6533b836fe1ef96bd12a0b48

RESEARCH PRODUCT

Trace cocharacters and the Kronecker products of Schur functions

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Abstract It follows from the theory of trace identities developed by Procesi and Razmyslov that the trace cocharacters arising from the trace identities of the algebra Mr(F) of r×r matrices over a field F of characteristic zero are given by TCr,n=∑λ∈Λr(n)χλ⊗χλ where χλ⊗χλ denotes the Kronecker product of the irreducible characters of the symmetric group associated with the partition λ with itself and Λr(n) denotes the set of partitions of n with r or fewer parts, i.e. the set of partitions λ=(λ1⩽⋯⩽λk) with k⩽r. We study the behavior of the sequence of trace cocharacters TCr,n. In particular, we study the behavior of the coefficient of χ(ν,n−m) in TCr,n as a function of n where ν=(ν1⩽⋯⩽νk) is some fixed partition of m and n−m⩾νk. Our main result shows that such coefficients always grow as a polynomial in n of degree r−1.