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}, pages = {1311--1327}, publisher = {MathOA foundation}, volume = {2}, number = {6}, year = {2019}, doi = {10.5802/alco.70}, zbl = {07140435}, mrnumber = {4049848}, 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/

[1] Differential posets, Cayley graphs, and critical groups, SĂ©min. Lothar. Comb. (2018), 12 pages

[2] Chip firing on Dynkin diagrams and McKay quivers, Math. Z., Volume 290 (2018) no. 1-2, pp. 615-648 | Article | MR 3848449 | Zbl 07031354

[3] Critical groups of group representations, Linear Algebra Appl., Volume 508 (2016), pp. 91-99 | Article | MR 3542983 | Zbl 1346.05298

[4] Critical Groups of McKay-Cartan matrices (2016) (Ph. D. Thesis)

[5] Dual graded graphs and Bratteli diagrams of towers of groups, Electron. J. Combin., Volume 26 (2019) no. 1, 12 pages | MR 3919618 | Zbl 1409.05212

[6] Path counting and rank gaps in differential posets (2018) (https://arxiv.org/abs/1806.03509)

[7] Chip-firing and rotor-routing on directed graphs, In and out of equilibrium. 2 (Progr. Probab.) Volume 60, BirkhĂ¤user, Basel, 2008, pp. 331-364 | Article | MR 2477390 | Zbl 1173.82339

[8] The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, Volume 16, Addison-Wesley Publishing Co., Reading, Mass., 1981, xxviii+510 pages | MR 644144 | Zbl 0491.20010

[9] Symmetric functions and Hall polynomials, Oxford Classic Texts in the Physical Sciences, The Clarendon Press, Oxford University Press, New York, 2015, xii+475 pages | MR 3443860

[10] Differential posets and Smith normal forms, Order, Volume 26 (2009) no. 3, pp. 197-228 | Article | MR 2544610 | Zbl 1228.05096

[11] Differential posets have strict rank growth: a conjecture of Stanley, Order, Volume 30 (2013) no. 2, pp. 657-662 | Article | MR 3063211 | Zbl 1283.06006

[12] Wreath products by the symmetric groups and product posets of Youngâ€™s lattices, J. Combin. Theory Ser. A, Volume 55 (1990) no. 1, pp. 14-32 | Article | MR 1070012 | Zbl 0707.05062

[13] Critical groups of covering, voltage and signed graphs, Discrete Math., Volume 318 (2014), pp. 10-40 | Article | MR 3141623 | Zbl 1281.05072

[14] Smith Normal Form of Matrices Associated with Differential Posets (2015) (https://arxiv.org/abs/1510.00588)

[15] Differential posets, J. Amer. Math. Soc., Volume 1 (1988) no. 4, pp. 919-961 | Article | MR 941434 | Zbl 0658.05006

[16] Enumerative combinatorics. Volume 1, Cambridge Studies in Advanced Mathematics, Volume 49, Cambridge University Press, Cambridge, 2012, xiv+626 pages | MR 2868112 | Zbl 1247.05003

[17] Smith normal form in combinatorics, J. Combin. Theory Ser. A, Volume 144 (2016), pp. 476-495 | Article | MR 3534076 | Zbl 1343.05026

[18] Functoriality of critical groups (2002) (Ph. D. Thesis)

[19] Three involutions on the set of 6-box Young diagrams, MathOverflow, 2015 (https://mathoverflow.net/q/215329 (version: 2015-08-21))