Character Polynomials, their q-Analogs and the Kronecker Product

Abstract

The numerical calculation of character values as well as Kronecker coefficients can efficently be carried out by means of character polynomials. Yet these polynomials do not seem to have been given a proper role in present day literature or software. To show their remarkable simplicity we give here an "umbral" version and a recursive combinatorial construction. We also show that these polynomials have a natural counterpart in the standard Hecke algebra ${\cal H}_n(q\, )$. Their relation to Kronecker products is brought to the fore, as well as special cases and applications. This paper may also be used as a tutorial for working with character polynomials in the computation of Kronecker coefficients.