The intersection ring of matroids

TL;DR

This paper introduces a graded ring structure on loopfree matroids, linking tropical geometry, combinatorics, and algebraic geometry, and demonstrates its properties including generation, basis, and geometric interpretation.

Contribution

It establishes a new algebraic structure on matroids, showing it is generated by corank one matroids, with a basis given by nested matroids, and relates it to toric geometry.

Findings

Ring is generated in corank one

Basis for rank r matroids is nested matroids

Ring is isomorphic to the cohomology ring of a toric variety

Abstract

We study a particular graded ring structure on the set of all loopfree matroids on a fixed labeled ground set, which occurs naturally in tropical geometry. The product is given by matroid intersection and the additive structure is defined by assigning to each matroid the indicator vector of its chains of flats. We show that this ring is generated in corank one, more precisely that any matroid can be written as a linear combination of products of corank one matroids. Moreover, we prove that a basis for the graded part of rank r matroids is given by the set of nested matroids and that the total number of these is a Eulerian number. Derksen's G-invariant then defines a Z-linear map on this ring, which implies for example that the Tutte polynomial is linear on it as well. Finally we show that the ring is the cohomology ring of the toric variety of the permutohedron and thus fulfills Poincar\'e duality.