The Hopf structure of symmetric group characters as symmetric functions

TL;DR

This paper explores the algebraic structure of symmetric group characters through symmetric functions, establishing product and coproduct formulas and transition coefficients, thereby deepening understanding of their combinatorial and algebraic properties.

Contribution

It introduces new product and coproduct formulas for bases of symmetric functions related to symmetric group characters and computes transition coefficients to elementary symmetric functions.

Findings

Derived product and coproduct formulas for the character-based symmetric function bases.

Calculated transition coefficients between elementary symmetric functions and the irreducible character basis.

Enhanced the algebraic understanding of symmetric group characters within the symmetric functions framework.

Abstract

In arXiv:1605.06672 the authors introduced inhomogeneous bases of the ring of symmetric functions. The elements in these bases have the property that they evaluate to characters of symmetric groups. In this article we develop further properties of these bases by proving product and coproduct formulae. In addition, we give the transition coefficients between the elementary symmetric functions and the irreducible character basis.