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 ${\mathrm{\u0111\u0165\u201d\u2013}}_{n}$ given by the second author, and to enumerate the factors in such critical groups.

Revised : 2019-03-03

Accepted : 2019-04-29

Published online : 2019-12-04

DOI : https://doi.org/10.5802/alco.70

Classification: 06A11, 20C30,

Keywords: differential poset, chip firing, critical group

@article{ALCO_2019__2_6_1311_0, author = {Agarwal, Ayush and Gaetz, Christian}, title = {Differential posets and restriction in critical groups}, journal = {Algebraic Combinatorics}, publisher = {MathOA foundation}, volume = {2}, number = {6}, year = {2019}, pages = {1311-1327}, doi = {10.5802/alco.70}, language = {en}, url = {https://alco.centre-mersenne.org/item/ALCO_2019__2_6_1311_0} }

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/item/ALCO_2019__2_6_1311_0/

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