Abstracts
for the Seminar

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.