Discrete CATS Seminar
U N I V E R S I T Y   O F   K E N T U C K Y
WHERE CATS =
845 PATTERSON OFFICE TOWER
2008 - 2009
Department of Mathematics Colloquium
"Enumeration in Coxeter groups"
Thursday, November 13, 2008
4:00 pm, 110 Patterson Office Tower
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.
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.