Grothendieck Weights on Permutohedral Varieties and Matroids

TL;DR

This paper develops a framework for Grothendieck weights on permutohedral fans, characterizes them via linear conditions, and applies it to matroids to compute motivic Chern classes that depend solely on the matroid structure.

Contribution

It introduces a finite linear characterization of Grothendieck weights on permutohedral fans and extends motivic Chern class definitions to all loopless matroids.

Findings

Grothendieck weights are characterized by a finite system of linear equations.

An explicit product rule for the ring structure of Grothendieck weights is established.

Motivic Chern classes of hyperplane arrangement complements depend only on the matroid.

Abstract

Grothendieck weights, introduced by Shah, are $K$-theoretic analogues of Minkowski weights on smooth toric varieties. We study Grothendieck weights on the permutohedral fan and prove two main results: a $K$-balancing condition that characterizes Grothendieck weights by a finite system of linear equations, and an explicit product rule for the ring structure. We apply this framework to matroids, giving a combinatorial characterization of Grothendieck weights on matroidal fans. As the main application, we compute the motivic Chern class of the hyperplane arrangement complement in its wonderful compactification and show that the result depends only on the matroid, not on the realization. This allows us to extend the definition of the motivic Chern class to all loopless matroids.