Tensor algebras, words, and invariants of polynomials in non-commutative variables

Abstract

Consider a vector space V for which we specify a basis, then the tensor algebra T(V) corresponds to the non-commutative polynomials expressed in that basis. If V has an S_n module structure (more generally, for a finite group) then identifying the invariants of the non-commutative polynomials corresponds to calculating the multiplicity of the trivial representation in the repeated Kronecker product of the Frobenius image of the module V. We consider a general method of arriving at a combinatorial interpretation for the Kronecker coefficients by embedding the representation ring within a group algebra. This is joint work with Anouk Bergeron-Brlek and Christophe Hohlweg.