Chromatic symmetric function of graphs from Borcherds algebras

TL;DR

This paper establishes a Lie theoretic approach to compute the chromatic symmetric function of graphs derived from Borcherds algebras, linking graph invariants to root multiplicities and Weyl denominator identities.

Contribution

It provides a novel Lie algebraic proof of Stanley's formula for chromatic symmetric functions and relates graph invariants to root multiplicities of Borcherds algebras.

Findings

Chromatic symmetric function can be derived from Weyl denominator identity.

Graphs with different chromatic discriminants are distinguished by their chromatic symmetric functions.

Non-negativity of coefficients of G-power sum symmetric functions is proven using Lie theory.

Abstract

Let $\mathfrak g$ be a Borcherds algebra with the associated graph $G$. We prove that the chromatic symmetric function of $G$ can be recovered from the Weyl denominator identity of $\mathfrak g$ and this gives a Lie theoretic proof of Stanley's expression for chromatic symmetric function in terms of power sum symmetric function. Also, this gives an expression for chromatic symmetric function of $G$ in terms of root multiplicities of $\lie g$. The absolute value of the linear coefficient of the chromatic polynomial of $G$ is known as the chromatic discriminant of $G$. As an application of our main theorem, we prove that graphs with different chromatic discriminants are distinguished by their chromatic symmetric functions. Also, we find a connection between the Weyl denominators and the $G$-elementary symmetric functions. Using this connection, we give a Lie theoretic proof of non-negativity of coefficients of $G$-power sum symmetric functions.