# ALGEBRAIC COMBINATORICS

Differential posets and restriction in critical groups
Algebraic Combinatorics, Volume 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 ${\mathrm{đť”–}}_{n}$ given by the second author, and to enumerate the factors in such critical groups.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.70
Classification: 06A11, 20C30
Keywords: differential poset, chip firing, critical group
Agarwal, Ayush 1; Gaetz, Christian 2

1 Stanford University Stanford, CA, USA
2 Department of Mathematics Massachusetts Institute of Technology Cambridge, MA, USA
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Agarwal, Ayush; Gaetz, Christian. Differential posets and restriction in critical groups. Algebraic Combinatorics, Volume 2 (2019) no. 6, pp. 1311-1327. doi : 10.5802/alco.70. https://alco.centre-mersenne.org/articles/10.5802/alco.70/

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