  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.
