra generated by the generic traceless matrices Y1 , . . . , Ym . (ii) (Razmyslov [R1]) The algebra of the generic traceless matrices Wd is isomorphic to the factor algebra Khy1 , . . . , yd i/I(M2 (K), sl2 (K)) of the free associative algebra Khy1 , . . . , yd i, where sl2 (K) is the Lie algebra of all traceless 2 × 2 matrices and I(M2 (K), sl2 (K)) is the ideal of all polynomials in Khy1 , . . . , yd i which vanish under the substitutions of yi by elements in sl2 (K). (Such polynomials are called weak polynomial identities for the pair (M2 (K), sl2 (K)).) The ideal

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VESSELIN DRENSKY

I(M2 (K), sl2 (K)) is generated as a weak T-ideal by the weak polynomial identity [x21 , x2 ] = 0, i.e. I(M2 (K), sl2 (K)) is the minimal ideal of weak polynomial identities containing the element [x21 , x2 ]. An equivalent description is that as an ideal of the free algebra, I(M2 (K), sl2 (K)) is generated by all elements [uv + vu, w], where u, v, w are all possible commutators [yj1 , . . . , yjk ] in the variables y1 , . . . , yd . (iii) (Drensky and Koshlukov [DK]) The ideal I(M2 (K), sl2 (K)) is the minimal ideal of the free associative algebra Khy1 , . . . , yd i which is invariant under the diagonal action of GLd and contains the elements [y12 , y2 ] and X sign(σ)yσ(1) yσ(2) yσ(3) yσ(4) , s4 (y1 , y2 , y3 , y4 ) = σ∈S4

the second polynomial appears for d ≥ 4 only. Hence the algebra of 2 × 2 generic traceless matrices has a uniform set of defining relations for any d ≥ 2. (iv) (Procesi [P4]) As a GLd -module Wd has the description M W (λ1 , λ2 , λ3 ), Wd ∼ =

where the sum is on all partitions λ in at most three parts. (v) (Formanek [F1]) The GLd -module T2d has the decomposition M (λ1 − λ2 + 1)(λ2 − λ3 + 1)(λ3 − λ4 + 1)W (λ1 , λ2 , λ3 , λ4 ). T2d ∼ =

(vi) (See [D1, D2, F1, P4, LB1, LB2] for other descriptions of the Hilbert series.) The Hilbert series of T2d is H(T2d , t1 , . . . , td ) =

d Y

1 X S(λ1 ,λ2 ,λ3 ) (t1 , . . . , td ). 1 − ti i=1

Now we give some results on C2d . We follow the way we used in the previous theorem. For more details, especially for the Hilbert series and the GLd -module decomposition of C2d , see [F1, LB2, P4] or [DF]. Theorem 3.3. (i) The commutative trace algebra C2d is isomorphic to the tensor product of K-algebras K[tr(X1 ), . . . , tr(Xm )] ⊗K C(Wd ), where C(Wd ) is the centre of the algebra Wd defined above. (ii) As a subalgebra of Wd , its centre C(Wd ) is generated by Yi2 ,

i = 1, . . . , d, Yi Yj + Yj Yi , 1 ≤ i < j ≤ d, X sign(σ)Yσ(1) Yσ(2) Yσ(3) , 1 ≤ i < j < k ≤ d, s3 (Yi , Yj , Yk ) = σ∈S3

where the symmetric group S3 acts on {i, j, k}. (iii) As a GLd -module Wd has the decomposition M W (λ1 , λ2 , λ3 ), Wd ∼ =

where the sum is on all partitions λ in at most three parts such that both λ1 − λ2 and λ2 − λ3 are even. Concerning the defining relations of C2d , the case d = 2 is trivial. Formanek, Halpin and Li [FHL] showed that C22 is generated by the algebraically independent elements tr(X1 ), tr(X2 ), det(X1 ), det(X2 ), tr(X1 X2 ).

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For d = 3 Sibirskii [Si] found one relation between the generators of C23 and, using the Hilbert series of C23 , Formanek [F1] proved that there are no more relations. In the general case, the description of the defining relations of C2d is reduced to a similar description of the defining relations of the subalgebra of C2d generated by tr(Yi2 ),

i = 1, . . . , d, tr(Yi Yj Yk ),

tr(Yi Yj ),

1 ≤ i < j ≤ d,

1 ≤ i < j < k ≤ d,

where Y1 , . . . , Yd are generic traceless 2 × 2 matrices. Since GL2 acts on the generic matrices by conjugation, we may replace its action with the action of SL2 and even with the action of P SL2 . Since P SL2 (C) ∼ = SO3 (C), the special orthogonal group, we may apply invariant theory of special linear groups. (The restriction K = C is not essential in the final version of the result.) We consider the action of the special orthogonal group SO3 = SO3 (K), i.e. the group of orthogonal 3 × 3 matrices with determinant 1, on the polynomial algebra in 3d variables (3) (2) (1) K[ui , ui , ui | i = 1, . . . , d], (1)

(2)

(3)

induced by the action of SO3 on the three-dimensional vectors ui = (ui , ui , ui ). (j) It is a classical result that the algebra of invariants K[ui ]SO3 is generated by all scalar products (3) (3)

(2) (2)

(1) (1)

hui , uj i = ui uj + ui uj + ui uj , and all 3 × 3 determinants of the coordinates (1) (1) (1) u uk uj i (2) (2) ∆(ui , uj , uk ) = u(2) uk , uj i (3) (3) (3) u u u i

j

1 ≤ i ≤ j ≤ d,

1 ≤ i < j < k ≤ d.

k

The defining relations express the fact that the underlying vector space is threedimensional and every four vectors are linearly dependent. In particular, they use the properties of the Gram determinant: hui , up i hui , uq i hui , ur i hui , us i huj , up i huj , uq i huj , ur i huj , us i = 0, Γ4 (ui , uj , uk , ul ; up , uq , ur , us ) = huk , up i huk , uq i huk , ur i huk , us i hul , up i hul , uq i hul , ur i hul , us i 1 ≤ i < j < k < l ≤ d,

1 ≤ p < q < r < s ≤ d,

∆(ui , uj , uk )∆(up , uq , ur ) − Γ3 (ui , uj , uk ; up , uq , ur ) = 0, hup , ui i∆(uj , uk , ul ) − hup , uj i∆(ui , uk , ul ) +hup , uk i∆(ui , uj , ul ) − hup , ul i∆(ui , uj , uk ) = 0. In order to apply invariant theory of SO3 we need a scalar product (i.e. nondegenerate symmetric bilinear form) on sl2 (K). We use the trace and define hu, vi = tr(uv),

u, v ∈ sl2 (K).

The following result gives the generators and the defining relations of C2d for d ≥ 2. It is a translation of the description of the invariants of SO3 .

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Theorem 3.4. (i) The algebra C2d is generated by tr(Xi ),

tr(Yi2 ),

tr(Yi Yj ),

tr(s3 (Yi , Yj , Yk )),

where i, j, k = 1, . . . , d, and in the traces involving two or three traceless matrices we require i < j or i < j < k, respectively. (ii) Drensky [D3]) The defining relations of C2d with respect to the above generators are tr(Yi Yp ) tr(Yi Yq ) tr(Yi Yr ) tr(s3 (Yi , Yj , Yk ))tr(s3 (Yp , Yq , Yr )) + 18 tr(Yj Yp ) tr(Yj Yq ) tr(Yj Yr ) = 0, tr(Yk Yp ) tr(Yk Yq ) tr(Yk Yr )

tr(Yp Yi )tr(s3 (Yj , Yk , Yl )) − tr(Yp Yj )tr(s3 (Yi , Yk , Yl )) +tr(Yp Yk )tr(s3 (Yi , Yj , Yl )) − tr(Yp Yl )tr(s3 (Yi , Yj , Yk )) = 0, where, again, i, j, k, p, q, r = 1, . . . , d, and, where necessary, we require i < j < k < l and p < q < r.

In order to work efficiently with an algebra R = K[x1 , . . . , xp ]/I, it is not sufficient to know the generators of the ideal I. For computational purposes one needs also the Gr¨ obner basis of I with respect to some ordering on the monomials of K[x1 , . . . , xp ], see e.g. the book by Adams and Loustaunau [AL]. The Gr¨obner basis of C2d is given by Domokos and Drensky [DD], see their paper for more details. Now we consider two generic 3 × 3 matrices. Using the Molien-Weyl formula, Teranishi [T1] calculated the Hilbert series of C32 , namely, H(C32 , t1 , t2 ) =

1 + t31 t32 , (1 − t1 )(1 − t2 )q2 (t1 , t2 )q3 (t1 , t2 )(1 − t21 t22 )

where q2 (t1 , t2 ) = (1 − t21 )(1 − t1 t2 )(1 − t22 ), q3 (t1 , t2 ) = (1 − t31 )(1 − t21 t2 )(1 − t1 t22 )(1 − t32 ). He also found the following system of generators of C32 : tr(X1 ), tr(X2 ), tr(X12 ), tr(X1 X2 ), tr(X22 ), tr(X13 ), tr(X12 X2 ), tr(X1 X22 ), tr(X23 ), tr(X12 X22 ), tr(X12 X22 X1 X2 ), where X1 , X2 are generic 3 × 3 matrices. He showed that the first ten of these generators form a homogeneous system of parameters of C32 and C32 is a free module with generators 1 and tr(X12 X22 X1 X2 ) over the polynomial algebra on these ten elements. Abeasis and Pittaluga [AP] found a system of generators of C3d in terms of representation theory of the symmetric and general linear groups, in the spirit of its use in theory of PI-algebras. They showed that C3d has a minimal system of generators which spans a GLd -module isomorphic to G = W (1) ⊕ W (2) ⊕ W (3) ⊕ W (13 ) ⊕ W (22 ) ⊕ W (2, 12 ) ⊕W (3, 12 ) ⊕ W (22 , 1) ⊕ W (15 ) ⊕ W (32 ) ⊕ W (3, 13 ). (The partitions in [AP] are given in “Francophone” way, i.e., transposed to ours.) It follows from the description of the generators of C32 given by Teranishi [T1], that tr(X12 X22 X1 X2 ) satisfies a quadratic equation with coefficients depending on the other ten generators. The explicit (but very complicated) form of the equation

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17

was found by Nakamoto [N], over Z, with respect to a slightly different system of generators. A much simpler description of C32 was obtained by Aslaksen, Drensky, and Sadikova [ADS]. The following generators are in the spirit of the ideas of [AP]. Proposition 3.5. Let X1 , X2 and Y1 , Y2 be, respectively, two generic and two generic traceless 3 × 3 matrices. The algebra C32 is generated by tr(X1 ), tr(X2 ), tr(Y12 ), tr(Y1 Y2 ), tr(Y22 ), tr(Y13 ), tr(Y12 Y2 ), tr(Y1 Y22 ), tr(Y23 ), V = tr(Y12 Y22 ) − tr(Y1 Y2 Y1 Y2 ),

W = tr(Y12 Y22 Y1 Y2 ) − tr(Y22 Y12 Y2 Y1 ).

Now we define the following elements of C32 : tr(Y12 ) tr(Y1 Y2 ) U = tr(Y1 Y2 ) tr(Y22 ) W1 = U 3 ,

W6 =

,

W2 = U 2 V, W4 = U V 2 , W7 = V 3 , tr(Y12 ) tr(Y1 Y2 ) tr(Y22 ) tr(Y12 Y2 ) tr(Y1 Y22 ) , W5 = V tr(Y13 ) tr(Y12 Y2 ) tr(Y1 Y22 ) tr(Y23 ) 2 tr(Y23 ) tr(Y1 Y22 ) tr(Y13 ) tr(Y13 ) tr(Y1 Y22 ) −4 tr(Y1 Y22 ) tr(Y12 Y2 ) tr(Y12 Y2 ) tr(Y12 Y2 ) tr(Y23 ) tr(Y12 ) tr(Y1 Y2 ) tr(Y22 ) tr(Y12 Y2 ) tr(Y1 Y22 ) , W3′ = U tr(Y13 ) tr(Y12 Y2 ) tr(Y1 Y22 ) tr(Y23 )

tr(Y12 Y2 ) tr(Y1 Y22 )

,

where U, V are defined above. Finally, we define one more element W3′′ as follows. Recall that a linear mapping δ of an algebra R is a derivation if δ(rs) = δ(r)s+rδ(s) for all r, s ∈ R. We consider the derivation δ of C32 which commutes with the trace and satisfies the conditions δ(X1 ) = 0, Then

δ(X2 ) = X1 ,

δ(Y1 ) = 0,

δ(Y2 ) = Y1 .

6

W3′′ =

1 X (−1)i δ i (tr3 (Y22 ))δ 6−i (tr2 (Y23 )). 144 i=0

The following theorem gives the defining relation of C32 . It uses representation theory of GL2 , combinatorics, computations by hand and easy computer calculations with standard functions of Maple. Theorem 3.6. (Aslaksen, Drensky, Sadikova [ADS]) The algebra of invariants C32 of two 3×3 matrices is generated by the elements from the previous theorem, subject to the defining relation 2 4 1 1 2 1 4 1 W1 − W2 + W3′ + W3′′ + W4 − W5 − W6 − W7 = 0. W2 − 27 9 15 90 3 3 3 27 The calculation of the Hilbert series of Cnd and Tnd based on the Molien-Weyl formula is quite complicated because requires evaluations of multiple integrals. Van den Bergh [VB2] sujected a way which involves graph theory. As a consequence, he established important properties of H(C2d , t1 , . . . , td ) and H(T2d , t1 , . . . , td ).

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Berele and Stembridge [BS] applied the method of van den Bergh [VB2] and calculated the Hilbert series of T32 . Using the above results of Aslaksen, Drensky, and Sadikova on C32 and the explicit form of the Hilbert series of T32 , Benanti and Drensky [BD] found a polynomial subalgebra S of C32 and a finite set of generators of the free S-module T32 . They gave also a set of defining relations of T32 as an algebra and a Gr¨obner basis of the corresponding ideal. (See the survey article by Ufnarovski [U] for a background on Gr¨obner bases in the noncommutative case, as well as the paper by Mikhalev and Zolotykh [MZ] which is closer to the situation in [BD].) For two generic 4 × 4 matrices, the Hilbert series of C42 was calculated (with some typos) by Teranishi [T1, T2] and corrected by Berele and Stembridge [BS]. Teranishi found also a homogeneous system of parameters and a system of generators, in the spirit of the 3 × 3 case. Recently, Drensky and Sadikova [DS] have found another system of generators of C42 which is minimal and seems to be more convenient for concrete calculations. The Hilbert series of a graded vector space with GLd -module structure determines uniquely its decomposition into irreducible submodules. Hence, in principle, one may calculate the multiplicities m(λ) if one knows the concrete form of the Hilbert series. Berele [B1] used the Hilbert series of C32 found by Teranishi [T1] and described the asymptotics of m(λ1 , λ2 ). (Due to a technical error (an omitted summand) some of the coefficients of the polynomials in the asymptotics of Berele are slightly different from the real ones.) Another approach to the problem was suggested by Drensky and Genov [DG1]. Let X f (t1 , t2 ) = aij ti1 tj2 , i,j≥0

aij ∈ K, aij = aji , be a symmetric function in two variables which is a formal power series from K[[t1 , t2 ]]. We present it in the form X m(λ1 , λ2 )S(λ1 ,λ2 ) (t1 , t2 ) f (t1 , t2 ) = λ1 ≥λ2

and want to find the multiplicities m(λ1 , λ2 ). In most of the cases which we consider, f (t1 , t2 ) is given explicitly as a rational function. So, it is natural to express m(λ1 , λ2 ) not in terms of the coefficients aij but in a more direct way. We introduce the generating function of the multiplicities X m(λ1 , λ2 )tλ1 uλ2 ∈ K[[t, u]] M (f, t, u) = λ1 ≥λ2

and call it the multiplicity series of f (t1 , t2 ). It is more convenient to introduce a new variable v = tu and to consider the series X m(λ1 , λ2 )tλ1 −λ2 v λ2 ∈ K[[t, v]], M ′ (f, t, v) = λ1 ≥λ2

because the mapping M ′ : K[[t1 , t2 ]]S2 → K[[t, v]] is a bijective linear mapping which is continuous with respect to the formal power series topology. It is easy to see that f (t1 , t2 ) and M ′ (f, t, v) are related by f (t1 , t2 ) =

t1 M ′ (f, t1 , t1 t2 ) − t2 M ′ (f, t2 , t1 t2 ) . t1 − t2

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Hence, if we have a potential candidate h(t, v) for M ′ (f, t, v), it is easy to verify whether h(t, v) = M ′ (f, t, v). Also, the elementary symmetric function e2 = t1 t2 behaves like a constant, M ′ (g(t1 t2 )f (t1 , t2 ), t, v) = g(v)M ′ (f, t, v), and this simplifies the calculations. Applying quite complicated (also technically) arguments, Drensky and Genov [DG1] found the multiplicity series of the Hilbert series of C32 . They also corrected the technical errors in [B1]. Theorem 3.7. (i) [DG1] The multiplicity series of the Hilbert series of the algebra C32 of invariants of two 3 × 3 matrices is 1 × M ′ (H(C32 , t1 , t2 ), t, v) = 2 (1 − v )(1 − v 3 )2 (1 + v 2 + v 4 )((1 + v 2 )(1 − t2 v) + 2tv(1 − v)) + × 3(1 − v)(1 − v 2 )3 (1 − t)2 (1 − t2 ) (1 − v 2 )(1 − tv) (1 − v)(1 + tv) + + 3(1 − v 2 )(1 − t)(1 − t2 ) 3(1 − v 3 )(1 − t3 ) v 3 ((1 − v + v 2 )(1 − t2 v 2 ) + tv(1 − v 2 )) . − (1 − v)(1 − v 2 )2 (1 − v 4 )(1 − t)(1 − t2 )(1 − tv) (ii) [B1, DG1] Let λ = (p, q) and let m(p, q) be the multiplicity of S(p,q) (t1 , t2 ) in H(C32 , t1 , t2 ). Then for p > 2q ≥ 0 q7 (p − q)q 6 (p − q)2 q 5 + + + O((p + q)6 ) 5 2 4 2 7!2 .3 6!2 .3 2!5!23 32 p2 q 5 11pq 6 71q 7 = − + + O((p + q)6 ); 17280 103680 1451520

m(p, q) =

for 2q ≥ p ≥ q ≥ 0 q7 (p − q)q 6 (p − q)2 q 5 (2q − p)7 + + − + O((p + q)6 ) 5 2 4 2 3 2 7!2 .3 6!2 .3 2!5!2 3 7!25 .32 p7 p6 q p5 q 2 p4 q 3 p3 q 4 7p2 q 5 7pq 6 19q 7 = − + − + − + − +O((p+q)6 ). 1451520 103680 17280 5184 2592 17280 34560 483840 Later the methods for calculating the multiplicity series of symmetric functions of special kinds were significantly improved [DG2]. The Hilbert series of C42 calculated by Teranishi [T1, T2] (with some typos corrected in [BS]) and the Hilbert series of T32 and T42 calculated by Berele and Stembridge [BS] allowed to express their multiplicity series and to determine the asympotics of the multiplicities. We shall state simplified versions of the results: m(p, q) =

Theorem 3.8. (i) (Drensky, Genov, Valenti [DGV]) The multiplicities m(λ1 ,λ2 ) (C32 ) and m(λ1 ,λ2 ) (T32 ) of the Hilbert series of C32 and T32 , respectively, are related by m(λ1 ,λ2 ) (T32 ) ≈ 9m(λ1 ,λ2 ) (C32 ). (ii)( Drensky and Genov [DGV]) Let λ = (λ1 , λ2 ). The multiplicities mλ (C42 ) of the Hilbert series of C42 satisfy the condition 13 if λ1 > 3λ2 , m1 + O((λ1 + λ2 ) ), mλ (C42 ) = m1 + m2 + O((λ1 + λ2 )13 ), if 3λ2 ≥ λ1 > 2λ2 , 13 m1 + m2 + m3 + O((λ1 + λ2 ) ), if 2λ2 ≥ λ1 ,

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where m1 =

(λ1 − λ2 )2 λ12 127(λ1 − λ2 )λ13 305λ14 (λ1 − λ2 )3 λ11 2 2 2 2 − + − , 8 2 8 3 10 4 11!3!2 3 12!2!2 3 13!2 3 14!29 35

(3λ2 − λ1 )14 , 14!210 35 52 7(2λ2 − λ1 )14 (λ1 − λ2 )(2λ2 − λ1 )13 . − m3 = − 13!210 32 5 14!293 · 52 The multiplicities mλ (T42 ) satisfy m2 =

mλ (T42 ) = 16mλ (C42 ) + O((λ1 + λ2 )13 ). We want to mention that Berele and Stembridge [BS] computed also the Hilbert series of C33 and T33 but the methods of [DG1, DG2, DG3, DGV] do not work successfully for symmetric functions in three variables. One can introduce the multiplicity series of a symmetric function in any number of variables, generalizing in an obvious way the case of symmetric functions in two variables. A recent theorem of Berele [B2] gives the rationality of the multiplicity series of a class of rational symmetric functions in any number of variables, including the Hilbert series of Cnd and Tnd . Unfortunately, it is not clear how to perform the concrete calculations, even for the Hilbert series of C33 and T33 . References [AP] S. Abeasis, M. Pittaluga, On a minimal set of generators for the invariants of 3 × 3 matrices, Commun. Algebra 17 (1989), 487-499. [AL] W.W. Adams, P. Loustaunau, An Introduction to Gr¨ obner Bases, Graduate Studies in Math. 3, AMS, Providence, R.I., 1994. [ADS] H. Aslaksen, V. Drensky, L. Sadikova, Defining relations of invariants of two 3×3 matrices, J. Algebra (to appear). Announcement: C.R. Acad. Bulg. Sci. 58 (2005), No. 6, 617-622. [BD] F. Benanti, V. Drensky, Defining relations of noncommutative trace algebra of two 3 × 3 matrices, Adv. Appl. Math. (to appear). Preprint, http://xxx.lanl.gov/abs/math.RA/0501219. [B1] A. Berele, Approximate multiplicities in the trace cocharacter sequence of two three-by-three matrices, Commun. Algebra 25 (1997), 1975-1983. [B2] A. Berele, Applications of Belov’s theorem to the cocharacter sequence of P.I. algebras, J. Algebra (to appear). [BS] A. Berele, J.R. Stembridge, Denominators for the Poincar´ e series of invariants of small matrices, Israel J. Math. 114 (1999), 157-175. [Ch] C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 67 (1955), 778-782. [DC] J.A. Dieudonn´ e, J.B. Carrell, Invariant Theory, Old and New, Academic Press, New YorkLondon, 1971. [DD] M. Domokos, V. Drensky, Gr¨ obner bases for the rings of invariants of special orthogonal ans 2 × 2 matrix invariants, J. Algebra 243 (2001), 706-716. [D1] V. Drensky, Codimensions of T-ideals and Hilbert series of relatively free algebras, J. Algebra 91 (1984), 1-17. [D2] V. Drensky, Free Algebras and PI-Algebras, Springer-Verlag, Singapore, 1999. [D3] V. Drensky, Defining relations for the algebra of invariants of 2 × 2 matrices, Algebras and Representation Theory 6 (2003), 193-214. [DF] V. Drensky, E. Formanek, Polynomial Identity Rings, Advanced Courses in Mathematics, CRM Barcelona, Birkh¨ auser, Basel-Boston, 2004. [DG1] V. Drensky, G.K. Genov, Multiplicities of Schur functions in invariants of two 3×3 matrices, J. Algebra 264 (2003), No. 2, 496-519. [DG2] V. Drensky, G.K. Genov, Multiplicities of Schur functions with applications to invariant theory and PI-algebras, C.R. Acad. Bulg. Sci. 57 (2004), No. 3, 5-10.

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[W2] H. Weyl, The Classical Groups, Their Invariants and Representations, Princeton Univ. Press, Princeton, N.J., 1946, New Edition, 1997. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria E-mail address: [email protected]