2026-09-19T00:42:27Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/1177802026-06-17T14:47:15Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Alexander Postnikov.
Hopkins, Samuel F
Massachusetts Institute of Technology. Department of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2018-09-17T14:49:19Z
2018-09-17T14:49:19Z
2018
2018
http://hdl.handle.net/1721.1/117780
1051190416
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018.
This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Cataloged from student-submitted PDF version of thesis.
Includes bibliographical references (pages 195-200).
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.
by Samuel Francis Hopkins.
Ph.D.
200 pages
eng
Massachusetts Institute of Technology
MIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission.
http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Root system chip-firing
Thesis
application/pdf
Publication
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<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.</Abstract>
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