Matroid Shellability, β-Systems, and Affine Hyperplane Arrangements

Content Provider	Semantic Scholar
Author	Ziegler, Günter M.
Copyright Year	1992
Abstract	AbstractThe 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.
Starting Page	283
Ending Page	300
Page Count	18
File Format	PDF HTM / HTML
DOI	10.1023/A:1022492019120
Volume Number	1
Alternate Webpage(s)	http://www.maths.tcd.ie/EMIS/journals/JACO/Volume1_3/t547g12057821376.fulltext.pdf
Alternate Webpage(s)	http://www.mi.fu-berlin.de/math/groups/discgeom/ziegler/Preprintfiles/030PREPRINT.pdf
Alternate Webpage(s)	https://doi.org/10.1023/A%3A1022492019120
Language	English
Access Restriction	Open
Content Type	Text
Resource Type	Article