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
@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 = {http://www.numdam.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 - http://www.numdam.org/articles/10.5802/alco.70/ DO - 10.5802/alco.70 LA - en ID - ALCO_2019__2_6_1311_0 ER -
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. http://www.numdam.org/articles/10.5802/alco.70/
[1] Differential posets, Cayley graphs, and critical groups, SĂ©min. Lothar. Comb. (2018), 80B, 12 pages
[2] Chip firing on Dynkin diagrams and McKay quivers, Math. Z., Volume 290 (2018) no. 1-2, pp. 615-648 | DOI | MR | Zbl
[3] Critical groups of group representations, Linear Algebra Appl., Volume 508 (2016), pp. 91-99 | DOI | MR | Zbl
[4] Critical Groups of McKay-Cartan matrices, Ph. D. Thesis, University of Minnesota Honors Thesis (B.S.) (2016) (http://www-users.math.umn.edu/~reiner/HonorsTheses/Gaetz_thesis.pdf)
[5] Dual graded graphs and Bratteli diagrams of towers of groups, Electron. J. Combin., Volume 26 (2019) no. 1, P1.25, 12 pages | MR | Zbl
[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, Birkhäuser, Basel, 2008, pp. 331-364 | DOI | MR | Zbl
[8] The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, 16, Addison-Wesley Publishing Co., Reading, Mass., 1981, xxviii+510 pages | MR | Zbl
[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
[10] Differential posets and Smith normal forms, Order, Volume 26 (2009) no. 3, pp. 197-228 | DOI | MR | Zbl
[11] Differential posets have strict rank growth: a conjecture of Stanley, Order, Volume 30 (2013) no. 2, pp. 657-662 | DOI | MR | Zbl
[12] Wreath products by the symmetric groups and product posets of Young’s lattices, J. Combin. Theory Ser. A, Volume 55 (1990) no. 1, pp. 14-32 | DOI | MR | Zbl
[13] Critical groups of covering, voltage and signed graphs, Discrete Math., Volume 318 (2014), pp. 10-40 | DOI | MR | Zbl
[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 | DOI | MR | Zbl
[16] Enumerative combinatorics. Volume 1, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, Cambridge, 2012, xiv+626 pages | MR | Zbl
[17] Smith normal form in combinatorics, J. Combin. Theory Ser. A, Volume 144 (2016), pp. 476-495 | DOI | MR | Zbl
[18] Functoriality of critical groups, Ph. D. Thesis, University of Minnesota Honors Thesis (B.S.) (2002) (http://www-users.math.umn.edu/~reiner/HonorsTheses/Treumann_thesis.pdf)
[19] Three involutions on the set of 6-box Young diagrams, MathOverflow, 2015 https://mathoverflow.net/q/215329 (version: 2015-08-21)
Cited by Sources: