Root system chip-firing

Abstract

This thesis investigates an extension of the classical chip-firing process to "other Cartan-Killing types." In Chapter 1 we review the classical chip-firing game: the states of this process are configurations of chips on the vertices of a graph; the transition moves are firings whereby a vertex with at least as many chips as neighbors may send one chip to each neighbor. A fundamental property of chip-firing is that it is confluent: from any initial configuration, all sequences of firings lead to the same terminal configuration. In Chapter 2 we discuss Propp's labeled chip-firing process on the infinite path, for which confluence becomes a subtler question. We prove that labeled chip-firing is confluent starting from an even number of chips at the origin (but not from an odd number). In Chapter 3 we reinterpret labeled chip-firing as a process on the weight lattice of a root system, where the firing moves consist of adding a positive root whenever the weight we are at is orthogonal to that root. We call this the central-firing process. We give conjectures about certain initial weights from which central-firing is confluent. We also prove that central-firing is always confluent from all initial weights if we mod out by the action of the Weyl group, thereby giving a generalization of unlabeled chip firing on the infinite path to other types. In Chapter 4 we introduce some remarkable deformations of the central-firing process which we call the symmetric and truncated interval-firing processes. These are analogous to the Catalan and Shi hyperplane arrangements. We prove that these interval-firing processes are always confluent from all initial weights. In Chapter 5 we study the set of weights with given interval-firing stabilization. We show that the number of weights with given stabilization is a polynomial in our deformation parameter. We call these polynomials the symmetric and truncated Ehrhart-like polynomials, because they are analogous to the Ehrhart polynomial of a polytope. We conjecture that the Ehrhart-like polynomials have nonnegative integer coefficients. In Chapter 6 we prove "half" of this positivity conjecture by providing an explicit, positive formula for the symmetric Ehrhart-like polynomials.