Abstract
For any Coxeter system (W, S) of rank n, we introduce an abstract boolean complex (simplicial poset) of dimension 2n − 1 which contains the Coxeter complex as a relative subcomplex. Faces are indexed by triples (J,w,K), where J and K are subsets of the set S of simple generators, and w is a minimal length representative for the double parabolic coset WJ wWK . 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 provide a “two-sided” analogue of the Coxeter complex by considering double cosets of parabolic subgroups
Summary
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 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. It is well known that the set of cosets of parabolic subgroups forms an abstract simplicial complex known as the Coxeter complex, and denoted by. 2. The facets (maximal faces) of Ξ are in bijection with the elements of W , and the Coxeter complex Σ is a relative subcomplex of Ξ. There are two different approaches to proving the topological results for the Coxeter complex listed in Theorem 1. Bjorner showed in [5] how to use poset-theoretic tools to study the topology of the complex with only the abstract definition of the face poset. We hope to uncover a more geometric description of Ξ in the future
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