Chow Rings of Matroids as Permutation Representations

TL;DR

This paper investigates how automorphisms of a matroid induce permutation actions on its Chow ring, extending known algebraic properties to these actions and proposing new conjectures.

Contribution

It demonstrates that automorphism groups induce permutation actions on the Chow ring and extends key algebraic properties to these actions, suggesting new conjectures.

Findings

Automorphism groups induce permutation actions on the Chow ring.

Poincaré duality and Hard Lefschetz properties extend to these permutation actions.

Proposes new conjectures related to these permutation representations.

Abstract

Given a matroid and a group of its matroid automorphisms, we study the induced group action on the Chow ring of the matroid. This turns out to always be a permutation action. Work of Adiprasito, Huh and Katz showed that the Chow ring satisfies Poincar\'e duality and the Hard Lefschetz theorem. We lift these to statements about this permutation action, and suggest further conjectures in this vein.