Abstracts for the Seminar
 Discrete Geometry and Combinatorics
 Fall 2014

Speaker:  Marcelo Aguiar, Cornell University
Title: The Steinberg torus and the Coxeter complex of a Weyl group
Time: 2:30 PM, Monday, September 29, 2014
Place:  Malott 206

Abstract: Given an irreducible crystallographic root system Φ, consider the torus obtained as the quotient of the ambient space by the coroot lattice of Φ. There is a certain cell complex structure on this torus, introduced by Steinberg and studied by Dilks, Petersen, and Stembridge. In joint work with Kyle Petersen, we exhibit a module structure on (the set of faces of) this complex over the (set of faces of the) Coxeter complex of Φ. The latter is a monoid under the Tits product of faces. The module structure is obtained from geometric considerations involving affine hyperplane arrangements. As a consequence, we obtain a module structure on the space spanned by affine descent classes of a Weyl group, over the classical descent algebra of Solomon. We provide combinatorial models when Φ is of type A or C.