## Coxeter Groups, The Waldspurger Decomposition, and Alternating Sign Matrices

- April 21, 2017
- 2:30 p.m.
- LeConte 312

## Abstract

Building on the work of Cartan, Killing, Lie, Weyl, and others, a classification of finite groups generated by reflections was completed by H.S.M. Coxeter in 1935. In 2005, however, J.L. Waldspurger proved a remarkable theorem giving a decomposition of the closed cone over the positive roots defined by an action of the Weyl group. In this talk I will state Coxeter's classification theorem but then focus mainly on Type A, where the Weyl group is the symmetric group. I will show some nice pictures of the Waldspurger decomposition in low dimensions, and provide a new combinatorial description. Specifically, I associate every permutation with a corresponding "Waldspurger Matrix". We will see that componentwise comparison of Waldspurger matrices is actually the Bruhat order on the symmetric group, and that the MacNeille completion here has a particularly nice description. Among other things, this provides a new way of thinking about Alternating Sign Matrices