Symmetric group characters as symmetric functions (extended abstract)

TL;DR

This paper introduces a new basis of symmetric functions derived from symmetric group characters, connecting representation theory, combinatorics, and symmetric functions, with potential implementation in computational algebra systems.

Contribution

It defines the irreducible character basis of symmetric functions, linking it to character polynomials, the partition algebra, and providing combinatorial expressions.

Findings

Defined the irreducible character basis in terms of induced trivial characters.

Connected the basis to character polynomials and the partition algebra.

Provided combinatorial formulas for change of basis coefficients.

Abstract

The irreducible characters of the symmetric group are a symmetric polynomial in the eigenvalues of a permutation matrix. They can therefore be realized as a symmetric function that can be evaluated at a set of variables and form a basis of the symmetric functions. This basis of the symmetric functions is of non-homogeneous degree and the (outer) product structure coefficients are the stable Kronecker coefficients. We introduce the irreducible character basis by defining it in terms of the induced trivial characters of the symmetric group which also form a basis of the symmetric functions. 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. This document is an extended abstract which can be used as a review reference so that this basis can be implemented in Sage. A more complete version of the results in this abstract can be found in {\tt arXiv:1605.06672}.