Differential posets and restriction in critical groups

Agarwal, Ayush; Gaetz, Christian

Algebraic Combinatorics, Tome 2 (2019) no. 6, pp. 1311-1327.

In recent work, Benkart, Klivans, and Reiner defined the critical group of a faithful representation of a finite group $G$, which is analogous to the critical group of a graph. In this paper we study maps between critical groups induced by injective group homomorphisms and in particular the map induced by restriction of the representation to a subgroup. We show that in the abelian group case the critical groups are isomorphic to the critical groups of a certain Cayley graph and that the restriction map corresponds to a graph covering map. We also show that when $G$ is an element in a differential tower of groups, as introduced by Miller and Reiner, critical groups of certain representations are closely related to words of up-down maps in the associated differential poset. We use this to generalize an explicit formula for the critical group of the permutation representation of ${𝔖}_n$ given by the second author, and to enumerate the factors in such critical groups.

Classification : 06A11, 20C30
Mots clés : differential poset, chip firing, critical group

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