On the Monoids Associated to the Coxeter Complex and the Bruhat Order of a Coxeter Group
Time and place
4 PM on Friday, October 15th, 2010; GC 5417
Stuart Margolis (Bar-Ilan University and CAISS, CCNY)
Abstract
Abstract: Coxeter groups are ubiquitous in mathematics arising in many parts of algebra, geometry, topology and other fields, providing deep and surprising connections between diverse areas of mathematics and computer science. There are a number of geometric and combinatorial objects associated to a Coxeter group, two of the most important being the Coxeter complex and the Bruhat order. While these have played an extremely important role in the theory, it was only within the last years that it was realized that each of these has the natural structure of a monoid.
The monoid structures have both been “frequently rediscovered” and have only slowly been realized to give very important information about the Coxeter group itself, that is not available from a purely group theoretic perspective. For example, the first edition of Kenneth Brown’s book Buildings does not mention the monoid structure on the Coxeter Complex at all, whereas the monoid plays a prominent role in the second edition. Also, while no finite group has a non-trivial multiplicative partial order, Bruhat order turns out to be a multiplicative order on the associated monoid.
The purpose of this talk is to give an introduction to these monoids for both group and semigroup theorists. I will discuss their structure and how they fit into two very important classes of monoids, the so called left regular bands and J -trivial monoids. I will look at their representation theory as well as how they are used to give deeper information on the associated Coxeter group. Surprisingly, the algebras of these monoids are intimately related to the Solomon Descent Algebra and the 0-Hecke algebra of the Coxeter groups as well.
