Chromatic symmetric function of graphs from Borcherds algebras
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by
G. Arunkumar
2019
Abstract
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 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.
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