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Chip firing on Dynkin diagrams and McKay quivers. (English) Zbl 1448.17015
Summary: This paper establishes new connections between the representation theory of finite groups and sandpile dynamics. 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.

MSC:
17B22 Root systems
05E10 Combinatorial aspects of representation theory
14E16 McKay correspondence
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