Chip firing on Dynkin diagrams and McKay quivers

TL;DR

This paper explores the properties of chip-firing dynamics on matrices derived from root systems and McKay quivers, revealing explicit configurations and connections to group abelianizations.

Contribution

It introduces explicit descriptions of recurrent configurations and links critical groups to group abelianizations in the context of Dynkin diagrams and McKay quivers.

Findings

Recurrent configurations are explicitly identified in root system cases.

Critical groups relate to the abelianization of finite subgroups G.

In the classical McKay correspondence, the critical group and abelianization are isomorphic.

Abstract

Two classes of avalanche-finite matrices and their critical groups (integer cokernels) are studied from the viewpoint of chip-firing/sandpile dynamics, namely, the Cartan matrices of finite root systems and the McKay-Cartan matrices for finite subgroups G of general linear groups. In the root system case, the recurrent and superstable configurations are identified explicitly and are related to minuscule dominant weights. In the McKay-Cartan case for finite subgroups of the special linear group, the cokernel is related to the abelianization of the subgroup G. In the special case of the classical McKay correspondence, the critical group and the abelianization are shown to be isomorphic.