A Two-Sided Analogue of the Coxeter Complex

T Kyle Petersen

doi:10.37236/8015

Abstract

For any Coxeter system $(W,S)$ of rank $n$, we study an abstract boolean complex (simplicial poset) of dimension $2n-1$ that contains the Coxeter complex as a relative subcomplex. For finite $W$, this complex is first described in work of Hultman. Faces are indexed by triples $(I,w,J)$, where $I$ and $J$ are subsets of the set $S$ of simple generators, and $w$ is a minimal length representative for the parabolic double coset $W_I w W_J$. There is exactly one maximal face for each element of the group $W$. The complex is shellable and thin, which implies the complex is a sphere for the finite Coxeter groups. In this case, a natural refinement of the $h$-polynomial is given by the "two-sided" $W$-Eulerian polynomial, i.e., the generating function for the joint distribution of left and right descents in $W$.

Highlights

Coxeter groups were developed to study symmetries of regular polytopes, and they play a major role in the study of Lie algebras
The Coxeter complex is a simplicial complex associated with the reflection representation of the group, but which can be defined abstractly via cosets of parabolic subgroups
The goal of this paper is to study a “two-sided” analogue of the Coxeter complex by considering double cosets of parabolic subgroups

Summary

Introduction

Coxeter groups were developed to study symmetries of regular polytopes, and they play a major role in the study of Lie algebras (the Weyl group of a root system is a Coxeter group). The faces of the Coxeter complex are identified with left cosets of parabolic subgroups, wWJ. For our two-sided analogue, we consider elements from all double quotients WI\W/WJ , so the faces will be related to double cosets of parabolic subgroups WIwWJ , where I and J are subsets of S. A first guess to define a two-sided Coxeter complex is to consider the set of all double cosets WIwWJ , ordered by reverse inclusion. Such a poset does exist, but it is difficult to analyze. We partially order the elements of Ξ by reverse inclusion of the index sets I and J as well as the corresponding double coset, i.e.,. It would be interesting to look at other special cases of ∆θ for W , e.g., with a different choice of involution θ

Ξ is boolean

Ξ is balanced

Σ is a relative subcomplex of Ξ

Ξ is partitionable

Ξ is shellable

Consequences of shelling