On the homology of the real complement of the k-parabolic subspace arrangement

In this paper, we study k-parabolic arrangements, a generalization of the k-equal arrangement for any finite real reflection group. When k=2, these arrangements correspond to the well-studied Coxeter arrangements. We construct a cell complex Permk(W) that is homotopy equivalent to the complement. We then apply discrete Morse theory to obtain a minimal cell complex for the complement. As a result, we give combinatorial interpretations for the Betti numbers, and show that the homology groups are torsion-free. We also study a generalization of the Independence Complex of a graph, and show that this generalization is shellable when the graph is a forest. This result is used in studying Permk(W) using discrete Morse theory.