Chromatic symmetric function of graphs from Borcherds algebras

Abstract

The Weyl denominator identity has interesting combinatorial properties for several classes of Lie algebras. Along these lines, we prove that given a finite graph G, the chromatic symmetric function XG can be recovered from the Weyl denominator identity of a Borcherds-Kac-Moody Lie algebra g whose associated graph is G. This gives a connection between (a) the coefficients appearing when the chromatic symmetric function XG is expressed in terms of the power sum symmetric functions, and (b) the root multiplicities of the Borcherds algebra g. From this result, we deduce a Lie theoretic proof of various alternate expressions of the chromatic symmetric function obtained by Stanley. Examples using small rank Lie algebras are provided to illustrate our results.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 identify a coefficient appearing in XG, which equals the chromatic discriminant. We also find a connection between the Weyl denominator and the G-elementary symmetric functions. Using this connection, we give a Lie theoretic proof of the non-negativity of coefficients of G-power sum symmetric functions.