Link complexes of subspace arrangements

Given a simplicial hyperplane arrangement HH and a subspace arrangement AA embedded in HH, we define a simplicial complex ΔA,HΔA,H as the subdivision of the link of AA induced by HH. In particular, this generalizes Steingrímsson’s coloring complex of a graph.We do the following: (1)When AA is a hyperplane arrangement, ΔA,HΔA,H is shown to be shellable. As a special case, we answer affirmatively a question of Steingrímsson on coloring complexes.(2)For HH a Coxeter arrangement of type AA or BB we obtain a close connection between the Hilbert series of the Stanley–Reisner ring of ΔA,HΔA,H and the characteristic polynomial of AA. This extends results of Steingrímsson and provides an interpretation of chromatic polynomials of hypergraphs and signed graphs in terms of Hilbert polynomials.