Abstract
The behavior of the Kirillov-Schilling-Shimozono bijection is examined under the similarity map on Kirillov-Reshetikhin crystals. It enables us to define this bijection over \(\mathbb {Q}\). Conjectures on the extension to \(\mathbb {R}\) are also presented.
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Berenstein, A., Kazhdan, D.: Geometric and unipotent crystals, GAFA 2000 (Tel Aviv 1999). Geom. Funct. Anal. Special Volume, Part I, 188–236 (2000)
Hatayama, G., Kuniba, A., Okado, M., Takagi, T., Tsuboi, Z.: Paths, Crystals and Fermionic Formulae, MathPhys Odyssey 2001, 205–272, Prog. Math. Phys, 23, Birkhäuser, Boston (2002)
Hatayama, G., Kuniba, A., Okado, M., Takagi, T., Yamada, Y.: Remarks on fermionic formula. Contemp. Math. 248, 243–291 (1999)
Kac, V.: Infinite dimensional Lie algebras, 3rd ed. Cambridge University Press (1990)
Kang, S.-J., Kashiwara, M., Misra, K.C., Miwa, T., Nakashima, T., Nakayashiki, A.: Affine crystals and vertex models. Int. J. Mod. Phys. A 7(suppl. 1A), 449–484 (1992)
Kang, S.-J., Kashiwara, M., Misra, K.C., Miwa, T., Nakashima, T., Nakayashiki, A.: Perfect crystals of quantum affine Lie algebras. Duke. Math. J 68, 499–607 (1992)
Kashiwara, M., Nakashima, T.: Crystal graphs for representations of the q-analogue of classical Lie algebras. J. Alg. 165, 295–345 (1994)
Kashiwara, M., Nakashima, T., Okado, M.: Affice geometric crystals and limit of perfect crystals. Trans. Amer. Math. Soc. 360(7), 3645–3686 (2008)
Kerov, S.V., Kirillov, A.N., Reshetikhin, N.Y.: Combinatorics, the Bethe ansatz and representations of the symmetric group Zap. Nauchn. Sem. (LOMI) 155 (1986) 50–64. (English translation: J. Sov. Math. 41 916–924.) (1988)
Kirillov, A.N., Reshetikhin, N.Y.u.: The Bethe ansatz and the combinatorics of Young tableaux. J. Sov. Math 41, 925–955 (1988)
Kirillov, A.N., Schilling, A., Shimozono, M.: A bijection between Littlewood-Richardson tableaux and rigged configurations. Selecta Math.(N.S.) 8(1), 67–135 (2002)
Kuniba, A., Okado, M., Sakamoto, R., Takagi, T., Yamada, Y.: Crystal interpretation of Kerov-Kirillov-Reshetikhin bijection. Nucl. Phys. B740, 299–327 (2006)
Kuniba, A., Okado, M., Takagi, T., Yamada, Y.: Geometric crystals and tropical \(\mathcal {R}\) for D n(1). IMRN 48, 2565–2620 (2003)
Kuniba, A., Sakamoto, R., Yamada, Y.: Tau functions in combinatorial Bethe ansatz. Nucl. Phys. B786, 207–266 (2007)
Lam, T., Pylyavskyy, P.: R Sakamoto, Rigged configurations and cylindric loop Schur functions. arXiv:1410.4455
Okado, M.: X = M Conjecture. MSJ Memoirs 17, 43–73 (2007)
Okado, M.: Simplicity and similarity of Kirillov-Reshetikhin crystals. Contemp. Math. 602, 183–194 (2013)
Okado, M., Schilling, A.: Existence of Kirillov-Reshetikhin crystals for nonexceptional types. Replace. Ther. 12, 186–207 (2008)
Okado, M., Schilling, A., Shimozono, M.: A tensor product theorem related to perfect crystals. J. Algebra 267, 212–245 (2003)
Schilling, A.: X = m theorem: Fermionic formulas and rigged configurations under review. MSJ Memoirs 17, 75–104 (2007)
Shimozono, M.: Affine type A crystal structure on tensor products of rectangles, Demazure characters, and nilpotent varieties. J. Algebraic Combin. 15(2), 151–187 (2002)
Takagi, T.: Combinatorial aspects of the conserved quantities of the tropical periodic Toda lattice. J. Phys. A: Math. Theor. 47(395201), 25 (2014)
Takahashi, D., Satsuma, J.: A soliton cellular automaton. J. Phys. Soc. Jpn. 59, 3514–3519 (1990)
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Presented by Anne Schilling.
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Okado, M. Similarity and Kirillov-Schilling-Shimozono Bijection. Algebr Represent Theor 19, 975–989 (2016). https://doi.org/10.1007/s10468-016-9607-6
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DOI: https://doi.org/10.1007/s10468-016-9607-6