In:
International Mathematics Research Notices, Oxford University Press (OUP), Vol. 2021, No. 13 ( 2021-06-28), p. 10037-10072
Abstract:
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.
Type of Medium:
Online Resource
ISSN:
1073-7928
,
1687-0247
Language:
English
Publisher:
Oxford University Press (OUP)
Publication Date:
2021
detail.hit.zdb_id:
1465368-0
SSG:
17,1
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