Seminar

This talk is divided into two parts: Let g be a Borcherds-Kac-Moody-Lie (super)algebra with the associated Quasi-Dynkin diagram G. In the first part, I will prove that the generalized chromatic polynomial of the graph G can be recovered from the Weyl denominator identity of g. From this result, I will deduce a closed formula for certain root multiplicities of g. Also, we construct a basis for these root spaces of g. In the second part, we are interested in the chromatic symmetric function of the graph G. I will prove an expression for the chromatic symmetric function of G in terms of root multiplicities of g. As an application, we will see Lie theoretic proof of many results of Stanley on chromatic symmetric functions. Stanley's tree conjecture is an important conjecture in the theory of chromatic symmetric functions which states that non-isomorphic trees are distinguished by their chromatic symmetric functions. We propose a Lie theoretic method to approach this conjecture. Finally, I will briefly discuss some future directions.