Revised : 2018-08-09

Accepted : 2018-09-23

Published online : 2019-03-05

DOI : https://doi.org/10.5802/alco.43

Keywords: birational rowmotion, dynamical algebraic combinatorics, homomesy, periodicity, toggling.

@article{ALCO_2019__2_2_275_0, author = {Musiker, Gregg and Roby, Tom}, title = {Paths to Understanding Birational Rowmotion on Products of Two Chains}, journal = {Algebraic Combinatorics}, publisher = {MathOA foundation}, volume = {2}, number = {2}, year = {2019}, pages = {275-304}, doi = {10.5802/alco.43}, language = {en}, url = {https://alco.centre-mersenne.org/item/ALCO_2019__2_2_275_0} }

Musiker, Gregg; Roby, Tom. Paths to Understanding Birational Rowmotion on Products of Two Chains. Algebraic Combinatorics, Volume 2 (2019) no. 2, pp. 275-304. doi : 10.5802/alco.43. alco.centre-mersenne.org/item/ALCO_2019__2_2_275_0/

[1] A uniform bijection between nonnesting and noncrossing partitions, Trans. Am. Math. Soc., Volume 365 (2013) no. 8, pp. 4121-4151 | Article | MR 3055691 | Zbl 1271.05011

[2] On the period of an operator, defined on antichains, Stichting Mathematisch Centrum. Zuivere Wiskunde, Volume ZW 24/74 (1974), pp. 1-13 | Zbl 0282.06003

[3] Orbits of antichains revisited, Eur. J. Comb., Volume 16 (1995) no. 6, pp. 545-554 | Article | MR 1356845 | Zbl 0831.06001

[4] $T$-systems with boundaries from network solutions, Electron. J. Comb., Volume 20 (2013) no. 1, Paper 3, 62 pages | MR 3015686 | Zbl 1266.05176

[5] Combinatorial, piecewise-linear, and birational homomesy for products of two chains (2013) (https://arxiv.org/abs/1310.5294v1 )

[6] Piecewise-linear and birational toggling, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), Discrete Mathematics & Theoretical Computer Science (DMTCS) (Discrete Mathematics and Theoretical Computer Science) (2014), pp. 513-524 | MR 3466399 | Zbl 1394.06005

[7] Interlacing networks: birational RSK, the octahedron recurrence, and Schur function identities, J. Comb. Theory, Ser. A, Volume 133 (2015), pp. 339-371 | Article | MR 3325638 | Zbl 1315.05144

[8] Orbits of antichains in ranked posets, Eur. J. Comb., Volume 14 (1993) no. 1, pp. 17-22 | Article | MR 1197471 | Zbl 0777.06002

[9] The geometric $R$-matrix for affine crystals of type $A$ (2017) (https://arxiv.org/abs/1710.07243 )

[10] Bijective proofs for Schur function identities (2009) (https://arxiv.org/abs/0909.5334 )

[11] Bijective proofs for Schur function identities which imply Dodgson’s condensation formula and Plücker relations, Electron. J. Comb., Volume 8 (2001) no. 1, 16, 22 pages | MR 1855857 | Zbl 0978.05005

[12] $R$-systems (2017) (https://arxiv.org/abs/1709.00578 )

[13] Donaldson-Thomas transformations of moduli spaces of G-local systems, Adv. Math., Volume 327 (2018), pp. 225-348 | Article | MR 3761995 | Zbl 06842126

[14] Quadratic forms of skew Schur functions, Eur. J. Comb., Volume 9 (1988) no. 2, pp. 161-168 | Article | MR 939866 | Zbl 0651.05011

[15] Iterative properties of birational rowmotion (2014) (https://arxiv.org/abs/1402.6178 )

[16] Iterative properties of birational rowmotion II: rectangles and triangles, Electron. J. Comb., Volume 22 (2015) no. 3, 3.40, 49 pages | MR 3414186 | Zbl 1339.06001

[17] Iterative properties of birational rowmotion I: generalities and skeletal posets, Electron. J. Comb., Volume 23 (2016) no. 1, 1.33, 40 pages | MR 3484738 | Zbl 1338.06003

[18] A periodicity theorem for the octahedron recurrence, J. Algebr. Comb., Volume 26 (2007) no. 1, pp. 1-26 | Article | MR 2335700 | Zbl 1125.05106

[19] Groups generated by involutions, Gelfand-Tsetlin patterns, and combinatorics of Young tableaux, Algebra Anal., Volume 7 (1995) no. 1, pp. 92-152 | MR 1334154 | Zbl 0848.20007

[20] On orbits of antichains of positive roots, Eur. J. Comb., Volume 30 (2009) no. 2, pp. 586-594 | Article | MR 2489252 | Zbl 1165.06001

[21] Homomesy in products of two chains, Electron. J. Comb., Volume 22 (2015) no. 3, 3.4, 29 pages | MR 3367853 | Zbl 1319.05151

[22] The cyclic sieving phenomenon, J. Comb. Theory, Ser. A, Volume 108 (2004) no. 1, pp. 17-50 | Article | MR 2087303

[23] What is $...$ cyclic sieving?, Notices Am. Math. Soc., Volume 61 (2014) no. 2, pp. 169-171 | Article | MR 3156682 | Zbl 1338.05012

[24] Dynamical algebraic combinatorics and the homomesy phenomenon, Recent trends in combinatorics, Springer (The IMA Volumes in Mathematics and its Applications) Volume 159 (2016), pp. 619-652 (Also available at http://www.math.uconn.edu/~troby/homomesyIMA2015Revised.pdf) | Article | MR 3526426 | Zbl 1354.05146

[25] On orbits of order ideals of minuscule posets, J. Algebr. Comb., Volume 37 (2013) no. 3, pp. 545-569 | Article | MR 3035516

[26] On Orbits of Order Ideals of Minuscule Posets II: Homomesy (2015) (https://arxiv.org/abs/1509.08047 )

[27] Perfect matchings and the octahedron recurrence, J. Algebr. Comb., Volume 25 (2007) no. 3, pp. 309-348 | Article | MR 2317336 | Zbl 1119.05092

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

[29] Enumerative combinatorics. Vol. 1, Cambridge University Press, Cambridge Studies in Advanced Mathematics, Volume 49 (2012), xiii+626 pages (Also available at http://math.mit.edu/~rstan/ec/ec1/.) | Zbl 1247.05003

[30] Rowmotion and generalized toggle groups, Discrete Math. Theor. Comput. Sci., Volume 20 (2018) no. 1, 17, 26 pages | MR 3811480 | Zbl 06991639

[31] Promotion and rowmotion, Eur. J. Comb., Volume 33 (2012) no. 8, pp. 1919-1942 | Article | MR 2950491 | Zbl 1260.06004

[32] SageMath, the Sage Mathematics Software System (Version 7.3) (2016) (http://www.sagemath.org/ )

[33] Rowmotion in slow motion (2017) (https://arxiv.org/abs/1712.10123 )

[34] On the periodicity conjecture for $Y$-systems, Commun. Math. Phys., Volume 276 (2007) no. 2, pp. 509-517 | Article | MR 2346398 | Zbl 1136.82011

[35] The Coxeter transformation on Cominuscule Posets (2017) (https://arxiv.org/abs/1710.10632 )