Alladi Ramakrishnan Hall

Root Multiplicities for Borcherds-Kac-Moody Algebras and Graph Coloring (Ph.D thesis defence)

G. Arunkumar

IISER Mohali

We explore the connection between the root multiplicities for Borcherds–Kac–Moody algebras and graph coloring. We 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 natural basis for the corresponding root spaces.

Next, we specialize to Borcherds algebras in which all simple roots are imaginary. The positive part of these algebras are isomorphic to free partially commutative Lie algebras. We determine the Hilbert series of the universal enveloping algebra of these algebras and as an application we give a Lie theoretic proof of Stanley’s reciprocity theorem of chromatic polynomials.

Finally, we consider the chromatic discriminant of a graph. The absolute value of the coefficient of the linear term in the chromatic polynomial of a graph is known as its chromatic discriminant. We give a fully combinatorial (bijective) proof, using acyclic orientations and spanning trees, for a recurrence formula for the chromatic discriminant that comes from the theory of Kac-Moody Lie algebras.