The University of Michigan Combinatorics Seminar


Abstract 

For a Coxeter group W, the Coxeter complex is the simplicial complex whose faces are Wcosets of parabolic subgroups of W. For an irreducible affine Weyl group, we have a realization of the Coxeter complex as a triangulation of Euclidean space with a translational symmetry given by the coroot lattice. The quotient of an affine Coxeter complex by this (full rank) lattice is a triangulation of a torus, which we call the Steinberg torus. The are only finitely many facets in this complex, in bijection with the underlying finite Weyl group. I will describe how the hvectors of both Coxeter complexes and Steinberg tori have nice combinatorial interpretations in terms of descentlike statistics for the finite Coxeter (Weyl) group elements. Moreover, the hvectors in both cases have a nonnegative expansion in the symmetric and unimodal ``gamma" basis suggested by Gal, the combinatorics here governed by peaklike permutation statistics. This is joint work with Kevin Dilks and John Stembridge. 