Chromatic symmetric function of graphs from Borcherds algebras [article]

G. Arunkumar
2021 arXiv   pre-print
Let 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 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 g. The absolute value of the linear coefficient of the chromatic polynomial of G is known as
more » ... 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.
arXiv:1908.08198v2 fatcat:ukroh7oi4jf2nct7bsorwoob5y