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What is a complex matroid?

Abstract

Following an “ansatz” of Björner and Ziegler [BZ], we give an axiomatic development of finite sign vector systems that we callcomplex matroids. This includes, as special cases, the sign vector systems that encode complex arrangements according to [BZ], and the complexified oriented matroids, whose complements were considered by Gel'fand and Rybnikov [GeR].

Our framework makes it possible to study complex hyperplane arrangements as entirely combinatorial objects. By comparing complex matroids with 2-matroids, which model the more general 2-arrangements introduced by Goresky and MacPherson [GoM], the essential combinatorial meaning of a “complex structure” can be isolated.

Our development features 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 Dress and Wenzel [D1], [DW1], but this fails in general.

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Ziegler, G.M. What is a complex matroid?. Discrete Comput Geom 10, 313–348 (1993). https://doi.org/10.1007/BF02573983

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  • DOI: https://doi.org/10.1007/BF02573983

Keywords

  • Sign Vector
  • Cohomology Algebra
  • Complex Arrangement
  • Oriented Matroids
  • Geometric Lattice