Symmetric group characters as symmetric functions

Rosa Orellana, Mike Zabrocki

Published 2016-05-21Version 1

We show that the irreducible characters of the symmetric group are symmetric polynomials evaluated at the eigenvalues of permutation matrices. In fact, these characters can be realized as symmetric functions that form a non-homogeneous basis for the ring of symmetric functions. We call this basis the {\it irreducible character basis}. Further, the structure coefficients for the (outer) product of these functions are the stable Kronecker coefficients. The induced trivial characters also give rise to a non-homogeneous basis of symmetric functions. We introduce the irreducible character basis by defining it in terms of the {\it induced trivial character basis}. In addition, we show that the irreducible character basis is closely related to character polynomials and we obtain some of the change of basis coefficients by making this connection explicit. Other change of basis coefficients come from a representation theoretic connection with the partition algebra, and still others are derived by developing combinatorial expressions.