Link complexes of subspace arrangements

Abstract

Given a simplicial hyperplane arrangement H and a subspace arrangement A embedded in H, we define a simplicial complex Delta_{A,H} as the subdivision of the link of A induced by H. In particular, this generalizes Steingrimsson's coloring complex of a graph. We do the following: (1) When A is a hyperplane arrangement, Delta_{A,H} is shown to be shellable. As a special case, we answer affirmatively a question of Steingrimsson 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 Delta_{A,H} and the characteristic polynomial of A. This extends results of Steingrimsson and provides an interpretation of chromatic polynomials of hypergraphs and signed graphs in terms of Hilbert polynomials.