A New Symmetric Function Identity With an Application to symmetric group character values

TL;DR

This paper introduces a new symmetric function identity related to symmetric group character sums, revealing a surprising equality and a zero-sum result for certain character sums over a novel set.

Contribution

It presents a novel symmetric function identity and applies it to symmetric group characters, establishing new equalities and zero-sum results for character sums over the set $Ev()$.

Findings

Equality between alternating sums of power sum symmetric polynomials and monomial symmetric polynomials.

A special case where the alternating sum of characters over $Ev()$ equals zero.

New conjecture related to symmetric functions and group characters.

Abstract

Symmetric functions show up in several areas of mathematics including enumerative combinatorics and representation theory. Tewodros Amdeberhan conjectures equalities of $\Sigma_n$ characters sums over a new set called $Ev(\lambda)$. When investigating the alternating sum of characters for $Ev(\lambda)$ written in terms of the inner product of Schur functions and power sum symmetric functions, we found an equality between the alternating sum of power sum symmetric polynomials and a product of monomial symmetric polynomials. As a consequence, a special case of an alternating sum of $\Sigma_n$ characters over the set $Ev(\lambda)$ equals $0$.