Multiplicities in the Trace Cocharacter Sequence of Two 4× 4 Matrices

We find explicitly the generating functions of the multiplicities in the pure and mixed trace cocharacter sequences of two 4×4 matrices over a field of characteristic 0. We determine the asymptotic behavior of the multiplicities and show that they behave as polynomials of 14th degree. Introduction Let us fix an arbitrary field F of characteristic 0 and two integers n, d ≥ 2. Consider the d generic n× n matrices X1, . . . , Xd. We denote by C the pure trace algebra generated by the traces of all products tr(Xi1 · · ·Xik), and by T the mixed trace algebra generated by X1, . . . , Xd and C, regarding the elements of C as scalar matrices. The algebra C coincides with the algebra of invariants of the general linear group GLn(F ) acting by simultaneous conjugation on d matrices of size n. The algebra T is the algebra of matrix concominants under a suitable action of GLn(F ). See e.g. the books [12], [11], or [5] as a background on C and T and their numerous applications. The algebras C and T are graded by multidegree. The Hilbert (or Poincaré) series of C is H(C) = H(C, t1, . . . , td) = ∑ dimCt1 1 · · · t kd d , where C is the homogeneous component of multidegree k = (k1, . . . , kd). In the same way one defines the Hilbert series of T . These series are symmetric functions and decompose as infinite linear combinations of Schur functions Sλ(t1, . . . , td), H(C) = ∑ mλ(C)Sλ(t1, . . . , td), where λ = (λ1, . . . , λd) is a partition in not more than d parts, and the mλ(C)s are nonnegative integers which are 0 if d > n and λn2+1 > 0. A similar expression holds for the Hilbert series of T . The multiplicities mλ(C) and mλ(T ) have important combinatorial properties and ring theoretical meanings. In particular, they are equal to the multiplicities of the irreducible Sk-characters in the sequences of pure and mixed trace cocharacters, respectively, and give estimates for the multiplicities in the “ordinary” cocharacters of the polynomial identities of the n × n matrix algebra. 2000 Mathematics Subject Classification. Primary: 16R30; Secondary: 05E05.