Chromatic symmetric function of graphs from Borcherds algebras

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.