Multiplicities in the Mixed Trace Cocharacter Sequence of Two 3 × 3 Matrices

We find explicitly the multiplicities in the (mixed) trace cocharacter sequence of two 3 × 3 matrices over a field of characteristic 0 and show that asymptotically they behave as polynomials of seventh degree. As a consequence we obtain also the multiplicities of certain irreducible characters in the cocharacter sequence of the polynomial identities of 3× 3 matrices. Introduction All considerations in this paper are over an arbitrary field F of characteristic 0. Let n ≥ 2 be a fixed integer. Consider the d generic n × n matrices X1, . . . , Xd, d ≥ 2. The main results of our paper concern the case n = 3 and d = 2. There are several important algebras related to X1, . . . , Xd. Among them are the algebra Rd generated by X1, . . . , Xd, the pure (or commutative) trace algebra Cd generated by the traces of all products tr(Xi1 · · ·Xik), and the mixed (or noncommutative) trace algebra Td generated by Rd and Cd regarding the elements of Cd as scalar matrices. We denote by C,R, and T the corresponding algebras related to a countable set {X1, X2, . . .} of generic matrices. The algebra R is one of the most important objects in the theory of algebras with polynomial identities. It is isomorphic to the factor algebra F 〈x1, x2, . . .〉/I(Mn(F )) of the free associative algebra F 〈x1, x2, . . .〉 modulo the ideal I(Mn(F )) of the polynomial identities of the n × n matrix algebra Mn(F ). The algebra Cd has a natural interpretation in classical invariant theory, as the algebra of invariants of the general linear group GLn(F ) acting by simultaneous conjugation on d matrices of size n. The algebra Td is known as the algebra of matrix concominants and also consists of the invariant functions under a suitable action of GLn(F ). See e.g. the books [12], [11], or [6] as a background on Rd, Cd, and Td and their application to invariant theory, structure theory of PI-algebras, and theory of finite dimensional division algebras. The algebras Rd, Cd, Td as well as R,C, T are graded by multidegree. The symmetric group Sk acts naturally on the multilinear elements of degree k of R, T,C. The corresponding Sk-characters χk(R) = χk(Mn(F )), χk(C), χk(T ) are called, respectively, the cocharacter of the polynomial identities, the pure trace cocharacter, 2000 Mathematics Subject Classification. Primary: 16R30; Secondary: 05E05.