In a previous work, 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 reduced homology of such a subcomplex is concentrated in degree k-1. This homology module supports a natural action of the Coxeter group W(Dn) of type D. In this paper, we explicitly determine the characters (over 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 Sn agree (over C) with the representations of Snon the (k-2)-nd homology of the complement of the k-equal real hyperplane arrangement. © 2009 Elsevier Inc.
Green, R. M. (2010). Homology representations arising from the half cube, II. Journal of Combinatorial Theory. Series A, 117(8), 1037–1048. https://doi.org/10.1016/j.jcta.2009.10.005