Birational and noncommutative lifts of antichain toggling and rowmotion
Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 955-984.

The rowmotion action on order ideals or on antichains of a finite partially ordered set has been studied (under a variety of names) by many authors. Depending on the poset, one finds unexpectedly interesting orbit structures, instances of (small order) periodicity, cyclic sieving, and homomesy. Many of these nice features still hold when the action is extended to [0,1]-labelings of the poset or (via detropicalization) to labelings by rational functions (the birational setting).

In this work, we parallel the birational lifting already done for order-ideal rowmotion to antichain rowmotion. We give explicit equivariant bijections between the birational toggle groups and between their respective liftings. We further extend all of these notions to labellings by noncommutative rational functions, setting an unpublished periodicity conjecture of Grinberg in a broader context.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.125
Classification: 05E18,  06A07,  12E15
Keywords: Antichain, birational rowmotion, dynamical algebraic combinatorics, graded poset, homomesy, isomorphism, noncommutative algebra, periodicity, rowmotion, toggle group, transfer map.
Joseph, Michael 1; Roby, Tom 2

1 Department of Technology and Mathematics Dalton State College 650 College Dr. Dalton, GA 30720, USA
2 University of Connecticut 341 Mansfield Road Storrs, CT 06269-1009, USA
@article{ALCO_2020__3_4_955_0,
     author = {Joseph, Michael and Roby, Tom},
     title = {Birational and noncommutative lifts of antichain toggling and rowmotion},
     journal = {Algebraic Combinatorics},
     pages = {955--984},
     publisher = {MathOA foundation},
     volume = {3},
     number = {4},
     year = {2020},
     doi = {10.5802/alco.125},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.125/}
}
TY  - JOUR
TI  - Birational and noncommutative lifts of antichain toggling and rowmotion
JO  - Algebraic Combinatorics
PY  - 2020
DA  - 2020///
SP  - 955
EP  - 984
VL  - 3
IS  - 4
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.125/
UR  - https://doi.org/10.5802/alco.125
DO  - 10.5802/alco.125
LA  - en
ID  - ALCO_2020__3_4_955_0
ER  - 
%0 Journal Article
%T Birational and noncommutative lifts of antichain toggling and rowmotion
%J Algebraic Combinatorics
%D 2020
%P 955-984
%V 3
%N 4
%I MathOA foundation
%U https://doi.org/10.5802/alco.125
%R 10.5802/alco.125
%G en
%F ALCO_2020__3_4_955_0
Joseph, Michael; Roby, Tom. Birational and noncommutative lifts of antichain toggling and rowmotion. Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 955-984. doi : 10.5802/alco.125. https://alco.centre-mersenne.org/articles/10.5802/alco.125/

[1] Armstrong, Drew; Stump, Christian; Thomas, Hugh A uniform bijection between nonnesting and noncrossing partitions, Trans. Amer. Math. Soc., Volume 365 (2013) no. 8, pp. 4121-4151 | Article | MR: 3055691 | Zbl: 1271.05011

[2] Brouwer, Andries E.; Schrijver, Alexander On the period of an operator, defined on antichains, Stichting Mathematisch Centrum, 1974, 13 pages (Stichting Mathematisch Centrum. Zuivere Wiskunde ZW 24/74) | Zbl: 0282.06003

[3] Cameron, Peter J.; Fon-Der-Flaass, Dmitriĭ G. Orbits of antichains revisited, European J. Combin., Volume 16 (1995) no. 6, pp. 545-554 | Article | MR: 1356845 | Zbl: 0831.06001

[4] Dilks, Kevin; Pechenik, Oliver; Striker, Jessica Resonance in orbits of plane partitions and increasing tableaux, J. Combin. Theory Ser. A, Volume 148 (2017), pp. 244-274 | Article | MR: 3603321 | Zbl: 1355.05266

[5] Dilks, Kevin; Striker, Jessica; Vorland, Corey Rowmotion and increasing labeling promotion, J. Combin. Theory Ser. A, Volume 164 (2019), pp. 72-108 | Article | MR: 3894190 | Zbl: 1407.05245

[6] Einstein, David; Propp, James Combinatorial, piecewise-linear, and birational homomesy for products of two chains (2018) (Preprint, https://arxiv.org/abs/1310.5294)

[7] Etienne, Gwihen Linear extensions of finite posets and a conjecture of G. Kreweras on permutations, Discrete Math., Volume 52 (1984) no. 1, pp. 107-111 | Article | MR: 765289 | Zbl: 0546.05003

[8] Grinberg, Darij; Roby, Tom Iterative properties of birational rowmotion II: rectangles and triangles, Electron. J. Combin., Volume 22 (2015) no. 3, Paper no. Paper 3.40, 49 pages | MR: 3414186 | Zbl: 1339.06001

[9] Grinberg, Darij; Roby, Tom Iterative properties of birational rowmotion I: generalities and skeletal posets, Electron. J. Combin., Volume 23 (2016) no. 1, Paper no. Paper 1.33, 40 pages | MR: 3484738 | Zbl: 1338.06003

[10] Haddadan, Shahrzad Some Instances of Homomesy Among Ideals of Posets (2016) (Preprint, https://arxiv.org/abs/1410.4819)

[11] Hopkins, Sam Minuscule doppelgängers, the coincidental down-degree expectations property, and rowmotion (2019) (Preprint, https://arxiv.org/abs/1902.07301)

[12] Joseph, Michael Antichain toggling and rowmotion, Electron. J. Combin., Volume 26 (2019) no. 1, 43 pages | MR: 3919614 | Zbl: 1409.05218

[13] Joseph, Michael; Roby, Tom Toggling independent sets of a path graph, Electron. J. Combin., Volume 25 (2018) no. 1, Paper no. Paper No. 1.18, 31 pages | MR: 3761932 | Zbl: 1380.05202

[14] Joseph, Michael; Roby, Tom A birational lifting of the Stanley–Thomas word on products of two chains (2020) (Preprint, https://arxiv.org/abs/2001.03811)

[15] Kirillov, Anatol N.; Berenstein, Arkadiy D. Groups generated by involutions, Gelʼfand–Tsetlin patterns, and combinatorics of Young tableaux, Algebra i Analiz, Volume 7 (1995) no. 1, pp. 92-152 (Translation available at http://pages.uoregon.edu/arkadiy/bk1.pdf.) | MR: 1334154 | Zbl: 0848.20007

[16] Musiker, Gregg; Roby, Tom Paths to understanding birational rowmotion on products of two chains, Algebr. Comb., Volume 2 (2019) no. 2, pp. 275-304 | Article | MR: 3934831 | Zbl: 07049526

[17] Okada, Soichi Birational rowmotion and Coxeter-motion on minuscule posets (2020) (Preprint, https://arxiv.org/abs/2004.05364)

[18] Panyushev, Dmitri I. On orbits of antichains of positive roots, European J. Combin., Volume 30 (2009) no. 2, pp. 586-594 | Article | MR: 2489252 | Zbl: 1165.06001

[19] Propp, James; Roby, Tom Homomesy in products of two chains, Electron. J. Combin., Volume 22 (2015) no. 3, Paper no. Paper 3.4, 29 pages | MR: 3367853 | Zbl: 1319.05151

[20] Reiner, Victor; Stanton, Dennis; White, Dennis The cyclic sieving phenomenon, J. Combin. Theory Ser. A, Volume 108 (2004) no. 1, pp. 17-50 | Article | MR: 2087303

[21] Roby, Tom Dynamical algebraic combinatorics and the homomesy phenomenon, Recent trends in combinatorics (IMA Vol. Math. Appl.), Volume 159, Springer, Cham, 2016, pp. 619-652 | Article | MR: 3526426 | Zbl: 1354.05146

[22] Rush, David B.; Wang, Kevin On orbits of order ideals of minuscule posets II: Homomesy (2015) (Preprint, https://arxiv.org/abs/1509.08047)

[23] Stanley, Richard P. Two poset polytopes, Discrete Comput. Geom., Volume 1 (1986) no. 1, pp. 9-23 | Article | MR: 824105 | Zbl: 0595.52008

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

[25] Striker, Jessica The toggle group, homomesy, and the Razumov–Stroganov correspondence, Electron. J. Combin., Volume 22 (2015) no. 2, Paper no. Paper 2.57, 17 pages | MR: 3367300 | Zbl: 1319.05015

[26] Striker, Jessica Rowmotion and generalized toggle groups, Discrete Math. Theor. Comput. Sci., Volume 20 (2018) no. 1, Paper no. Paper No. 17, 26 pages | MR: 3811480 | Zbl: 1401.05315

[27] Striker, Jessica; Williams, Nathan Promotion and rowmotion, European J. Combin., Volume 33 (2012) no. 8, pp. 1919-1942 | Article | MR: 2950491

[28] Thomas, Hugh; Williams, Nathan Rowmotion in slow motion, Proc. Lond. Math. Soc. (3), Volume 119 (2019) no. 5, pp. 1149-1178 | Article | MR: 3968720 | Zbl: 07143125

[29] Vorland, Corey Homomesy in products of three chains and multidimensional recombination (2017) (Preprint, https://arxiv.org/abs/1705.02665v2) | Zbl: 1427.05231

[30] Vorland, Corey Multidimensional Toggle Dynamics (2018) (Ph. D. Thesis)

[31] Wieland, Benjamin A large dihedral symmetry of the set of alternating sign matrices, Electron. J. Combin., Volume 7 (2000), Paper no. Research Paper 37, 13 pages | MR: 1773294 | Zbl: 0956.05015

[32] Yıldırım, Emine The Coxeter transformation on cominuscule posets, Algebr. Represent. Theory, Volume 22 (2019) no. 3, pp. 699-722 | Article | MR: 3951768 | Zbl: 07075207

Cited by Sources: