Homology representations arising from the half cube, II

TL;DR

This paper explicitly determines the characters of homology representations arising from certain subcomplexes of the half cube, revealing their multiplicity-free nature and connections to symmetric group representations and hyperplane arrangements.

Contribution

It provides explicit character formulas for these homology representations, showing they are multiplicity free and relate to induced representations of symmetric groups.

Findings

Homology representations are multiplicity free.

Representations are sums of induced representations from parabolic subgroups.

Connections established with homology of hyperplane arrangement complements.

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 ${\Bbb C}$) 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 ${\Bbb C}$) with the representations of $\sym_n$ on the $(k-2)$-nd homology of the complement of the $k$-equal real hyperplane arrangement.