``What is a complex matroid?''
type: Item
instance of: scholarly article
title: ``What is a complex matroid?''
zbMATH Open document ID: 0782.52005
DOI: 10.1007/BF02573983
author: Günter M. Ziegler
published in: Discrete \& Computational Geometry
publication date: 1994-02-28T00:00:00Z
full work available at URL: https://eudml.org/doc/131278
review text: Following an `Ansatz' of \textit{A. Björner} and the author [J. Am. Math. Soc. 5, No. 1, 105-149 (1992; Zbl 0754.52003)] we give an axiomatic development of finite sign vector systems that we call ``complex matroids''. This includes, as special cases, the sign vector systems that encode complex arrangements according to Björner and the author, and the complexified oriented matroids, whose complements were considered by \textit{I. M. Gel'fand} and \textit{G. L. Rybnikov} [Sov. Math., Dokl. 40, No. 1, 148-152 (1990); translation from Dokl. Akad. Nauk SSSR 307, No. 4, 791-795 (1989; Zbl 0717.33009)]. Our framework makes it possible to study complex hyperplane arrangements as entirely combinatorial objects. By comparing with the structure of 2- matroids, which model the more general 2-arrangements introduced by \textit{M. Goresky} and \textit{R. MacPherson} [`Stratified Morse theory' (1988; Zbl 0639.14012)], one can isolate the essential combinatorial meaning of a ``complex structure''. This paper produces a topological representation theorem for 2-matroids and complex matroids, and the computation of the cohomology of the complement of a 2-arrangement, including its multiplicative structure in the complex case. Duality is established in the cases of complexified oriented matroids, and for realizable complex matroids. Complexified oriented matroids are shown to be matroids with coefficients in the sense of \textit{A. W. M. Dress} [Adv. Math. 59, 97-123 (1986; Zbl 0656.05025)], but this fails in general.
reviewed by: Günter M. Ziegler
Mathematics Subject Classification ID: 52C35
Mathematics Subject Classification ID: 51M20
Mathematics Subject Classification ID: 51D20
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 437649
zbMATH Keywords: complex structure
zbMATH Keywords: duality
zbMATH Keywords: hyperplane arrangements
zbMATH Keywords: oriented matroids
describes a project that uses: GeM
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 437649