Homology representations arising from the half cube, II

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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 ( D n ) 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 S n by restriction, the homology representations turn out to be direct sums of certain representations induced from parabolic subgroups. The latter representations of S n agree (over C ) with the representations of S n on the ( k − 2 ) -nd homology of the complement of the k-equal real hyperplane arrangement.