Root system chip-firing II: Central-firing

TL;DR

This paper generalizes chip-firing processes using root systems, proving confluence in a broad setting and connecting it to Dynkin diagrams, with conjectures on specific initial configurations.

Contribution

It introduces a root system framework for chip-firing, proving confluence universally after symmetry reduction, and relates the process to Dynkin diagram number games.

Findings

Central-firing is confluent from any initial weight after symmetry reduction.

For simply-laced systems, the process is described as a number game on Dynkin diagrams.

Conjectures are proposed for confluence from specific initial configurations.

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.