On the Schur function expansion of a symmetric quasi-symmetric function

TL;DR

This paper provides a simplified proof of a recent reformulation relating Schur function expansions of symmetric functions to fundamental quasi-symmetric functions, utilizing a sign-reversing involution for clarity.

Contribution

It offers a straightforward proof of Garsia and Remmel's reformulation, enhancing understanding of the Schur function expansion in symmetric and quasi-symmetric functions.

Findings

Simplified proof of Garsia and Remmel's reformulation

Use of sign-reversing involution for proof

Clarification of the relationship between Schur and quasi-symmetric functions

Abstract

Egge, Loehr, and Warrington proved a formula for the Schur function expansion of a symmetric function in terms of its expansion in fundamental quasi-symmetric functions. Their formula involves the coefficients of a modified inverse Kostka matrix. Recently Garsia and Remmel gave a simpler reformulation of Egge, Loehr, and Warrington's result, with a new proof. We give here a simple proof of Garsia and Remmel's version, using a sign-reversing involution.