In recent work, Benkart, Klivans, and Reiner defined the critical group of a faithful representation of a finite group , 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 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 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
Agarwal, Ayush 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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