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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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