Computational Approach to Polynomial Identities of Matrices a Survey

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1 Computational Approach to Polynomial Identities of Matrices a Survey FRANCESCA BENANTI 1 Dipartimento di Matematica ed Applicazioni, Università di Palermo, via Archirafi 34, Palermo, Italy, (fbenanti@dipmat.math.unipa.it). JAMES DEMMEL Department of Mathematics, Computer Science Division, University of California at Berkeley, Berkeley, CA 94720, U.S.A., (demmel@cs.berkeley.edu) VESSELIN DRENSKY 2 Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Akad. G. Bonchev Str., Block 8, 1113 Sofia, Bulgaria, (drensky@math.bas.bg). PLAMEN KOEV Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720, U.S.A., (plamen@math.berkeley.edu) Abstract We present a survey on polynomial identities of matrices over a field of characteristic 0 from computational point of view. We describe several computational methods for calculation with polynomial identities of matrices and related objects. Among the other applications, these methods have been successfully used: 1 The work of this author was partially supported by MURST of Italy. 2 The work of this author was partially supported by Grant MM-1106/2001 of the Bulgarian Foundation for Scientific Research.

2 (i) to show that all polynomial identities of degree 2n + 2 for the n n matrix algebra for n = 3, 4, 5 are consequences of the standard identity s 2n ; (ii) to obtain upper bounds for the multiplicities of the irreducible S 9 -characters of the multilinear identities of degree 9 in 3 3 matrices; (iii) in the discovery of new central polynomials of low degree for matrices of order 3 and 4; (iv) to obtain upper bounds for the multiplicities of the irreducible S 9 -characters of -polynomial identities of degree 9 in symmetric variables only for 6 6 matrices with symplectic involution. Key words: polynomial identities of matrices, computer programs, cocharacters of T-ideals. MSC 2000: 16-02; 16R30; 16R10; 16R50; 15-04; 16-04; 20C30. 1 INTRODUCTION The polynomial identities of the matrix algebra M n (K) over a field K of characteristic 0 are among the most intriguing objects in the theory of algebras with polynomial identities. Since 1950, when Amitsur and Levitzki proved their famous theorem that M n (K) satisfies the standard identity of degree 2n, a lot of results have been established. Nevertheless, the picture is far from a satisfactory form and only the structure of the polynomial identities of the 2 2 matrices is well known. There are a lot of open problems and conjectures concerning the bases of polynomial identities of M n (K), the minimal degree of identities which do not follow from the standard identity, the minimal degree of central polynomials, the numerical invariants of polynomial identities, etc. There are similar problems to consider for matrix algebras with additional structure as involution or group gradings. Some of the facts in the theory have been obtained, or at least conjectured, as a result of heavy calculations by hand or computer. The main purpose of the present paper is to survey some computational techniques for working with the polynomial identities, weak polynomial identities, central polynomials and -polynomial identities of n n matrices. The paper is organized as follows. In Section 2 we present the main objects and give a short survey on some results on polynomial identities for matrices obtained without computers but from computational point of view. In Section 3 we give the necessary algebraic and combinatorial background and in Section 4 we describe several computational approaches. Section 5 is devoted to concrete results of the calculations. First we discuss the result that all polynomial identities of degree 2n + 2 for n n matrices, n = 3, 4, 5, are consequences of the standard identity of degree 2n confirming for small n the conjecture that for n 3 all such identities follow from the stan-

3 dard identity. We also give an upper bound for the multiplicities of the irreducible S 9 -characters and the dimension of the polynomial identities of degree 9 for M 3 (K). It seems that the obtained numerical results are equal or very close to the real ones. In a similar way we study the identities in symmetric variables of degree 9 for the matrix algebra M 6 (K) with symplectic involution. Finally we discuss central polynomials of low degree for matrices and the difficulties in their computational study. In the last section we deal with computations by hand which seems to be perspective for computerization. The authors do not pretend that the present paper is an exhausted survey on all computational methods used to study polynomial identities of matrices. The choice of the topics is influenced very much by the taste and experience of the authors. We do not enter in details of the computer programs. We want only to mention that the surveyed results have been obtained using different computer systems and the programs have been written in different languages. In particular, we do not discuss the existing packages for computing with polynomial identities and combinatorial objects related with representation theory of the symmetric and the general linear groups. 2 THE MAIN OBJECTS Let K be a fixed field of characteristic 0. We denote by A = K X the free associative algebra with a countable set of free generators X = {x 1, x 2,...}. An element f(x 1,..., x m ) A is called a polynomial identity for a K- algebra R if f(r 1,..., r m ) = 0 for any r 1,..., r m R. The set of all polynomial identities of R, denoted by T (R), is a T-ideal, i.e. a twosided ideal of A which is invariant under all endomorphisms of A. When T (R) (0), R is called a PI-algebra. Any set of generators of T (R) as a T-ideal is called a basis of the polynomial identities of R. The T-ideal of the polynomial identities of the matrix algebra M n (K) is one of the most interesting objects in the theory of PI-algebras and its applications. We now briefly give some results on the polynomial identities of M n (K). For further reading we refer to the books by Procesi [56], Jacobson [40], Rowen [68], Bahturin [3], Formanek [32] and Drensky [20]. In 1950, Amitsur and Levitzki [1] proved that M n (K) satisfies the standard identity of degree 2n s 2n = s 2n (x 1,..., x 2n ) = (signσ)x σ(1)... x σ(2n) σ S 2n and that s 2n is the only polynomial identity of degree 2n for M n (K). Moreover, from the Cayley-Hamilton theorem, one obtains that the powers

4 1, r, r 2,..., r n of any matrix r M n (K) are linearly dependent and this easily implies that M n (K) satisfies the identity of algebraicity a n = a n (x, y 1,..., y n ) = (signσ)x σ(0) y 1 x σ(1) y 2 x σ(2)... y n x σ(n), σ S n+1 where the symmetric group S n+1 acts on the symbols 0, 1,..., n. It can be shown that a n, n 2, does not follow from the standard polynomial s 2n. One of the central problems in the theory of PI-algebras is to find a basis of the polynomial identities for the algebra of n n matrices, n 2. (In 1986, Kemer [45] (see also [46]) gave a positive solution to the famous Specht problem [73] whether every T-ideal has a finite basis. This result justifies the search for a finite basis of T (R), for any K-algebra R.) The complete answer is known for 2 2 matrices only. Till the early 70 s, the only known identities for M 2 (K) were s 4 and the Hall identity [[x, y] 2, z]. It expresses the fact that the trace of the commutator of two matrices is equal to 0 and, by the Cayley-Hamilton theorem, [r 1, r 2 ] 2 is a scalar matrix for all r 1, r 2 M 2 (K). In 1973, Razmyslov [61] found a finite basis of T (M 2 (K)) consisting of 9 polynomials of degree 6. Drensky [14] in 1981 showed that the standard identity s 4 (x 1,..., x 4 ) and the Hall identity in two variables [[x, y] 2, x] (which up to a sign is equal to the identity of algebraicity a 2 (x, y, y)) form a minimal basis for the T-ideal of the polynomial identities of M 2 (K). He used the fact that the identities of M 2 (K) follow from these of degree 6 (and did not use the concrete basis found by Razmyslov). Up till now, no explicit set of generators of T (M n (K)), n > 2, is known. In 1973, Leron [50] proved that for n > 2 all polynomial identities of degree 2n + 1 for M n (K) are consequences of the standard identity s 2n. For n = 3 this result was improved by Drensky and Kasparian [24]. In 1983, they found all identities of degree 8 and showed that they are consequences of the standard identity s 6. These calculations (done by hand) were checked by computer by Bondari [11] in 1997 and by Vishne [75] in 2002, see the last section of the present paper. There was a hope that the standard identity s 2n and the identity of algebraicity a n form a basis for T (M n (K)) also for n > 2. But for n = 3 several new identities of degree 9 have been found by Okhitin [54] and Domokos [12]. For n > 3 no new identities of degree < dega n = n(n + 3)/2 are known. The following problems related to the polynomial identities of M n (K) are well known: Let n 4. (i) What is the minimal degree of the polynomial identities for M n (K) which do not follow from s 2n? (For n = 2, 3 this degree is equal to dega n.) (ii) Are all polynomial identities of degree 2n + 2 for the matrix algebra M n (K) consequences of the standard identity s 2n? Usually the results on polynomial identities are expressed in the language

5 of the representation theory of the symmetric group S k. It is well known that over a field of characteristic zero every T-ideal is generated by its multilinear elements. If V k is the vector space of all multilinear polynomials of degree k in x 1,..., x k, the properties of the T-ideal T (R) are determined by the properties of V k T (R) for every k. The symmetric group S k acts from the left on V k by permuting of the variables. Under this action V k is an S k -module and the space V k T (R) is a submodule of V k. For a partition λ of k we denote by χ λ the irreducible S k -character corresponding to λ. Then χ Sk (V k T (R)) = m λ (R)χ λ, where m λ (R) is the multiplicity of χ λ. In 1977, Olsson and Regev [55] computed the S k+1 -character of the multilinear consequences of degree k + 1 of the standard polynomial s k : χ Sk+1 ( s k T V k+1 ) = χ (3,1 k 2 ) + χ (2 2,1 k 3 ) + 3χ (2,1 k 1 ) + χ (1 k+1 ), where s k T denotes the T-ideal of A generated by s k. Recently, Benanti and Drensky [7] found all consequences of degree k + 2 of s k and calculated the S k+2 -character of s k T, for k 5: χ Sk+2 (V k+2 s k T ) = 2χ (4,2,1 k 4 ) + 4χ (4,1 k 2 ) + 2χ (3,2 2,1 k 5 ) +8χ (3,2,1 k 3 ) + 9χ (3,1 k 1 ) + 4χ (2 3,1 k 4 ) + 8χ (2 2,1 k 2 ) + 5χ (2,1 k ) + χ (1 k+2 ). The method in [7] is based on the representation theory of the general linear group GL m (K) which is equivalent to the representation theory of S k. Often, it is more useful to study the S k -module V k /(V k T (R)). The S k - character of this module, denoted by χ k (R), is called the k-th cocharacter of R and the sequence {χ k (R)} k 1 is the cocharacter sequence of R. The k-cocharacter of R is decomposed as χ k (R) = m λ (R)χ λ and the main quantitative problem is to determine the multiplicities m λ (R). In the case of matrix algebras the first result in this direction was proved in 1979 by Regev [65]: χ k (M n (K)) = m λ (M n (K))χ λ, λ Λ n 2(k) where Λ n 2(k) = {λ = (λ 1,..., λ r ) k r n 2 }. Bergman [10] described all polynomial identities for M n (K) of the form f(x, y 1,..., y n ) which are linear in each variable y 1,..., y n. Using the result of [10], Drensky and Kasparian [23] calculated the multiplicities m λ (M n (K)) in the special case λ = (λ 1, λ 2,..., λ r ) when λ λ r = n. (It is easy to see that m λ (M n (K)) = 0 when λ λ r < n.) It turns out that all these identities are consequences of the identity of algebraicity. On the other hand, Formanek [30, 31] showed that m λ (M n (K)) = m µ (M n (K)) for λ = (λ 1,..., λ n 2), µ = (λ 1,..., λ n 2 1) and λ n 2 2.

6 For n = 2 and n = 3 the information is more complete. In 1984, the explicit expression of χ k (M 2 (K)) was determined by Formanek [30] and Drensky [15]. Benanti [4] studied the k-cocharacter of M 3 (K) and determined all partitions λ of k such that m λ (M 3 (K)) is not zero. We now briefly continue our exposition on PI-algebras with central polynomials and weak polynomial identities for M n (K). A homogeneous polynomial of positive degree f(x 1,..., x m ) A is called a central polynomial for the algebra R if f(r 1,..., r m ) is in the center of R for all r 1,..., r m R, and f(x 1,..., x m ) is not a polynomial identity for R. By the Cayley-Hamilton theorem it is easy to show that [x 1, x 2 ] 2 is a central polynomial for the 2 2 matrix algebra. In 1956, Kaplansky [42], see also the revised version [43] in 1970, asked his famous 12 problems which motived a significant research activity in ring theory. One of this problems was whether there exists a central polynomial for the matrix algebra M n (K), n 3. The answer to the problem of Kaplansky was given in by Formanek [29] and Razmyslov [62] who constructed central polynomials for M n (K), n 2, with two different methods. The existence of central polynomials was crucial in many considerations and constructions in PI-theory and for applications to other branches of mathematics, see the book by Jacobson [40]. The construction of Formanek yields a central polynomial of degree n 2. The original polynomial of Razmyslov [62] is of degree 3n 2 1 but Halpin [39] showed that the method of Razmyslov also gives a central polynomial of degree n 2. Although it is of the same degree as the polynomial constructed by Formanek, the polynomial of Halpin is different. For long time it was thought that the minimal degree of the central polynomials for the n n matrices is equal to n 2 and this is true for n = 1 and n = 2. But, in , Drensky and Kasparian [24, 25] found a central polynomial of degree 8 for 3 3 matrices and showed that for n = 3 the minimal degree of the central polynomials is equal to 8. Formanek [32] in 1991 conjectured that the minimal degree of the central polynomials for M n (K) is equal to (n 2 + 3n 2)/2. In 1994, Drensky and Piacentini Cattaneo [26] found a new central polynomial of degree 13, for n = 4, and this agrees with the conjecture. The result was generalized by Drensky [19] who constructed new central polynomials of degree (n 1) for M n (K), n > 2. For n = 3 and n = 4 this degree coincides with the degree prescribed in the conjecture of Formanek. The construction of Razmyslov is based on two ideas of independent interest, the Razmyslov transform [62] and weak polynomial identities [61, 62] (see also the book by Razmyslov [63]). The notion of a weak polynomial identity due to Razmyslov plays a crucial role in his approach to the polynomial identities of 2 2 matrices and central polynomials for matrix algebras of any size. The polynomial f(x 1,..., x m ) A is called a weak

7 polynomial identity for M n (K) if f(r 1,..., r m ) = 0 for all traceless matrices r 1,..., r m M n (K) (i.e. r 1,..., r m belong to the Lie subalgebra sl n (K) of M n (K)). A weak polynomial identity for M n (K) which is not a polynomial identity for M n (K) is called an essential weak polynomial identity. Once we have at our disposal an essential weak polynomial identity, the method of Razmyslov gives a central polynomial. In this way the existence of the weak identity [x 2 1, x 2] of degree three for M 2 (K) gives rise to a central polynomial of degree four. Similarly the central polynomials for M n (K) of degree n 2 constructed by Halpin [39] used weak polynomial identities of degree n(n + 1)/2. Initially, the central polynomial of degree 8 for M 3 (K) had been found [25] by a direct construction. Later, Drensky and Rashkova [27] described all weak polynomial identities of degree 6 for M 3 (K) and showed that one of them gives a central polynomial of degree 8. Also, for M 4 (K), in [26] the authors obtained a central polynomial of degree 13 from an essential weak polynomial identity of degree 9 for 4 4 matrices. The lower degree of the weak polynomial identities in the construction of central polynomials is a big advantage from the computational point of view, see Section 5.3 of the present paper. Finally we report results about polynomial identities for matrix algebras with involution. We denote by K X, the free associative algebra with involution generated by X over K. If R is an associative K-algebra with involution, then an element f(x 1, x 1,..., x m, x m) of K X, is a -polynomial identity for R if f(r 1, r1,..., r m, rm) = 0 for all substitutions r 1,..., r m R. The set T (R, ) of all -polynomial identities of R is a -T-ideal of K X, i.e., an ideal invariant under all endomorphisms of K X, commuting with (see [35]). In case R = M n (K), n 2, two involutions play a very important role in the study of -polynomial identities: the transpose involution, denoted = t, and the canonical symplectic involution, denoted = s and defined only in case n = 2m is even. In fact, it is well known (see [68], Theorem ) that, if K is an infinite field and is an involution in M n (K), then either T (M n (K), ) = T (M n (K), t) or T (M n (K), ) = T (M n (K), s). For i = 1, 2,..., define s i = x i + x i and k i = x i x i. Obviously, s i = s i, ki = k i. We can regard K X, as generated by the symmetric variables s i and the skew-symmetric variables k i and write K X, = K s 1, k 1, s 2, k 2,.... The theorem of Amitsur and Levitzki [1] can be improved if we consider only polynomials in symmetric or skew-symmetric variables. In 1974, Rowen extending a result of Kostant [49] proved that s 2n 2 is a -polynomial identity for (M n (K), t) in skew-symmetric variables (see [67]). In 1982, also he proved that s 2n 2 is a -polynomial identity for (M n (K), s) in sym-

8 metric variables [69]. No -polynomial identities of degree lower than 2n 2 are known for (M n (K), t) and (M n (K), s) for arbitrary n. In 1990, Giambruno [34] showed that if f is a -polynomial identity for (M n (K), s) and n > 2, then degf n + 1. This result was improved by Drensky and Giambruno [22] who proved in 1995 that if n > 4, then (M n (K), s) does not satisfy identities of degree n + 1 in symmetric variables only. Therefore the minimal degree of the identities in symmetric variables for (M n (K), s) is greater than n + 1. In 1996 Rashkova [58], see also [59], made one more step in this direction, showing that (M 6 (K), s) has no identities of degree 8 in symmetric variables. In 1995, it was proved by Giambruno and Valenti (see [36]) that if f is a -polynomial identity in skew-symmetric variables for (M n (K), s) then degf > n + n/2. It is easy to show that s 2n is a - polynomial identity for (M n (K), t) of minimal degree among -polynomial identities in symmetric variables. But in [51] and [36] it was shown that in general -polynomial identities of minimal degree for matrices with involution need not resemble standard identities. One of the first problems to study is to determine the minimal degree of a -polynomial identity for (M n (K), ). Concerning general results, Rashkova [60] has recently established that a polynomial f(x, y 1,..., y m ) which is linear in each y i, i = 1,..., m, is a -polynomial identity for (M 2m (K) +, s) if and only if it is an ordinary polynomial identity for M m (K). The -polynomial identities of an algebra in characteristic zero are determinated by the multilinear ones. Hence, if we denote by V k ( ) the space of all multilinear polynomials of degree k in x 1, x 1,..., x k, x k, to study T (R, ) is equivalent to study V k ( ) T (R, ), for any k 1. The space V k ( ) T (R, ) has a natural structure of B k -module, where B k is the hyperoctahedral group of degree k (see [35] and also [33] for the equivalent approach based on representation theory of the direct product of two copies of the general linear group). The B k -character of V k ( ) T (R, ) is decomposed as χ Bk (R, ) = k r=0 λ r µ k r m λ,µ (R, )χ λ,µ, r=0 λ r µ k r where χ λ,µ denotes the irreducible B k -character corresponding to the pair of partitions (λ, µ) and m λ,µ (R, ) 0 are the corresponding multiplicities. The main problem is to determine m λ,µ (R, ), for any k or, equivalently, the multiplicities m λ,µ (R, ) in the k-th -cocharacter of (R, ) in k the -cocharacter sequence χ k (R, ) = m λ,µ (R, )χ λ,µ. A complete study of T (M 2 (K), t) and of T (M 2 (K), s), in characteristic zero, has

9 been made by Drensky and Giambruno in [21]. They determined the multiplicities m λ,µ (M 2 (K), ), for any k. Recently Benanti and Campanella [5] investigated the -cocharacter sequence of (M 3 (K), t) and determined all the pairs of partitions such that m λ,µ (M 3 (K), t) = 0. 3 ALGEBRAIC AND COMBINATORIAL BACKGROUND We consider associative, not necessarily unitary K-algebras only, where K is a fixed field of characteristic 0. All vector spaces and tensor products are also over K. We denote by A = K X the free associative algebra of countable rank with a set of free generators X = {x 1, x 2,...}. An ideal U of A is a T-ideal if it is invariant under all endomorphisms of A. It is well known that U is a T-ideal of A if and only if U = T (R) for some algebra R, where T (R) is the set of all polynomial identities for an associative algebra R. Let H be a vector subspace of an associative algebra R which generates R as an algebra. The polynomial f(x 1,..., x m ) is called a weak polynomial identity for the pair (R, H) if f(h 1,..., h m ) = 0 for arbitrary h 1,..., h m H. The weak identity f is called an essential weak polynomial identity if it is not a polynomial identity for R. We assume that H has a structure of a Lie algebra with respect to the multiplication [h 1, h 2 ] = h 1 h 2 h 2 h 1. Then the weak identities for (R, H) form an ideal T (R, H) of A which is invariant under all substitutions of the free generators of A by elements of the free Lie algebra L, considered as a subalgebra of A. We consider the free algebra A m = K x 1,..., x m of rank m as a subalgebra of A. The algebras A and A m are graded, A = A (k), where A (k) is the space of all homogeneous polynomials of degree k and A m = A (k) m, with A (k) m = A m A (k). Moreover, there is a more precise multigrading of A m counting the degree of each variable in the monomials from A m, A m = A (k 1,...,k m) m. In the sequel, all graded vector spaces are subspaces, vector spaces or tensor products of subspaces of A and A m with these particular gradings. Let V k = A (1,...,1) k, k = 1, 2,..., be the vector space of all multilinear polynomials of degree k in A k. The polynomial identities for an algebra R are determined by the multilinear ones, i.e. as a T-ideal T (R) is generated by V k T (R), k = 1, 2,.... For many problems, the study of T (R) is equivalent to the study of V k T (R) for every k. Let S k be the symmetric group of degree k acting on the set of symbols {1,..., k}. The vector space V k is a left S k -module with action of S k defined by σ(x i1... x ik ) = x σ(i1 )... x σ(ik ),

10 where σ S k, x i1... x ik V k. As an S k -module V k is isomorphic to the group algebra KS k equipped with the left regular S k -action (by multiplication from the left). For any algebra R, the vector space V k T (R) is an S k -submodule of V k, k = 1, 2,.... The irreducible representations of the symmetric group S k are described by partitions of k and Young diagrams. Let λ = (λ 1,..., λ r ), λ 1... λ r 0, λ λ r = k be a partition of k (notation λ k). The partition is of height h if λ h > 0 and λ h+1 = 0. We relate to the partition λ = (λ 1,..., λ r ) k its Young diagram D λ with r rows of λ 1,..., λ r boxes. For example, if λ = (4, 2 2, 1) then D λ = We denote by M λ and χ λ respectively the irreducible S k -module related with the partition λ and its character. Then V k T (R) = λ m λ (R)M λ, (1) χ Sk (V k T (R)) = λ m λ (R)χ λ and one of the main problems in the quantitative study of the polynomial identities of R is to describe the multiplicities m λ (R). A result of Regev [64] gives that the degree of the S k -character of V k T (R) (i.e. the dimension of the vector space V k T (R)) is very close to dimv k = k! and it is more reasonable instead of the S k -character of V k T (R) to study the k-th cocharacter of R which has the decomposition χ k (R) = χ Sk (V k /(V k T (R))), k = 1, 2,..., χ k (R) = λ m λ (R)χ λ, where m λ (R) = d λ m λ (R) with d λ = degχ λ = dimm λ. As a first approximation to the problem one considers the k-th codimension of R c k (R) = degχ k (R) = dim(v k /(V k T (R))), k = 1, 2,....

11 The celebrated theorem of Regev [64] states that there exists a positive constant a such that c k (R) a k for all k = 1, 2,.... Recently, Giambruno and Zaicev [37, 38] have proved that the limit lim k k c k (R) exists for any PI-algebra R and is a nonnegative integer. In the special case of the matrix algebra the exact asymptotics of the codimension sequence is obtained by Regev, see [66]: For any n > 1 c k (M n (K)) (2π) (1 n)/2 2 (1 n2 )/2 1!2!... (n 1)!n (n2 +4)/2 k (1 n2 )/2 n 2k+2. The exact values of c k (M n (K)) (for all k) is known for n = 2 only. Procesi [57] proved that c k (M 2 (K)) = 1 ( ) ( ) 2k + 2 k k. k + 2 k The degrees of the irreducible S k -characters are given by the hook formula. Let λ = (λ 1,..., λ r ) be a partition of k and let λ 1,..., λ λ 1 be the lengths of the columns of the diagram D λ. Then d λ = n! i,j h ij where h ij = λ i j + λ j i + 1 is the hook length of the (i, j)-th box. For example, if λ = (4, 2 2, 1) then the hook length of the box in position (2, 1) given below is h 21 = 4: X X X X The vector space W m with basis x 1,..., x m is a left GL m -module with the canonical action of the general group GL m = GL m (K). The free algebra A m is isomorphic to the tensor algebra of W m. Therefore A m is a GL m -module with the diagonal action of GL m defined by g(x i1... x ik ) = g(x i1 )... g(x ik ), g GL m, x i1... x ik A m, and A m T (R) is a GL m -submodule of A m. The irreducible polynomial representations of GL m are also described by partitions. We denote by W m,λ the irreducible GL m -module related with λ. Then A (k) m T (R) = λ ˇm λ (R)W m,λ. (2)

12 It is known that for any T-ideal T (R) the action of S k on V k T (R) and of GL m on A (k) m T (R), k = 1, 2,..., are equivalent in the following way. PROPOSITION 1 (see [13] and [9]) If the S k -module V k T (R) has the decomposition given in (1) and the GL m -module A (k) m T (R) has the decomposition (2), then m λ (R) = ˇm λ (R) for each λ of height m. This assertion holds also for the ideal T (R, H) of the weak identities for the pair (R, H). In particular, V k T (R, H) and A (k) m T (R, H) are S k - and GL m -modules, respectively, with the same multiplicities of the corresponding irreducible components. For concrete computations we need explicit generators of the irreducible submodules of the S k -module V k and the GL m -module A (k) m. We fix a partition λ = (λ 1,..., λ r ) k. A Young tableau of shape λ (or a λ-tableau) is a Young diagram whose boxes are filled in with the integers 1, 2,..., k in an arbitrary way (and each 1, 2,..., k appears in the tableau exactly once). The tableau is standard if its entries increase from top to bottom in the columns and from left to right in the rows. It is known that the number of the standard λ-tableaux is equal to d λ. For τ S k we denote by D τ = Dλ τ the Young tableau such that the first column of Dτ is filled in consequently from top to bottom with the integers τ(1),..., τ(λ 1 ), the second column is filled in with τ(λ 1 + 1),..., τ(λ 1 + λ 2 ), etc. We denote by ST λ the set of permutations τ S k corresponding to standard tableaux D τ. Since the S k -modules V k and KS k are naturally isomorphic, for our purposes it is sufficient to describe the generators of M λ KS k. The group S k acts on the set of all λ-tableaux: If the (i, j)-th box of the tableau D τ contains the integer p, then the (i, j)-th box of σ(d τ ) contains σ(p), σ S k. The row stabilizer R(D τ ) of D τ is the subgroup of S k which fixes the sets of entries of each row of D τ. Similarly one defines the column stabilizer C(D τ ) of the tableau. The element of the group algebra KS k e(d τ ) = (signγ)ργ ρ R(D τ ) γ C(D τ ) generates an irreducible S k -module (i.e. a minimal left ideal of KS k ) which is isomorphic to M λ. Up to a multiplicative constant, e(d τ ) is equal to a minimal idempotent of the group algebra KS k. The automorphisms of the S k -module M λ are indexed by non-zero elements of K. Such an automorphism ψ α, 0 α K, is defined by ψ α (v) = αv, v M λ. Hence, for each pair of λ-tableaux D τ 1 and D τ 2 there is an isomorphism φ τ1 τ 2 between the S k -modules KS k e(d τ 1 ) and

13 KS k e(d τ 2 ) such that all isomorphisms are αφ τ1 τ 2, 0 α K. The concrete form of φ τ1 τ 2 is the following: Let π be the element in S k which sends D τ 1 to D τ 2. Then e(d τ 1 ) = π 1 e(d τ 2 )π and φ τ1 τ 2 a σ σ e(d τ 1 ) = a σ σ π 1 e(d τ 2 ). σ S k σ S k In the S k -module KS k = λ k d λ M λ we can choose the d λ direct summands M λ as M τ λ = KS k e(d τ ), where D τ, τ ST λ, are all standard λ-tableaux. Hence, if ε S k is the identity permutation and e 1 = e(d ε ), then every submodule of KS k isomorphic to M λ is generated by an element e = b τ e 1 τ 1 = b τ τ 1 e(d τ ). τ ST λ τ ST λ Two submodules M = M = M λ of KS k coincide if and only if the ST λ - tuples of coefficients (b τ τ ST λ ) and (b τ τ ST λ ) related with M and M, respectively, are equal up to a multiplicative constant. Up to a multiplicative constant there exists a unique generator w λ of W m,λ which is multihomogeneous of degree (λ 1,..., λ m ). This element is called the highest weight vector of W m,λ and can be described in the following way. The symmetric group S k acts from the right on the homogeneous component of degree k of A m by place permutation (x i1... x ik )σ 1 = (x iσ(1)... x iσ(k) ), σ S k, x i1... x ik A (k) m. Every nonzero element w λ =w λ (x 1,..., x m ), where λ 1 w λ (x 1,..., x m )= s λ j (x 1,..., x λ j ) b σ σ= b σ wλ, σ b σ K (3) j=1 σ S k σ S k is the highest weight vector of a submodule W m,λ of A (k) m. The complete linearization of w λ generates an irreducible S k -module M λ and every submodule M λ of V k can be obtained in this way. The highest weight vector w λ of W m,λ can be expressed uniquely as a linear combination of the polynomials w τ λ where the τ s are such that Dτ λ are standard tableaux (see [27]) w λ = ξ τ wλ, τ ξ τ K. (4) τ ST λ

14 We refer to the books by Weyl [76], Macdonald [52], James and Kerber [41] and Sagan [70] for the representation theory of the symmetric and the general linear groups. For further reading for applications to PI-algebras we refer to the survey articles of Regev [66] and one of the authors [17, 18]. The study of polynomial identities can be carried out also for K-algebras with involution. By definition, an involution of an algebra R (notation (R, )) is a linear operator of R such that (r ) = r and (r 1 r 2 ) = r2 r 1 for all r, r 1, r 2 R. Hence, the involution is an antiisomorphism of the algebra with square equal to the identity map. We denote by K X, = K x 1, x 1, x 2, x 2,... the free associative algebra with involution generated by X over K. As in the case of ordinary polynomial identities, in characteristic zero, the -polynomial identities of an algebra are determinated by the multilinear ones. Hence, if we denote by V k ( ) the space of all multilinear polynomials of degree k in x 1, x 1,..., x k, x k, the study of T (R, ) is equivalent to the study of V k ( ) T (R, ) for any k 1. It is useful to regard the free associative algebra with involution K X, as generated by symmetric and skewsymmetric variables. We set s i = x i + x i and k i = x i x i, for i = 1, 2,..., and then K X, = K s 1, k 1, s 2, k 2,.... Moreover, for any K-algebra R with involution, R = R + R where R + = {r R r = r} and R = {r R r = r} are the spaces of symmetric and skew-symmetric elements of R, respectively. Therefore f(s 1,..., s p, k 1,..., k q ) K X, is a -polynomial identity for R if f(r 1 +,..., r+ p, r1,..., r q ) = 0 for all r i + R +, rj R, i = 1,..., p, j = 1,..., q. We can write the space of all multilinear polynomials of degree k as V k ( ) = Span K {w σ(1) w σ(k) σ S k, w i = s i or w i = k i, i = 1,... k}. Recall that if Z 2 = {1, } is the cyclic group of order 2, then B k is the wreath product Z 2 S k = {(a 1,..., a k ; σ) a i Z 2, σ S k } with multiplication given by (a 1,..., a k ; σ) (b 1,..., b k ; τ) = (a 1 b σ 1 (1),..., a k b σ 1 (k); στ). The space V k ( ) has a structure of a left B k -module induced by the action of h = (a 1,..., a k ; σ) B k defined by hs i = s σ(i), hk i = k a σ(i) σ(i) = ±k σ(i). There is a one-to-one correspondence between irreducible B k -characters and pairs of partitions (λ, µ), where λ r, µ k r, for all r = 0, 1,..., k. If M λ,µ and χ λ,µ denote the irreducible B k -module corresponding to (λ, µ) and its character, we write V k ( ) T (R, ) = λ m λ,µ (R, )M λ,µ,

15 χ Bk (R, ) = k r=0 λ r µ k r m λ,µ (R, )χ λ,µ, (5) where m λ,µ (R, ) 0 are the corresponding multiplicities. For every K-algebra R with involution the vector space V k ( ) T (R, ) is invariant under the above action of B k, hence the space V k (R, ) = V k ( )/ (V k ( ) T (R, )) has a structure of left B k -module. Its character χ k (R, ) is called the k-th -cocharacter of R and it has the following decomposition χ k (R, ) = k r=0 λ r µ k r m λ,µ (R, )χ λ,µ. Again, as in the case of ordinary polynomial identities, one of the main problems for the -polynomial identities of R is to describe the multiplicities m λ (R, ) or, equivalently, m λ (R, ). For r fixed we let V r,k r ( ) be the space of multilinear polynomials in s 1,..., s r, k r+1,..., k k. Now, if we let S r act on the symmetric variables s 1,..., s r and S k r on the skew-symmetric ones k r+1,..., k k, then we obtain an action of S r S k r on V r,k r ( ). Since T-ideals are invariant under permutations of symmetric and skew-symmetric variables, we obtain that V r,k r (R, ) = V r,k r ( ) T (R, ) has the induced structure of a left S r S k r -module. We denote by χ r,k r (R, ) its character. It is well known that there is a one-to-one correspondence between irreducible S r S k r - characters and pairs of partition (λ, µ) such that λ r, and µ k r. Hence, by the complete reducibility, we have the decomposition χ r,k r (R, ) = λ r µ k r m λ,µ (R, )(χ λ χ µ ), (6) where χ λ (resp. χ µ ) denotes the usual S r -character (resp. S k r -character), χ λ χ µ is the irreducible S r S k r -character associated with the pair (λ, µ) and m λ,µ (R, ) 0 is the corresponding multiplicity. The relation between the B k -character and the S r S k r -character is expressed in the following: PROPOSITION 2 ([21], Theorem 1.3) If the k-th -character of R has the decomposition given in (5) and the S r S k r -character of V r,k r (R, ) has the decomposition (6), then m λ,µ (R, ) = m λ,µ (R, ), for all λ and µ. Let us denote by K m X, = K x 1, x 1,..., x m, x m the free associative algebra with involution and of rank m. We can assume that it is

16 generated by the symmetric variables s i and the skew-symmetric variables k i, i = 1,..., m, i.e., K m X, = K s 1, k 1,..., s m, k m. Let U = Span K {s 1,..., s m } and V = Span K {k 1,..., k m }. The group GL(U) GL(V ) = GL m GL m acts from the left on the space U V and we can extend this action diagonally to an action on K m X,. For every -Tideal T (R, ), K m (R, ) = K m X, T (R, ) is a GL m GL m -module. Let K m (k) (R, ) be its homogeneous component of degree k; it is a GL m GL m - submodule of K m (R, ) and we denote its character by ψ k (R, ). The irreducible polynomial GL m GL m -characters are described by pairs of partitions (λ, µ), where λ r and µ k r, for all r = 0,..., k. If we denote by ψ λ,µ the irreducible GL m GL m -character associated to the pair (λ, µ), then we have the following decomposition ψ k (R, ) = k r=0 λ r µ k r ˆm λ,µ (R, )ψ λ,µ. (7) The B k -module structure of V k (R, ) and the GL m GL m -module structure of K (k) m (R, ) are related by the following result: PROPOSITION 3 ([33], Theorem 3) If the k-th -character of R has the decomposition given in (5) and the GL m GL m -character of K m (k) (R, ) has the decomposition (7) then m λ,µ (R, ) = ˆm λ,µ (R, ), for all λ, µ. It is well known that every irreducible submodule of K (k) m X, corresponding to the pair (λ, µ) is cyclic and is generated by a nonzero polynomial called the highest weight vector w λ,µ of the form w λ,µ = w λ,µ (s 1,..., s p, k 1,..., k q ) = w λ (s 1,..., s p ) w µ (k 1,..., k q ) τ S k a τ τ, where a τ K, w λ = w λ (s 1,..., s p ) and w µ = w µ (k 1,..., k q ) are as in (3). As in the ordinary case w λ,µ can be expressed as a linear combination of the polynomials (w ρ λ wσ µ)τ where the ρ s and σ s are such that the tableaux of the pair (Dλ σ, Dσ µ) are standard and the action of τ S k from the right determines the positions of the variables s 1,..., s p and k 1,..., k q : w λ,µ = ρ ST λ,σ STµ, τ S k ρ ξ ρστ (wλ µ)τ. (8) Notice that if we consider symmetric (respectively skew-symmetric) variables only, then µ = (resp. λ = ) and w λ,µ (s 1,..., s p, k 1,..., k q ) = w λ (s 1,..., s p ) (resp. w λ,µ (s 1,..., s p, k 1,..., k q ) = w µ (k 1,..., k q )). For a complete description of the representation theory of the group B k = Z 2 S k on V k ( ) and of GL m GL m on K m X, we refer to [35] and [33].

17 4 COMPUTATIONAL APPROACHES In this section we describe several computational methods to study PIalgebras. All of them are based on the following idea. Let H be a subspace of the finite dimensional algebra R and let P be a finite dimensional subspace of the free algebra A. We want to find all polynomials in P which vanish on H. We fix a basis {w j (x 1,..., x m ) j = 1,..., d} of P and consider the polynomial d w(x 1,..., x m ) = ξ j w j (x 1,..., x m ) P, ξ j K, (9) j=1 where we consider ξ j, j = 1,..., d, as unknowns. Giving different values h (s) 1,..., h(s) m H to x 1,..., x m, we obtain a homogeneous linear system w(h (s) 1,..., h(s) m ) = p j=1 ξ j w j (h (s) 1,..., h(s) m ) = 0, with respect to the d unknowns ξ j and with coefficients w j (h (s) 1,..., h(s) m ) in R. Fixing a basis of R, and assuming that the coordinates of the above sums are equal to 0, each equation gives rise to dim R equations with coefficients in K. In this way, the total number of the equations in the system is equal to dim R (number of experiments ), where under an experiment and perform the calculations. The solutions of this system are potential candidates for polynomials in P vanishing on H. The number n P (R, H) of the linear independent solutions gives an upper bound for the dimension m P (R, H) of the vector space of such polynomials. One of the main difficulties is to decide how far is the bound n P (R, H) from the real value m P (R, H). In principle, this problem can be solved with additional arguments and reasonable amount of additional calculations if the degree k of the identities is not too high. We shall discuss it in any concrete case which we consider. Another possibility is to evaluate x 1,..., x m in (9) with generic objects. Fixing a basis {h 1,..., h p } of H, we define the generic elements we mean to fix the values of h (s) 1,..., h(s) m y i = z i1 h z ip h p, i = 1,..., m, where the z ij s are algebraically independent commuting variables. Then w(x 1,..., x m ) P vanishes on H if and only if w(y 1,..., y m ) = 0. Clearly, w(y 1,..., y m ) is a linear combination of the basis elements of R with coefficients which are polynomials in the z ij s. If w(x 1,..., x m ) is in the form (9), we obtain a system with unknowns the ξ j s and polynomial coefficients. Unfortunately, in most of the cases the calculations are too big.

18 Typical examples of generic objects are the generic matrices. A version of the above scheme was used by Schelter [71] to find the first new polynomial identities for 2 2 matrices over a field of characteristic 2 which do not originate from the identities of M 2 (Z) and the identity 2x = 0. See also [28] for a multilinear version of this result (obtained by calculations by hand) and [2]. First we shall consider the problem to find all polynomial identities of degree k for M n (K). The simplest (but not the best) algorithm uses that the polynomial identities of any algebra are equivalent to the multilinear identities and is the following. We consider the multilinear polynomial f(x 1,..., x k ) = σ S k ξ σ x σ(1)... x σ(k) with unknown coefficients ξ σ, σ S k. Since f(x 1,..., x k ) is a polynomial identity for M n (K) if and only if f(e i1 j 1,..., e ik j k ) = 0 for all matrix units e ipj p, p = 1,..., k, we consider each of the n 2 entries of f(e i1 j 1,..., e ik j k ) = σ S k ξ σ e iσ(1) j σ(1)... e iσ(k) j σ(k) = 0 (10) as a linear homogeneous equation. In this way, we obtain a system with k! unknowns ξ σ, σ S k, and n 2 (n 2 ) k equations. The set of the solutions coincides with the set of multilinear identities of degree k. For n = 5 and k = 12 (the most difficult case handled up till now, but with other methods) we have 12! = unknowns, equations and the number of independent solutions is equal to Although many of the equations are trivial, the number of the unknowns and the equations which we essentially need (at least c 12 (M 5 (K)) = 12! 8491) is so big that the task is out of reach for the contemporary computers. There is a nice graph theoretic approach to eliminate the trivial equations which was initially created and used for purely theoretic considerations. It is based on arguments used in the Swan proof of the Amitsur-Levitzki theorem [74], see also [20]. For every set of matrix units {e i1 j 1,..., e ik j k } M n (K) we associate an oriented graph with a set of vertices {1,..., n} and a set of oriented edges {(i 1, j 1 ),..., (i k, j k )}. Then the product e i1 j 1... e ik j k is nonzero if and only if (i 1, j 1 ),..., (i k, j k ) is a path of the graph. Oriented graphs which have paths going exactly once through all edges are called Eulerian. Hence, it is sufficient to consider only those equations (10) and only those summands there which correspond to Eulerian graphs and Eulerian paths, respectively. Of course, one may use some symmetries (e.g. to consider the graphs up to an isomorphism) in order to decrease additionally the amount of calculations. Although these graph theoretic arguments cannot bring the number of the equations in the above mentioned case below the bound 12! 8491, they can be useful for other considerations.

19 Instead of considering all multilinear identities of a given degree, it is better to act following the Roman Divide et impera! (Divide and conquer!). In our case, this means to fix a partition λ of k and to consider polynomials which belong to the direct sum of all irreducible S k -modules M λ V k or all irreducible GL m -submodules W m,λ of A (k) m. We shall pay more attention to the method based on the representations of GL m. If the highest weight vector w λ is a polynomial identity for a M n (K), then w λ (r 1,..., r m ) = 0, for all r 1,..., r m M n (K). Therefore, from (4), ξ τ wλ(r τ 1,..., r m ) = 0, (11) τ ST λ where we consider the ξ τ s as unknowns. There are d λ standard tableaux. The solutions of this system give potential candidates for polynomial identities of M n (K). The number n λ (M n (K)) of the linear independent solutions gives an upper bound for m λ (M n (K)): n λ (M n (K)) m λ (M n (K)). The approach described above can be also used to investigate the weak polynomial identities. (Its earlier versions were used in [27] and [26] to study weak polynomial identities of degree 6 for M 3 (K) and of degree 9 for M 4 (K), respectively.) If R = M n (K) is the n n matrix algebra and H = sl n (K), then, by the method of Razmyslov, we can study the central polynomials of R. If R is a K-algebra with involution we can use this method also to study -polynomial identities of R in symmetric or skew-symmetric variables only. In fact, if w λ,µ is a -polynomial identity for (R, ), then w λ,µ (r + 1,..., r+ p, r 1,..., r q ) = 0, r + 1,..., r+ p R +, r 1,..., r q R. Hence, from (8), ξ ρστ (wλ (r+ 1,..., r+ p )wµ(r σ 1,..., r q ))τ = 0. (12) τ S k ρ ST λ,σ STµ, ρ If we consider symmetric variables only, then w λ,µ (r + 1,..., r+ p, r 1,..., r q ) = w λ (r + 1,..., r+ p ) and (12) becomes in the form (11) σ S k,σ ST λ ξ σ w σ λ(r + 1,..., r+ p ) = 0, (13) with r 1 +,..., r+ p R +. Similarly, for skew-symmetric variables we obtain ξ σ wµ(r σ 1,..., r q ) = 0, r1,..., r q R. σ ST µ

20 In principle, the described calculations can be performed only over a constructive base field K. This is not a problem if the algebra R is sufficiently good. If for a fixed basis {e 1,..., e s } all products e i e j are linear combinations of the basis elements with coefficients of a subfield K 0 of K, then the K 0 -algebra R 0 with K 0 -basis {e 1,..., e s } and the K-algebra R have the same polynomial identities in the following sense. If we embed canonically K 0 X into K X and T K0 (R 0 ) and T K (R) are the T-ideals of R 0 and R in K 0 X and K X, respectively, then the K-vector space T K (R) is spanned by T K0 (R 0 ). The multiplicities m λ (R 0 ) and m λ (R) of the irreducible K 0 S n - and KS n -characters are also the same. In the special case of matrices, we can restrict our calculations to the case K = Q only. Since the ring M n (Z) and the Q-algebra M n (Q) have the same identities in the above sense, and we have to solve a homogeneous linear system with rational coefficients, we may replace the variables of wλ σ with matrices with integer entries and consider systems with integer unknowns. Since we are interested in upper bounds of the number of independent solutions of the systems, as in [26] we may work modulo some sufficiently large prime number. A similar idea, but in the language of multilinear polynomial identities and representations of the symmetric group was used by Bondari [11]. In order to reduce the amount of calculations he used probabilistic methods. Recently Vishne [75] has presented a new computational method for the study of the multilinear polynomial identities for matrix algebras with entries from the base field or from the Grassmann algebra and for their important subalgebras. As in [11], his method is based on representation theory of the symmetric group combined with ideas from graph theory, see [75] for more details. Fortunately, in some of the considerations (e.g. for the polynomial identities of degree 2n + 2 for n n matrices), in both the cases of usage of representations of GL m and S k, we have lower theoretical bounds for m λ (M n (K)), which coincide with the upper bounds n λ (M n (K)) obtained experimentally. This gives that n λ (M n (K)) = m λ (M n (K)). For other ways to be sure that the multiplicities m λ and n λ coincide, see [27] and [26]. For example, the exact values of m λ (M 3 (K), sl 3 (K)), λ 6, were found in [27] with computer only, combined with Eulerian graphs arguments. The proof of the existence and the concrete form of the weak identity w λ, λ = (5, 1 4 ), for (M 4 (K), sl 4 (K)) which gave rise to a central polynomial of degree 13 for M 4 (K) was also performed by computer only and then, using this concrete form, a purely theoretical proof was found, see [26].

21 5 APPLICATIONS In this section we show some applications and concrete results of the computational approach using representations of GL m, for polynomial identities, weak polynomial identities, central polynomials and -polynomial identities of matrix algebras. 5.1 Polynomial identities of degree 2n + 2 The Hall identity [[x 1, x 2 ] 2, x 3 ] does not follow from the standard identity s 4 (x 1, x 2, x 3, x 4 ) and the minimal degree of the identities for M 2 (K) which do not follow from s 4 is equal to 5. On the other hand, by the result of Leron [50], for n 3 all identities of degree 2n + 1 for M n (K) follow from s 2n. By [24] the polynomial identities of degree 8 of M 3 (K) are consequences of the standard identity of degree 6. There are many reasons to believe that the identities for M n (K) of degree 2n + k are consequences of s 2n for small k. It seems that here k increases at least linearly with the size n of the matrices. In this subsection we shall verify for n = 4, 5 the following conjecture. CONJECTURE 1 For n 3 all polynomial identities of degree 2n + 2 for M n (K) are consequences of the standard identity s 2n. Our strategy is the following. From [1], s 2n is a polynomial identity for M n (K), i.e. s 2n T T (M n (K)) and m λ ( s 2n T ) m λ (M n (K)), for all λ 2n + 2. If we use the computational method described above, then we can determine n λ (M n (K)) linear independent solutions and we have n λ (M n (K)) m λ (M n (K)) m λ ( s 2n T ), λ 2n + 2. From [7], we know the exact values of m λ ( s 2n T ) for all partitions λ of 2n + 2, n 3. If our calculations show that n λ (M n (K)) = m λ ( s 2n T ) for some n 3, then we have the following results: 1. m λ (M n (K)) = m λ ( s 2n T ) for all λ 2n + 2; 2. All the polynomial identities of degree 2n + 2 for M n (K) and for the corresponding n, follow from s 2n. This strategy does not work for 2 2 matrices but in this case we have the complete quantitative information both for T (M 2 (K)) and s 4 T : The cocharacters of the T-ideal of M 2 (K) were calculated by Formanek [30] and one of the authors [15]. Kemer [44] found a basis for the multilinear elements of s 4 T modulo T (M 2 (K)). Restated in our language, see [16], the S k -character of (V k T (M 2 (K)))/(V k s 4 T ) is: χ(v 5 T (M 2 (K)))/(V 5 s 4 T ) = χ (3,2),

22 χ(v 6 T (M 2 (K)))/(V 6 s 4 T ) = χ (4,2) + 2χ (3 2 ) + χ (3,2,1), and in the general case k 7, χ(v k T (M 2 (K)))/(V k s 4 T ) = χ (k 2,2) + 2χ (k 3,3) + χ (k 3,2,1) +2χ (k 4,3,1) + χ (k 4,2 2 ) + 2χ (k 5,3,2) + χ (k 6,3 2 ), where for small k the summation is on all partitions which make sense (e.g. for (k 6, 3 2 ) we need k 9). Hence, if all polynomial identities of degree 2n + 2 for M n (K) are consequences of the standard identity s 2n, then everything depends on our possibilities for computer calculations. In order to find n λ (M n (K)) it is sufficient to perform the calculations only for those λ = (λ 1,..., λ r ) 2n + 2 such that λ 1 4. This follows from the fact in [7] that m λ ( s 2n T ) = 0 if λ 1 5 and from the the following assertion which is also of independent interest. (Nevertheless, in the calculations for n = 4, 5, see [6], we have worked out also these cases for λ 2n + 2 as a test of the computer programs.) PROPOSITION 4 [6] Let λ = (λ 1,..., λ r ) 2n + 2, n 2. If λ 1 5 then m λ (M n (K)) = 0. The methods for the calculations for the polynomial identities of degree 8 for M 3 (K) are discussed in the last section. In [6] we have performed the computer calculations for the polynomial identities for M n (K) of degree 2n + 2 for n = 4, 5 using 64 processors on the Cray T3E at the Lawrence Berkeley Lab. It took a total of about 8 hours to complete the computations for the identities of degree 12 for 5 5 matrices from start to finish. (In order to compare our results with the known facts, six hours of this time were spent for calculations which follow from other arguments.) The results show that our upper bounds n λ (M n (K)) coincide with the exact values of m λ ( s 2n T ) obtained theoretically in [7]. We shall state as a theorem the result obtained as a combination of theoretical conclusions and computer experiments: THEOREM 1 For n = 3, n = 4 and n = 5 all polynomial identities of degree 2n+2 for M n (K) are consequences of the standard identity of degree 2n. REMARK 1 The calculations for n = 5 and k = 12 as performed in our paper [6] demonstrate the possibilities and the advantages of the computational method based on representations of GL m. Instead of working with one system with 12! unknowns, as in the direct attempt to handle the multilinear identities, we have to consider 77 systems with up to 7700 unknowns