The rank-nullity ring of a matroid

TL;DR

This paper introduces the rank-nullity ring of a matroid, a subring of the Chow ring of a permutahedral toric variety, highlighting its structure, symmetry, and explicit computations for uniform matroids.

Contribution

It defines the rank-nullity ring of a matroid, shows its relation to tautological classes, and computes its Hilbert function and Gr"obner basis for uniform matroids.

Findings

The rank-nullity ring contains tautological Chern classes.

For uniform matroids, the ring matches the $S_n$-invariant subring.

Explicit Hilbert function and Gr"obner basis are provided.

Abstract

We introduce the rank-nullity ring of a matroid $M$, which is a subring of the Chow ring of the permutahedral toric variety. This subring contains the tautological Chern classes of $M$, a fact we deduce from a highly symmetric formula for these classes. When the matroid $M$ is a uniform matroid, the rank-nullity ring coincides with the subring of $S_n$-invariants of the Chow ring of the permutahedral toric variety. In this case, we compute its Hilbert function explicitly and provide a Gr\"obner basis for the ideal of relations among its generators.