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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 $Perm_k(W)$ 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 $Perm_k(W)$ using discrete Morse theory.