Plethysms of Chromatic and Tutte Symmetric Functions

TL;DR

This paper introduces a graph-theoretic interpretation of plethysm in symmetric functions, providing new proofs and insights into plethystic identities through chromatic symmetric functions.

Contribution

It offers the first combinatorial graph-based interpretation of plethysm in symmetric function theory, linking it with chromatic symmetric functions.

Findings

Graph-theoretic interpretation of plethysm

Simplified proofs of plethystic identities

New connections between chromatic and Tutte symmetric functions

Abstract

Plethysm is a fundamental operation in symmetric function theory, derived directly from its connection with representation theory. However, it does not admit a simple combinatorial interpretation, and finding coefficients of Schur function plethysms is a major open question. In this paper, we introduce a graph-theoretic interpretation for any plethysm based on the chromatic symmetric function. We use this interpretation to give simple proofs of new and previously known plethystic identities, as well as chromatic symmetric function identities.