UK Department of Mathematics Colloquium

"Enumeration in Coxeter groups"

Louis Billera
Cornell

Thursday, November 13, 2008
4:00 pm, 110 Patterson Office Tower

Abstract:

We study the strong Bruhat order of Coxeter groups from an enumerative perspective and encode the relevant information as a quasisymmetric function. This quasisymmetric function belongs to the subalgebra of peak functions and hence leads to an extension of the cd-index of convex polytopes. We call this the complete cd-index.

The holy grail of Coxeter groups is to understand the Kazhdan-Lusztig polynomials. These polynomials occur in topology and representation theory, but can be defined combinatorially. We show that there is a linear map from the cd-polynomials taking the complete cd-index to the Kazhdan-Lusztig polynomials, implying that the complete cd-index is a fundamental invariant that needs more study.

This is joint work with Francesco Brenti.

Bio

Louis Billera is a Professor of Mathematics at Cornell University. The common thread through much of his research is to study problems motivated by discrete and convex geometry. A sampling includes constructing polytopes to prove the sufficiency condition for the g-theorem (with Carl Lee), discovering fiber polytopes (with Bernd Sturmfels), and studying the space of phylogenetic trees (with Susan Holmes and Karen Vogtmann). In 1994 he won the Fulkerson prize for his paper, "Homology of smooth splines: Generic triangulations and a conjecture of Strang", Trans AMS 310 (1988) 325-340. This prize is given every three years to the best paper in Discrete Mathematics.

Louis Billera served as the first Associate Director of the National Science Foundation Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) at Rutgers. He has held visiting positions at Brandeis, Universite Catholique de Louvain in Belgium, MSRI and most recently, the Mittag-Leffler Institute in Sweden. He has graduated over 24 PhD students.