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Matroid Theory for Algebraic Geometers

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Nonarchimedean and Tropical Geometry

Part of the book series: Simons Symposia ((SISY))

Abstract

This article is an introduction to matroid theory aimed at algebraic geometers. Matroids are combinatorial abstractions of linear subspaces and hyperplane arrangements. Not all matroids come from linear subspaces; those that do are said to be representable. Still, one may apply linear algebraic constructions to non-representable matroids. There are a number of different definitions of matroids, a phenomenon known as cryptomorphism. In this survey, we begin by reviewing the classical definitions of matroids, develop operations in matroid theory, summarize some results in representability, and construct polynomial invariants of matroids. Afterwards, we focus on matroid polytopes, introduced by Gelfand–Goresky–MacPherson–Serganova, which give a cryptomorphic definition of matroids. We explain certain locally closed subsets of the Grassmannian, thin Schubert cells, which are labeled by matroids, and which have applications to representability, moduli problems, and invariants of matroids following Fink–Speyer. We explain how matroids can be thought of as cohomology classes in a particular toric variety, the permutohedral variety, by means of Bergman fans, and apply this description to give an exposition of the proof of log-concavity of the characteristic polynomial of representable matroids due to the author with Huh.

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  1. Department of Combinatorics & Optimization, University of Waterloo, Waterloo, ON, Canada, N2L 3G1

    Eric Katz

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  1. School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, USA

    Matthew Baker

  2. Department of Mathematics, Yale University, New Haven, Connecticut, USA

    Sam Payne

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Katz, E. (2016). Matroid Theory for Algebraic Geometers. In: Baker, M., Payne, S. (eds) Nonarchimedean and Tropical Geometry. Simons Symposia. Springer, Cham. https://doi.org/10.1007/978-3-319-30945-3_12

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