Stellahedral geometry of matroids

TL;DR

This paper explores the geometry of the stellahedral toric variety to analyze matroids, establishing new connections between matroid invariants and algebraic geometry, and proving a novel log-concavity property of the Tutte polynomial.

Contribution

It introduces the use of stellahedral geometry to unify matroid invariants and proves a new log-concavity result for the Tutte polynomial.

Findings

Valuative group of matroids identified with the cohomology ring of the stellahedral toric variety.

Established that valuative, homological, and numerical equivalences for matroids coincide.

Proved a new log-concavity property for the Tutte polynomial.

Abstract

We use the geometry of the stellahedral toric variety to study matroids. We identify the valuative group of matroids with the cohomology ring of the stellahedral toric variety, and show that valuative, homological, and numerical equivalence relations for matroids coincide. We establish a new log-concavity result for the Tutte polynomial of a matroid, answering a question of Wagner and Shapiro-Smirnov-Vaintrob on Postnikov-Shapiro algebras, and calculate the Chern-Schwartz-MacPherson classes of matroid Schubert cells. The central construction is the "augmented tautological classes of matroids," modeled after certain vector bundles on the stellahedral toric variety.