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arXiv · 1708.04849

Root system chip-firing II: Central-firing

Pavel Galashin·Sam Hopkins·Thomas McConville·Alexander Postnikov
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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.

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Pavel Galashin, Sam Hopkins, Thomas McConville, Alexander Postnikov. 2017-08-16. Root system chip-firing II: Central-firing. https://doi.org/10.1093/imrn/rnz112

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