Stellahedral geometry of matroids

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 toric vector bundles on the stellahedral toric variety.

Schubert matroids, Delannoy paths, and Speyer's invariant

“…The above result is possible to derive in an alternative way by building on the work of Eur, Huh and Larson in [EHL23]: they established an isomorphism between the cohomology of the stellahedral variety and the valuative group of matroids (which is defined in a very similar way). A consequence of the above result is that for every matroid M on E there exists a list of Schubert matroids on E, say M 1 , .…”