Journal of Algebraic Combinatorics

, Volume 1, Issue 3, pp 283–300 | Cite as

Matroid Shellability, β-Systems, and Affine Hyperplane Arrangements

  • Günter M. Ziegler


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 \(\overline {BC} (M)\), 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\(\overline {BC} (M)\) The intersection poset of any (real or complex) afflnehyperplane arrangement Α is a geometric semilattice. Thus the construction yields a set of basic cycles, indexed by βnbc(M), for the union ⋃Α of such an arrangement.

matroid β-invariant broken-circuit complex shellability affine hyperplane arrangement 


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Copyright information

© Kluwer Academic Publishers 1992

Authors and Affiliations

  • Günter M. Ziegler
    • 1
  1. 1.Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)BerlinGermany

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