ISSN:
1572-9192
Keywords:
matroid
;
β-invariant
;
broken-circuit complex
;
shellability
;
affine hyperplane arrangement
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1022492019120
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