Chip-Firing and the Sandpile Group of the $R_{10}$ Matroid

TL;DR

This paper explores chip-firing dynamics on the $R_{10}$ matroid, providing a simple description, an interactive app, and a classification of its sandpile group structure, contributing to the understanding of regular matroids.

Contribution

It offers a novel, straightforward description of chip-firing on $R_{10}$ and classifies its sandpile group, linking combinatorial dynamics with complex numbers.

Findings

Simple description of chip-firing on $R_{10}$ using complex numbers

Development of an interactive app for chip-firing dynamics

Classification of the 162 equivalence classes of the sandpile group

Abstract

A celebrated result of Seymour is that all regular matroids are built up from graphic matroids, cographic matroids, and a specific 10 element rank 5 matroid called $R_{10}$. In this article, we give a simple description of chip-firing on $R_{10}$ using complex numbers on the vertices of a pentagon, and link to an app where readers can play around with the combinatorial dynamics of the system. We also provide an easy to describe set of representatives for each of the 162 equivalence classes that make up the sandpile group of $R_{10}$.