Symmetric group characters as symmetric functions

TL;DR

This paper introduces a new basis of symmetric functions that directly encodes the irreducible characters of the symmetric group, providing novel characterizations and linking to stable Kronecker coefficients.

Contribution

It presents a new symmetric function basis that evaluates to symmetric group characters and establishes three characterizations, connecting to stable Kronecker coefficients.

Findings

The basis evaluates to irreducible symmetric group characters.

One characterization links the basis to stable Kronecker coefficients.

The paper develops fundamental properties of this new basis.

Abstract

We introduce a basis of the symmetric functions that evaluates to the (irreducible) characters of the symmetric group, just as the Schur functions evaluate to the irreducible characters of $GL_n$ modules. Our main result gives three different characterizations for this basis. One of the characterizations shows that the structure coefficients for the (outer) product of these functions are the stable Kronecker coefficients. The results in this paper focus on developing the fundamental properties of this basis.