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

Michael Ion, Alex McDonough

TL;DR

The paper addresses understanding chip-firing on the regular matroid $R_{10}$ to inform chip-firing on regular matroids in general. It adopts a Gaussian-integer formulation on a pentagon, derives that the sandpile group $S(R_{10})$ is isomorphic to $(\\mathbb{Z}/3)^4 \\\oplus (\\mathbb{Z}/2)$ and has order $162$, and provides an explicit, constructive method to represent each class with a canonical form and an accompanying algorithm, alongside an interactive web app. The work contributes a simple, concrete picture of chip-firing on $R_{10}$, a complete set of representatives, and a practical stabilization procedure, offering a tractable model for exploring sandpile dynamics in regular matroids and their Seymour-based decompositions. This advances both the theoretical understanding of sandpile groups for regular matroids and provides a tangible tool for intuition and further research into torsor actions on bases.

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 $R_{10}$.

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

TL;DR

The paper addresses understanding chip-firing on the regular matroid to inform chip-firing on regular matroids in general. It adopts a Gaussian-integer formulation on a pentagon, derives that the sandpile group is isomorphic to and has order , and provides an explicit, constructive method to represent each class with a canonical form and an accompanying algorithm, alongside an interactive web app. The work contributes a simple, concrete picture of chip-firing on , a complete set of representatives, and a practical stabilization procedure, offering a tractable model for exploring sandpile dynamics in regular matroids and their Seymour-based decompositions. This advances both the theoretical understanding of sandpile groups for regular matroids and provides a tangible tool for intuition and further research into torsor actions on bases.

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 . In this article, we give a simple description of chip-firing on 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 .

Paper Structure

This paper contains 5 sections, 8 theorems, 16 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Chip-firing on $R_{10}$
  4. Representatives of $S(R_{10})$
  5. Future directions

Key Result

Proposition 2.5

Multijection Let $M$ be a regular matroid with ground set $E$ that is represented over $\mathbb R$ by a totally unimodular matrix $A = $. Furthermore, define $\widehat{A}= $ and $K = = $. The sandpile group$S(M)$ is equal to $\mathop{\mathrm{coker}}\nolimits_{\mathbb Z} K$ (which is defined as $\ma

Figures (2)

  • Figure 1: Above is a screenshot from the app that we developed to help understand chip-firing on $R_{10}$https://pentagon-game.github.io/. The user can fire a vertex by clicking on it, or switch the kind of firing by clicking on the center pentagon. In this game mode, the chip configuration is chosen to be firing equivalent to the all zeros configuration, and the goal is to reach this configuration through a sequence of firings.
  • Figure 2: This figure shows one way to visualize chip-firing on $R_{10}$. Chip-configurations correspond to 5 labeled nodes on lattice points on the complex plane. For example, the initial configuration is $(1+i, -2 + 3i, -2, 1-2i, 2)$ and the final configuration is $(2+3i, -2 + 2i, -2 + i,-2i, 1-i)$. This final configuration is reached after an A firing of 0, a (-B) firing of 2, a (-A) firing of 4, a (-A) firing of 3, and a B firing of 2. Note that there are no restrictions in this construction about multiple nodes occupying the same position; we only avoided this for the example to make things easier to draw.

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • Remark 2.6
  • Remark 2.7
  • Proposition 2.8
  • Example 2.9
  • Definition 3.1
  • ...and 16 more