Trace cocharacters and the Kronecker products of Schur functions

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.