Homology representations arising from the half cube, II

Abstract

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 $W(D_n)$ of type $D$. In this paper, we explicitly determine the characters (over $\mathbb{C}$) of these homology representations, which turn out to be multiplicity free. Regarded as representations of the symmetric group $S_n$ by restriction, the homology representations turn out to be direct sums of certain representations induced from parabolic subgroups. The latter representations of $\sym_n$ agree (over $\mathbb{C}$) with the representations of $\sym_n$ on the $(k-2)$-nd homology of the complement of the $k$-equal real hyperplane arrangement.

19 pages AMSTeX. One figure. The Conjecture in the previous version is now a Theorem. This research was supported by NSF grant DMS-0905768

References in corpus (4)

• Lectures on 0/1-polytopes
• Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations
• Zigzag structure of complexes
• Representations of Lie algebras arising from polytopes