Abstract
We use the combinatorics of toric networks and the double affine geometric R-matrix to define a three-parameter family of generalizations of the discrete Toda lattice. We construct the integrals of motion and a spectral map for this system. The family of commuting time evolutions arising from the action of the R-matrix is explicitly linearized on the Jacobian of the spectral curve. The solution to the initial value problem is constructed using Riemann theta functions.
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Berenstein, A., Kazhdan, D.: Geometric and unipotent crystals. Geom. Funct. Anal. Special Volume, Part I, pp. 188–236 (2000)
Etingof P.: Geometric crystals and set-theoretical solutions to the quantum Yang–Baxter equation. Commun. Algebra 31(4), 1961–1973 (2003)
Fay, J.D.: Theta functions on Riemann surfaces. In: Lecture Notes in Mathematics, vol. 352. Springer, Berlin (1973)
Fock, V.V., Marshakov, A.: Loop Groups, Clusters, Dimers and Integrable Systems. ar**v:1401.1606
Gekhtman, M., Shapiro, M., Tabachnikov, S., Vainshtein, A.: Integrable cluster dynamics of directed networks and pentagram maps. Adv. Math. ar**v:1406.1883 (to appear)
Goncharov A.B., Kenyon R.: Dimers and cluster integrable systems. Ann. Sci. éc. Norm. Supér. (4) 46(5), 747–813 (2013)
Hirota, R., Tsujimoto, S., Imai, T.: Difference scheme of soliton equations. In: Future Directions of Nonlinear Dynamics in Physical and Biological Systems (Lyngby, 1992). NATO Adv. Sci. Inst. Ser. B Phys. 312, 7–15 (Plenum Press, New York, 1993)
Inoue, R., Kuniba, A., Takagi, T.: Integrable structure of box-ball systems: crystal, Bethe ansatz, ultradiscretization and tropical geometry. J. Phys. A 45(7), 073001 (2012)
Inoue, R., Takenawa, T.: Tropical spectral curves and integrable cellular automata. Int. Math. Res. Not. IMRN 27, Art ID. rnn019 (2008)
Inoue R., Takenawa T.: A tropical analogue of Fay’s trisecant identity and an ultradiscrete periodic Toda lattice. Commun. Math. Phys. 289, 995–1021 (2009)
Inoue, R., Lam, T., Pylyavskyy, P.: On the cluster nature and quantization of geometric R-matrices. ar**v:1607.00722
Iwao, S.: Solution of the generalized periodic discrete Toda equation. J. Phys. A 41(11), 115201 (2008)
Iwao, S.: Solution of the generalized periodic discrete Toda equation. II. Theta function solution. J. Phys. A 43(15), 155208 (2010)
Iwao, S.: Linearisation of the (M,K)-reduced non-autonomous discrete periodic KP equation. ar**v:0912.3333
Kirillov, A.N.: Introduction to tropical combinatorics. In: Kirillov, A.N., Liskova, N. (eds.) Physics and Combinatorics 2000. Proceedings of the Nagoya 2000 International Workshop, pp. 82–150. World Scientific, Singapore (2001)
Khovanskii A.: Three Lectures on Newton Polyhedra (Newton Polyhedrons and Singularities). RIMS Kokyuroku 1233, 1–17 (2001)
Kang, S.-J., Kashiwara, M., Misra, K.C., Miwa, T., Nakashima, T., Nakayashiki, A.: Affine crystals and vertex models. In: Infinite analysis Part A (Kyoto 1991), pp. 449–484. Adv. Ser. Math. Phys. 16, World Sci. Publishing, River Edge (1992)
Kashiwara M., Nakashima T., Okado M.: Affine geometric crystals and limit of perfect crystals (English summary). Trans. Am. Math. Soc. 360(7), 3645–3686 (2008)
Kashiwara M., Nakashima T., Okado M.: Tropical R maps and affine geometric crystals. Represent. Theory 14, 446–509 (2010)
Kajiwara K., Noumi M., Yamada Y.: Discrete dynamical systems with \({W(A^(1)_m-1 \times A^{(1)}_n-1)}\) symmetry. Lett. Math. Phys. 60(3), 211–219 (2002)
Lascoux, A.: Double crystal graphs. In: Studies in Memory of Issai Schur (Chevaleret/Rehovot, 2000). Progr. Math., vol. 210, pp. 95–114. Birkhäuser Boston, Boston (2003)
Lam T., Pylyavskyy P.: Affine geometric crystals in unipotent loop groups. Represent. Theory 15, 719–728 (2011)
Lam T., Pylyavskyy P.: Total positivity in loop groups. I: Whirls and curls. Adv. Math. 230(3), 1222–1271 (2012)
Lam T., Pylyavskyy P.: Inverse problem in electrical cylindrical networks. SIAM J. Appl. Math. 72, 767–788 (2012)
Lam T., Pylyavskyy P.: Crystals and total positivity on orientable surfaces. Selecta Math. (N.S.) 19(1), 173–235 (2013)
Mada, J., Idzumi, M., Tokihiro, T.: Path description of conserved quantities of generalized periodic box-ball systems. J. Math. Phys. 46(2), 022701 (2005)
Schützenberger, M.-P.: Promotion des morphismes d’ensembles ordonnes. Discrete Math. 2, 73–94 (1972)
Speyer D.: Perfect matchings and the octahedron recurrence. J. Algebr. Combin. 25(3), 309–348 (2007)
Takahashi D., Satsuma J.: A soliton cellular automaton. J. Phys. Soc. Japan 59(10), 3514–3519 (1990)
van Moerbeke P., Mumford D.: The spectrum of difference operators and algebraic curves. Acta Math. 143(1–2), 94–154 (1979)
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Communicated by A. Borodin
R. Inoue was partially supported by JSPS KAKENHI Grant Number 26400037.
T. Lam was partially supported by NSF Grants DMS-1160726, DMS-1464693, and a Simons Fellowship.
P. Pylyavskyy was partially supported by NSF Grants DMS-1148634, DMS-1351590, and Sloan Fellowship.
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Inoue, R., Lam, T. & Pylyavskyy, P. Toric Networks, Geometric R-Matrices and Generalized Discrete Toda Lattices. Commun. Math. Phys. 347, 799–855 (2016). https://doi.org/10.1007/s00220-016-2739-z
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DOI: https://doi.org/10.1007/s00220-016-2739-z