2008/12/05 by R. M. Green, R.M. Green, Green, R. M.
Mathematics · #05E25 #20C15 #52B11 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #math.CO #math.GT #math.RT #msc:05E25 #msc:20C15 #msc:52B11
paper · pdf · doi:10.48550/arxiv.0812.1208
19 pages AMSTeX. One figure. The Conjecture in the previous version is now a Theorem. This research was supported by NSF grant DMS-0905768
openalex publication_date 2008/12/05 · arxiv created 2010/05/28 · arxiv updated 2010/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In a previous work (arXiv:0806.1503v2), we defined a family of subcomplexes of the n-dimensional half cube by removing the interiors of all half cube shaped faces of dimension at least k, and we proved that the homology of such a subcomplex is concentrated in degree k-1. This homology group supports a natural action of the Coxeter group W(Dn) of type D. In this paper, we explicitly determine the characters (over \Bbb C) of these homology representations, which turn out to be multiplicity free. Regarded as representations of the symmetric group Sn by restriction, the homology representations turn out to be direct sums of certain representations induced from parabolic subgroups. The latter representations of \symn agree (over \Bbb C) with the representations of \symn on the (k-2)-nd homology of the complement of the k-equal real hyperplane arrangement.