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

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 $R_{10}$.