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Grothendieck Weights on Permutohedral Varieties and Matroids

  • Yiyu Wang
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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.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026grothendieck,
  title = {{Grothendieck Weights on Permutohedral Varieties and Matroids}},
  author = {Yiyu Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.09762},
  eprint = {2605.09762},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.09762},
}
RIS
TY  - UNPB
TI  - Grothendieck Weights on Permutohedral Varieties and Matroids
AU  - Yiyu Wang
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2605.09762
UR  - https://arxiv.org/abs/2605.09762
AB  - 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.
ER  -