Abstract
Let \(\mathfrak{g}\) be an affine Lie algebra with index set I={0,1,2,ā¦,n} and \(\mathfrak{g}^{L}\) be its Langlands dual. It is conjectured in Kashiwara et al. (Trans. Am. Math. Soc. 360(7):3645ā3686, 2008) that for each iāIā{0} the affine Lie algebra \(\mathfrak{g}\) has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for \(\mathfrak{g}^{L}\). We prove this conjecture for i=2 and \(\mathfrak{g} = A_{n}^{(1)}\).
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References
Berenstein, A., Kazhdan, D.: Geometric crystals and unipotent crystals. In: GAFA 2000, Tel Aviv, 1999. Geom Funct. Anal. Special Volume, Part I, pp. 188ā236 (2000)
Fourier, G., Okado, M., Schilling, A.: Kirillov-Reshetikhin crystals for nonexceptional types. Adv. Math. 222(3), 1080ā1116 (2009)
Fourier, G., Okado, M., Schilling, A.: Perfectness of Kirillov-Reshetikhin crystals for nonexceptional types. Contemp. Math. 506, 127ā143 (2010)
Hatayama, G., Kuniba, A., Okado, M., Takagi, T., Yamada, Y.: Remarks on fermionic formula. Contemp. Math. 248, 243ā291 (1999)
Hatayama, G., Kuniba, A., Okado, M., Takagi, T., Tsuboi, Z.: Paths, crystals and fermionic formulae. In: Kashiwara, M., Miwa, T. (eds.) MathPhys Odessey 2001āIntegrable Models and Beyond in Honor of Barry M.Ā McCoy, pp. 205ā272. BirkhƤuser, Basel (2002)
Igarashi, M., Nakashima, T.: Affine geometric crystal of type \(D^{(3)}_{4}\). Contemp. Math. 506, 215ā226 (2010)
Igarashi, M., Misra, K.C., Nakashima, T.: Ultra-discretization of the \(D^{(3)}_{4}\)-geometric crystals to the \(G^{(1)}_{2}\)-perfect crystals. Pac. J. Math. 255(1), 117ā142 (2012)
Kac, V.G.: Infinite Dimensional Lie Algebras, 3rd edn. Cambridge Univ. Press, Cambridge (1990)
Kac, V.G., Peterson, D.H.: Defining relations of certain infinite-dimensional groups. In: Artin, M., Tate, J. (eds.) Arithmetic and Geometry, pp. 141ā166. BirkhƤuser, Boston (1983)
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), 449ā484 (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(3), 499ā607 (1992)
Kang, S.-J., Kashiwara, M., Misra, K.C.: Crystal bases of Verma modules for quantum affine Lie algebras. Compos. Math. 92, 299ā345 (1994)
Kashiwara, M.: Crystallizing the q-analogue of universal enveloping algebras. Commun. Math. Phys. 133, 249ā260 (1990)
Kashiwara, M.: On crystal bases of the q-analogue of universal enveloping algebras. Duke Math. J. 63, 465ā516 (1991)
Kashiwara, M.: On level-zero representation of quantized affine algebras. Duke Math. J. 112, 499ā525 (2002)
Kashiwara, M.: Level zero fundamental representations over quantized affine algebras and Demazure modules. Publ. Res. Inst. Math. Sci. 41(1), 223ā250 (2005)
Kashiwara, M., Misra, K., Okado, M., Yamada, D.: Perfect crystals for \(U_{q}(D_{4}^{(3)})\). J. Algebra 317(1), 392ā423 (2007)
Kashiwara, M., Nakashima, T., Okado, M.: Affine geometric crystals and limit of perfect crystals. Trans. Am. Math. Soc. 360(7), 3645ā3686 (2008)
Kirillov, A.N., Reshetikhin, N.: Representations of Yangians and multiplicity of occurrence of the irreducible components of the tensor product of representations of simple Lie algebras. J. Sov. Math. 52, 3156ā3164 (1990)
Kumar, S.: Kac-Moody Groups, Their Flag Varieties and Representation Theory. Progress in Mathematics, vol. 204. BirkhƤuser, Boston (2002)
Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Am. Math. Soc. 3, 447ā498 (1990)
Misra, K.C., Mohamad, M., Okado, M.: Zero action on perfect crystals for \(U_{q}(G_{2}^{(1)})\). SIGMA 201, 022 (2010), 12Ā pages
Nakashima, T.: Geometric crystals on Schubert varieties. J. Geom. Phys. 53(2), 197ā225 (2005)
Nakashima, T.: Geometric crystals on unipotent groups and generalized Young tableaux. J. Algebra 293(1), 65ā88 (2005)
Nakashima, T.: Affine Geometric Crystal of Type \(G_{2}^{(1)}\). Contemporary Mathematics, vol. 442, pp. 179ā192. Amer. Math. Soc., Providence (2007)
Nakashima, T.: Ultra-discretization of the \(G^{(1)}_{2}\)-geometric crystals to the \(D^{(3)}_{4}\)-perfect crystals. In: Representation Theory of Algebraic Groups and Quantum Groups. Progr. Math., vol. 284, pp. 273ā296. BirkhƤuser/Springer, New York (2010)
Okado, M., Schilling, A.: Existence of Kirillov-Reshetikhin crystals for nonexceptional types. Represent. Theory 12, 186ā207 (2008)
Okado, M., Schilling, A., Shimozono, M.: A tensor product theorem related to perfect crystals. J. Algebra 267, 212ā245 (2003)
Peterson, D.H., Kac, V.G.: Infinite flag varieties and conjugacy theorems. Proc. Natl. Acad. Sci. USA 80, 1778ā1782 (1983)
Yamane, S.: Perfect crystals of \(U_{q}(G_{2}^{(1)})\). J. Algebra 210(2), 440ā486 (1998)
Acknowledgements
We thank the referee for valuable comments. K.C.M. acknowledges partial support by Simon Foundation Grant 208092 and NSA Grant H98230-12-1-0248 and T.N. acknowledges partial support by JSPS Grants in Aid for Scientific Research āÆ 22540031 during this work.
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Dedicated to Professor Michio Jimbo on the occasion of his 60th birthday.
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Misra, K.C., Nakashima, T. (2013). \(A^{(1)}_{n}\)-Geometric Crystal Corresponding to Dynkin Index i=2 and Its Ultra-Discretization. In: Iohara, K., Morier-Genoud, S., RĆ©my, B. (eds) Symmetries, Integrable Systems and Representations. Springer Proceedings in Mathematics & Statistics, vol 40. Springer, London. https://doi.org/10.1007/978-1-4471-4863-0_12
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