Matroid Theory for Algebraic Geometers

By Eric Katz. 2014

60 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/fe25c77de1b6622b173191c360ba8f19c34f6b99

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Citations in the corpus

MODULI SPACES OF RATIONAL WEIGHTED STABLE CURVES AND TROPICAL GEOMETRY (2014, background): “For a detailed introduction to matroids, see for instance [16].”
Simplicial generation of Chow rings of matroids (2019, methodology): “eir Chow rings. We assume familiarity with the basics of matroids, and refer to [Oxl11] as a general reference. For accounts tailored towards Chow ring of matroids, we recommend [Bak18], [Huh18b], or [Kat16]. Here we specialize the discussion above on tropical intersection theory to the setting of m”