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
Bibliographic Information
- Colin Defant
- Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08540
- MR Author ID: 1096299
- Email: cdefant@princeton.edu
- Received by editor(s): December 17, 2021
- Received by editor(s) in revised form: March 11, 2022
- Published electronically: September 30, 2022
- Additional Notes: The author was supported by a Fannie and John Hertz Foundation Fellowship and an NSF Graduate Research Fellowship (grant number DGE 1656466)
- Communicated by: Isabella Novik
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 45-57
- MSC (2020): Primary 05E18; Secondary 05C05
- DOI: https://doi.org/10.1090/proc/16079
- MathSciNet review: 4504606