Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Wang, Yiyu
Institution: The Ohio State University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7160962049