A Consistent Sandpile Torsor Algorithm for Regular Matroids

TL;DR

This paper proves the consistency of a class of sandpile torsor actions for regular matroids, extending previous results and advancing the classification of all such consistent actions.

Contribution

It establishes the consistency of a generalized class of sandpile torsor actions for regular matroids, building on prior work and extending the theoretical framework.

Findings

Proves the consistency of the generalized class of actions for regular matroids.

Extends the existence results for sandpile torsor actions beyond plane graphs.

Progresses towards classifying all consistent actions in this context.

Abstract

Every regular matroid is associated with a sandpile group, which acts simply transitively on the set of bases in various ways. Ganguly and the second author introduced the notion of consistency to describe classes of actions that respect deletion-contraction in a precise sense, and proved the consistency of rotor-routing torsors (and uniqueness thereof) for plane graphs. In this work, we prove that the class of actions introduced by Backman, Baker, and the fourth author, is consistent for regular matroids. More precisely, we prove the consistency of its generalization given by Backman, Santos and the fourth author, and independently by the first author. This extends the above existence assertion, as well as makes progress on the goal of classifying all consistent actions.