Matroid Shellability , β - Systems and Affine Hyperplane Arrangements

The broken circuit complex plays a fundamental role for the shellability and homology of matroids, geometric lattices and linear hyperplane arrangements. Here we introduce and study the β-system of a matroid, βnbc(M), whose cardinality is Crapo’s β-invariant. In studying the shellability and homology of base-pointed matroids, geometric semilattices and affine hyperplane arrangements, we find that the β-system acts as the ‘affine counterpart’ to the broken circuit complex. In particular, we show that the β-system indexes the homology facets for the lexicographic shelling of the reduced broken circuit complex BC(M), and explicitly construct the basic cycles. Similarly, we produce an EL-shelling for the geometric semilattice associated with M , and show the β-system labels its decreasing chains. Basic cycles can be carried over from BC(M). The intersection poset of any (real or complex) affine hyperplane arrangement A is a geometric semilattice. Thus our construction yields a set of basic cycles, indexed by βnbc(M), for the union ∪ A of such an arrangement.