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:

Accepted:

Published online:

DOI: 10.5802/alco.43

^{1}; Roby, Tom

^{2}

@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 = {http://www.numdam.org/articles/10.5802/alco.43/} }

TY - JOUR AU - Musiker, Gregg AU - Roby, Tom TI - Paths to Understanding Birational Rowmotion on Products of Two Chains JO - Algebraic Combinatorics PY - 2019 SP - 275 EP - 304 VL - 2 IS - 2 PB - MathOA foundation UR - http://www.numdam.org/articles/10.5802/alco.43/ DO - 10.5802/alco.43 LA - en ID - ALCO_2019__2_2_275_0 ER -

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. http://www.numdam.org/articles/10.5802/alco.43/

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

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

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

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

[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 | Zbl

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

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

[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 | Zbl

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

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

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

[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 | Zbl

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

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

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

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

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

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

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

[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) | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

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

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

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

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

[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

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

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

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