A two-sided analogue of the Coxeter complex

2016/07/01 by T. Kyle Petersen, Petersen, T. Kyle
Mathematics · #05E45 #20F55 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05E45 #msc:20F55

paper · pdf · doi:10.48550/arxiv.1607.00086

26 pages, several large tables and figures

arxiv created 2016/07/01 · arxiv updated 2016/07/04

Abstract

For any Coxeter system (W,S) of rank n, we introduce an abstract boolean complex (simplicial poset) of dimension 2n-1 that contains the Coxeter complex as a relative subcomplex. 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.

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