On matroids and Orlik-Solomon algebras
type: Item
instance of: scholarly article
title: On matroids and Orlik-Solomon algebras
zbMATH Open document ID: 1067.05018
author: Yukihito Kawahara
published in: Annals of Combinatorics
publication date: 2004-12-13T00:00:00Z
review text: Suppose \(\mathcal A\) is a finite set of hyperplanes in a fixed vector space \(V\) over a field \(\mathbb K,\) with intersection poset \(L\) and Orlik-Solomon algebra \(A({\mathcal A}).\) If \(\mathcal A\) is a central arrangement, then \(L\) is a geometric lattice, and is the lattice of flats of a simple matroid with ground set \(\mathcal A.\) In general the intersection poset \(L\) is a geometric semi-lattice, as introduced by \textit{M. L. Wachs} and \textit{J. W. Walker} [Order 2, 367--385 (1986; Zbl 0589.06005)]. There is a simple process, called coning, by which the arrangement \(\mathcal A\) is replaced by a central arrangement \(c{\mathcal A}\) in \(V \oplus {\mathbb K}\) that includes an additional hyperplane (``at infinity'') \(V \oplus 0.\) Central arrangements can be deconed once a hyperplane is specified to be the hyperplane at infinity. This process is reflected in the intersection posets: every geometric semi-lattice is a subposet of a geometric lattice with one additional atom \(a,\) as the set of elements not greater than \(a\). The category of geometric semi-lattices is equivalent to the category of pointed geometric lattices. On the algebraic side, the Orlik-Solomon algebras fit into a split short exact sequence \(0 \to A({\mathcal A}) \to A(c{\mathcal A}) \to A({\mathcal A})[-1] \to 0.\) This sequence is extracted from an acyclic chain complex \((A(c{\mathcal A}),\partial).\) In the paper under review this construction is carried out at the more fundamental level of matroids. The author gives an axiomatic definition of a ``quasi-matroid,'' to model the intrinsic combinatorics of non-central arrangements. A quasi-matroid consists of a simplicial complex \(\mathcal E\) and a matroid subcomplex \(\mathcal I,\) subject to a single axiom. An arrangement \(\mathcal A\) determines a quasi-matroid, with \({\mathcal E}\) recording the sets of hyperplanes with nonempty intersection, and \(\mathcal I\) consisting of the independent sets of hyperplanes. One…
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 52C35
Mathematics Subject Classification ID: 13D03
zbMATH DE Number: 2121815
zbMATH Keywords: quasi-matroid
zbMATH Keywords: arrangement
zbMATH Keywords: geometric semilattice
zbMATH Keywords: pointed matroid
zbMATH Keywords: hyperplane
zbMATH Keywords: flat
zbMATH Keywords: chain complexes
reviewed by: Michael Falk
MaRDI profile type: Publication
full work available at URL: s00026-004-0205-7
OpenAlex ID: W2044317152
DOI: 10.1007/S00026-004-0205-7
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description: scientific article; zbMATH DE number 2121815