Description
Speaker: G. Arunkumar
Date & Time : Monday Feb 26, at 11:30am
Venue: Ramanujan Hall
Title: Chromatic polynomials and Lie algebras
Abstract: In this talk, I will prove a connection between root
multiplicities for Borcherds-Kac-Moody
algebras and graph coloring. I will show that the generalized
chromatic polynomial of the graph
associated to a given Borcherds algebra can be used to give a closed
formula for certain root
multiplicities. As an application, using the combinatorics of Lyndon
words, we construct a basis for the root spaces corresponding to these
roots and determine
the Hilbert series in the case when all simple roots are imaginary.
In last ten minutes, We will talk about chromatic discriminant of a graph:
The absolute value of the coefficient of q in the chromatic polynomial
of a graph
G is known as the chromatic discriminant of G and is denoted
$\alpha(G)$. We start with a brief survey on many interesting
algebraic and combinatorial interpretations of $\alpha(G)$. We use two
of these interpretations (in terms of
acyclic orientations and spanning trees) to give two bijective proofs
for a recurrence formula
of $\alpha(G)$ which comes from the Peterson recurrence formula for
root multiplicities of Kac-Moody algebras.