Toric promotion

Defant, Colin

doi:10.1090/proc/16079

Toric promotion
HTML articles powered by AMS MathViewer

by Colin Defant

Proc. Amer. Math. Soc. 151 (2023), 45-57

DOI: https://doi.org/10.1090/proc/16079

Published electronically: September 30, 2022

HTML | PDF | Request permission

Abstract:

This article introduces toric promotion as a cyclic analogue of Schützenberger’s promotion operator. Toric promotion acts on the set of labelings of a graph $G$. We discuss connections between toric promotion and previously-studied notions such as toric posets and friends-and-strangers graphs. Our main theorem provides a surprisingly simple description of the orbit structure of toric promotion when $G$ is a forest.

References

Ron M. Adin, Victor Reiner, and Yuval Roichman, On cyclic descents for tableaux, Int. Math. Res. Not. IMRN 24 (2020), 10231–10276. MR 4190402, DOI 10.1093/imrn/rny280
Arvind Ayyer, Steven Klee, and Anne Schilling, Combinatorial Markov chains on linear extensions, J. Algebraic Combin. 39 (2014), no. 4, 853–881. MR 3199029, DOI 10.1007/s10801-013-0470-9
Joseph Bernstein, Jessica Striker, and Corey Vorland, $P$-strict promotion and $B$-bounded rowmotion, with applications to tableaux of many flavors, Comb. Theory 1 (2021), Paper No. 8, 46. MR 4396213, DOI 10.5070/C61055369
Colin Defant and Noah Kravitz, Friends and strangers walking on graphs, Comb. Theory 1 (2021), Paper No. 6, 34. MR 4396211, DOI 10.5070/C61055363
C. Defant and N. Kravitz, Promotion sorting, Order 14 (2022), DOI 10.1007/s11083-022-09603-9.
Mike Develin, Matthew Macauley, and Victor Reiner, Toric partial orders, Trans. Amer. Math. Soc. 368 (2016), no. 4, 2263–2287. MR 3449239, DOI 10.1090/tran/6356
Kevin Dilks, Oliver Pechenik, and Jessica Striker, Resonance in orbits of plane partitions and increasing tableaux, J. Combin. Theory Ser. A 148 (2017), 244–274. MR 3603321, DOI 10.1016/j.jcta.2016.12.007
Kevin Dilks, Jessica Striker, and Corey Vorland, Rowmotion and increasing labeling promotion, J. Combin. Theory Ser. A 164 (2019), 72–108. MR 3894190, DOI 10.1016/j.jcta.2018.12.007
Paul Edelman and Curtis Greene, Balanced tableaux, Adv. in Math. 63 (1987), no. 1, 42–99. MR 871081, DOI 10.1016/0001-8708(87)90063-6
Mark D. Haiman, Dual equivalence with applications, including a conjecture of Proctor, Discrete Math. 99 (1992), no. 1-3, 79–113. MR 1158783, DOI 10.1016/0012-365X(92)90368-P
Sam Hopkins and Martin Rubey, Promotion of Kreweras words, Selecta Math. (N.S.) 28 (2022), no. 1, Paper No. 10, 38. MR 4346507, DOI 10.1007/s00029-021-00714-6
Brice Huang, Cyclic descents for general skew tableaux, J. Combin. Theory Ser. A 169 (2020), 105120, 45. MR 3983097, DOI 10.1016/j.jcta.2019.105120
Claudia Malvenuto and Christophe Reutenauer, Evacuation of labelled graphs, Discrete Math. 132 (1994), no. 1-3, 137–143. MR 1297379, DOI 10.1016/0012-365X(92)00569-D
T. Kyle Petersen, Pavlo Pylyavskyy, and Brendon Rhoades, Promotion and cyclic sieving via webs, J. Algebraic Combin. 30 (2009), no. 1, 19–41. MR 2519848, DOI 10.1007/s10801-008-0150-3
Svetlana Poznanović and Kara Stasikelis, Properties of the promotion Markov chain on linear extensions, J. Algebraic Combin. 47 (2018), no. 3, 505–528. MR 3802326, DOI 10.1007/s10801-017-0783-1
Oliver Pretzel, On reorienting graphs by pushing down maximal vertices, Order 3 (1986), no. 2, 135–153. MR 865458, DOI 10.1007/BF00390104
Brendon Rhoades, Cyclic sieving, promotion, and representation theory, J. Combin. Theory Ser. A 117 (2010), no. 1, 38–76. MR 2557880, DOI 10.1016/j.jcta.2009.03.017
M. P. Schützenberger, Quelques remarques sur une construction de Schensted, Math. Scand. 12 (1963), 117–128 (French). MR 190017, DOI 10.7146/math.scand.a-10676
M. P. Schützenberger, Promotion des morphismes d’ensembles ordonnés, Discrete Math. 2 (1972), 73–94 (French). MR 299539, DOI 10.1016/0012-365X(72)90062-3
M. P. Schützenberger, Evacuations, Colloquio Internazionale sulle Teorie Combinatorie (Roma, 1973) Accad. Naz. Lincei, Rome, 1976, pp. 257–264 (English, with Italian summary). MR 476842
Richard P. Stanley, Promotion and evacuation, Electron. J. Combin. 16 (2009), no. 2, Special volume in honor of Anders Björner, Research Paper 9, 24. MR 2515772, DOI 10.37236/75