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Root System Chip-Firing II: Central-Firing

Author(s)
Galashin, Pavel; Hopkins, Sam; McConville, Thomas; Postnikov, Alexander
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Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/
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Abstract
<jats:title>Abstract</jats:title> <jats:p>Jim Propp recently proposed a labeled version of chip-firing on a line and conjectured that this process is confluent from some initial configurations. This was proved by Hopkins–McConville–Propp. We reinterpret Propp’s labeled chip-firing moves in terms of root systems; a “central-firing” move consists of replacing a weight $\lambda$ by $\lambda +\alpha$ for any positive root $\alpha$ that is orthogonal to $\lambda$. We show that central-firing is always confluent from any initial weight after modding out by the Weyl group, giving a generalization of unlabeled chip-firing on a line to other types. For simply-laced root systems we describe this unlabeled chip-firing as a number game on the Dynkin diagram. We also offer a conjectural classification of when central-firing is confluent from the origin or a fundamental weight.</jats:p>
Date issued
2019
URI
https://hdl.handle.net/1721.1/135573
Department
Massachusetts Institute of Technology. Department of Mathematics
Journal
International Mathematics Research Notices
Publisher
Oxford University Press (OUP)

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