Differential posets and restriction in critical groups.

*(English)*Zbl 07140435Summary: 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 \(\mathfrak{S}_n\) given by the second author, and to enumerate the factors in such critical groups.

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