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Root multiplicities for Borcherds algebras and graph coloring

Arunkumar, G and Kus, Deniz and Venkatesh, R (2018) Root multiplicities for Borcherds algebras and graph coloring. In: JOURNAL OF ALGEBRA, 499 . pp. 538-569.

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Official URL: http://dx.doi.org/10.1016/j.jalgebra.2017.11.050

Abstract

We establish a connection between 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. Using this connection we give a second interpretation, namely that the root multiplicity of a given root coincides with the number of acyclic orientations with a unique sink of a certain graph (depending on the root). Finally, 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. As an application we give a Lie theoretic proof of Stanley's reciprocity theorem of chromatic polynomials. (C) 2017 Elsevier Inc. All rights reserved.

Item Type: Journal Article
Additional Information: Copy right for the article belong to ACADEMIC PRESS INC ELSEVIER SCIENCE, 525 B ST, STE 1900, SAN DIEGO, CA 92101-4495 USA
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Depositing User: Id for Latest eprints
Date Deposited: 14 Mar 2018 17:40
Last Modified: 14 Mar 2018 17:40
URI: http://eprints.iisc.ac.in/id/eprint/59157

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