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 đť”– n given by the second author, and to enumerate the factors in such critical groups.

Received: 2018-02-25
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},
     zbl = {07140435},
     mrnumber = {4049848},
     language = {en},
     url = {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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