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Root multiplicities for Borcherds algebras and graph coloring
- Source :
- J. Algebra 499 (2018), 538-569
- Publication Year :
- 2016
-
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.<br />Comment: 23 pages, comments are welcome
Details
- Database :
- arXiv
- Journal :
- J. Algebra 499 (2018), 538-569
- Publication Type :
- Report
- Accession number :
- edsarx.1612.01320
- Document Type :
- Working Paper