Matroid theory for algebraic geometers

TL;DR

This survey introduces matroid theory to algebraic geometers, covering classical definitions, operations, invariants, and applications like matroid polytopes, Grassmannian subsets, and log-concavity of characteristic polynomials.

Contribution

It provides a comprehensive overview connecting matroid theory with algebraic geometry, emphasizing applications and recent results such as the proof of log-concavity.

Findings

Matroid polytopes offer a cryptomorphic perspective on matroids.

Thin Schubert cells relate matroids to Grassmannian subsets.

Proof of log-concavity of the characteristic polynomial for representable matroids.

Abstract

This article is a survey of 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.