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

Abstract

Abstract. 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.

Maximal multiplicities in cocharacter sequences

“…In all cases which have been calculated, the lower bound for mlt (h) is correct: We proved in [4] that mlt (2) (M 2 (F )) = 2 which equals (2 − 1)4 − 2+1 2 + 1; Drensky, Genov and Valenti proved in [16] thatm (2) n (M 3 (F )) is polynomial of degree 7. Since…”

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

“…Later they [7] suggested a method to find the coefficients of the Schur functions in the expansion of a class of rational symmetric functions in two variables. Jointly with Valenti [8] they have used the expression of the Hilbert series H(T ) found by Berele and Stembridge [3] and have calculated explicitly the multiplicities m (λ1,λ2) (T ) for n = 3 and d = 2.…”

“…It turns out that the multiplicities m (λ1,λ2) (C) and m (λ1,λ2) (T ) behave as polynomials of degree 14 in λ 1 and λ 2 . As in [8], our approach is to apply the methods of [7] to the explicit form of the Hilbert series of C and T found in [3]. As in [8], results of Formanek [9,10] imply that the values of the multiplicities m λ (M 4 (F )) for λ = (λ 1 , .…”

“…As in [8], our approach is to apply the methods of [7] to the explicit form of the Hilbert series of C and T found in [3]. As in [8], results of Formanek [9,10] imply that the values of the multiplicities m λ (M 4 (F )) for λ = (λ 1 , . .…”

Using Hook Schur Functions to Compute Matrix Cocharacters

“…These multiplicities have been the subject of too many papers for us to attempt a list. Recent ones include [3], [7], [9] and [13] . An important tool in the study of the m λ is the Poincaré series P n (x 1 , .…”