Homology representations arising from the half cube
Abstract
Description
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 $3 \leq k \leq n$, 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.
Approximately 35 pages, AMSTeX. Revised in light of referee's comments. To appear in Advances in Mathematics
Approximately 35 pages, AMSTeX. Revised in light of referee's comments. To appear in Advances in Mathematics