Link complexes of subspace arrangements

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