Link complexes of subspace arrangements

Given a simplicial hyperplane arrangement H and a subspace arrangement A embedded in H , we define a simplicial complex Δ A , H as the subdivision of the link of A induced by H . In particular, this generalizes Steingrímsson’s coloring complex of a graph. the following: (1) is a hyperplane arrangement, Δ A , H is shown to be shellable. As a special case, we answer affirmatively a question of Steingrímsson on coloring complexes. a Coxeter arrangement of type A or B we obtain a close connection between the Hilbert series of the Stanley–Reisner ring of Δ A , H and the characteristic polynomial of A . This extends results of Steingrímsson and provides an interpretation of chromatic polynomials of hypergraphs and signed graphs in terms of Hilbert polynomials.