Birational rowmotion is an action on the space of assignments of rational functions to the elements of a finite partially-ordered set (poset). It is lifted from the well-studied rowmotion map on order ideals (equivalently on antichains) of a poset $P$, which when iterated on special posets, has unexpectedly nice properties in terms of periodicity, cyclic sieving, and homomesy (statistics whose averages over each orbit are constant). In this context, rowmotion appears to be related to Auslander–Reiten translation on certain quivers, and birational rowmotion to $Y$-systems of type ${A}_{m}\times {A}_{n}$ described in Zamolodchikov periodicity.

We give a formula in terms of families of non-intersecting lattice paths for iterated actions of the birational rowmotion map on a product of two chains. This allows us to give a much simpler direct proof of the key fact that the period of this map on a product of chains of lengths $r$ and $s$ is $r+s+2$ (first proved by D. Grinberg and the second author), as well as the first proof of the birational analogue of homomesy along files for such posets.

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}, pages = {275--304}, publisher = {MathOA foundation}, volume = {2}, number = {2}, year = {2019}, doi = {10.5802/alco.43}, mrnumber = {3934831}, zbl = {07049526}, 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. https://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 and Theoretical Computer Science), Discrete Mathematics & Theoretical Computer Science (DMTCS), 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 ) | Zbl 07036397

[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 (The IMA Volumes in Mathematics and its Applications) Volume 159, Springer, 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 | Zbl 1284.06008

[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 Studies in Advanced Mathematics, Volume 49, Cambridge University Press, 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 1401.05315

[31] Promotion and rowmotion, Eur. J. Comb., Volume 33 (2012) no. 8, pp. 1919-1942 | Article | MR 2950491 | Zbl 1417.06003

[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 ) | Zbl 07143125

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

[35] The Coxeter transformation on Cominuscule Posets (2017) (https://arxiv.org/abs/1710.10632 ) | Zbl 07075207