Matroid Shellability, β-Systems, and Affine Hyperplane Arrangements

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The broken-circuit complex is fundamental to the shellability and homology of matroids, geometric lattices, and linear hyperplane arrangements. This paper introduces and studies 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 afflne hyperplane arrangements, it is found that the β-system acts as the afflne counterpart to the broken-circuit complex. In particular, it is shown that the β-system indexes the homology facets for the lexicographic shelling of the reduced broken-circuit complex Symbol(M), and the basic cycles are explicitly constructed. Similarly, an EL-shelling for the geometric semilattice associated with M is produced, and it is shown that the β-system labels its decreasing chains. Basic cycles can be carried over from Symbol(M) The intersection poset of any (real or complex) afflne hyperplane arrangement 𝒜 is a geometric semilattice. Thus the construction yields a set of basic cycles, indexed by βnbc(M), for the union ∪𝒜 of such an arrangement.

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