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:

Accepted:

Published online:

DOI: 10.5802/alco.70

Keywords: differential poset, chip firing, critical group

^{1}; Gaetz, Christian

^{2}

@article{ALCO_2019__2_6_1311_0, author = {Agarwal, Ayush and Gaetz, Christian}, title = {Differential posets and restriction in critical groups}, journal = {Algebraic Combinatorics}, pages = {1311--1327}, publisher = {MathOA foundation}, volume = {2}, number = {6}, year = {2019}, doi = {10.5802/alco.70}, mrnumber = {4049848}, zbl = {07140435}, language = {en}, url = {https://alco.centre-mersenne.org/articles/10.5802/alco.70/} }

TY - JOUR AU - Agarwal, Ayush AU - Gaetz, Christian TI - Differential posets and restriction in critical groups JO - Algebraic Combinatorics PY - 2019 SP - 1311 EP - 1327 VL - 2 IS - 6 PB - MathOA foundation UR - https://alco.centre-mersenne.org/articles/10.5802/alco.70/ DO - 10.5802/alco.70 LA - en ID - ALCO_2019__2_6_1311_0 ER -

%0 Journal Article %A Agarwal, Ayush %A Gaetz, Christian %T Differential posets and restriction in critical groups %J Algebraic Combinatorics %D 2019 %P 1311-1327 %V 2 %N 6 %I MathOA foundation %U https://alco.centre-mersenne.org/articles/10.5802/alco.70/ %R 10.5802/alco.70 %G en %F 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/articles/10.5802/alco.70/

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