K-classes of matroids and equivariant localization

TL;DR

This paper introduces a new K-theoretic class associated with matroids, interprets the Tutte polynomial geometrically, and extends known results to non-realizable matroids using equivariant localization.

Contribution

It defines a novel K-theoretic class for all matroids, provides a geometric interpretation of the Tutte polynomial, and generalizes previous results beyond realizable matroids.

Findings

Established a geometric interpretation of the Tutte polynomial.

Extended behavior results of matroid classes under various operations.

Applied equivariant localization to study matroid classes.

Abstract

To every matroid, we associate a class in the K-theory of the Grassmannian. We study this class using the method of equivariant localization. In particular, we provide a geometric interpretation of the Tutte polynomial. We also extend results of the second author concerning the behavior of such classes under direct sum, series and parallel connection and two-sum; these results were previously only established for realizable matroids, and their earlier proofs were more difficult.