Kronecher powers and character polynomials

Abstract

In this talk, I will present joint works with Cedric Chauve and Adriano Garsia. With C. Chauve, we studied Kronecker powers of the irreducible representation of Sn indexed with (n-1,1). We gave a combinatorial interpretation and a generating function for the coefficients of any irreducible representation in a k-th Kronecker power ( χ(n-1,1) )⊗k. With A. Garsia, we studied character polynomials qλ(x1,…,xn) which are polynomials in several variables with the fundamental property that their evaluation on the multiplicities (m1,m2, …,mn) of a partition µ of n gives the value of the irreducible character χ( n- | λ | , λ ) of the symmetric group Sn on the conjugacy class Cµ . Character polynomials are closely related to the problem of decomposition of Kronecker product of representations of Sn. They were defined by Specht in 1960. Since then they received little attention from the combinatorics community. I will show how character polyomials are related to Kronecker products, how to produce them, their algebraic structure and show some applications.