Homology representations arising from the half cube

2008/06/09 by R. M. Green, Green, R. M. · 2 citations
Computer Science · Mathematics · #05E25 #52B11 #57Q05 #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Topological and Geometric Data Analysis #math.CO #math.GT #msc:05E25 #msc:52B11 #msc:57Q05

paper · pdf · doi:10.48550/arxiv.0806.1503

Approximately 35 pages, AMSTeX. Revised in light of referee's comments. To appear in Advances in Mathematics

openalex publication_date 2008/06/09 · arxiv created 2008/12/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct a CW decomposition C_n of the n-dimensional half cube in a manner compatible with its structure as a polytope. For each 3 ≤ k ≤ n, the complex C_n has a subcomplex C_{n, k}, which coincides with the clique complex of the half cube graph if k = 4. The homology of C_{n, k} is concentrated in degree k-1 and furthermore, the (k-1)-st Betti number of C_{n, k} is equal to the (k-2)-nd Betti number of the complement of the k-equal real hyperplane arrangement. These Betti numbers, which also appear in theoretical computer science, numerical analysis and engineering, are the coefficients of a certain Pascal-like triangle (Sloane's sequence A119258). The Coxeter groups of type D_n act naturally on the complexes C_{n, k}, and thus on the associated homology groups.

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• Homology representations arising from the half cube, II
• Morse Matchings on a Hypersimplex

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