A categorification of the chromatic symmetric function

TL;DR

This paper introduces a homological framework that categorifies the chromatic symmetric function of a graph, extending its combinatorial properties into a richer algebraic and homological context.

Contribution

It constructs a homology theory for graded S_n-modules that generalizes the chromatic symmetric function and lifts known decomposition formulas to exact sequences.

Findings

Homology reduces to the chromatic symmetric function at q=t=1

Decomposition formulas are lifted to long exact sequences in homology

Provides a new algebraic perspective on graph invariants

Abstract

The Stanley chromatic symmetric function $X_G$ of a graph $G$ is a symmetric function generalization of the chromatic polynomial, and has interesting combinatorial properties. We apply the ideas of Khovanov homology to construct a homology of graded $S_n$-modules, whose graded Frobenius series $Frob_G(q,t)$ reduces to the chromatic symmetric function at $q=t=1$. This homology can be thought of as a categorification of the chromatic symmetric function, and provides a homological analogue of several familiar properties of $X_G$. In particular, the decomposition formula for $X_G$ discovered recently by Orellana and Scott, and Guay-Paquet is lifted to a long exact sequence in homology.