Tropical R maps and affine geometric crystals

Kashiwara, Masaki; Nakashima, Toshiki; Okado, Masato

doi:10.1090/S1088-4165-2010-00379-9

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-4165

Tropical R maps and affine geometric crystals

Authors: Masaki Kashiwara, Toshiki Nakashima and Masato Okado
Journal: Represent. Theory 14 (2010), 446-509
MSC (2010): Primary 17B37, 17B67; Secondary 22E65, 14M15
Posted: July 7, 2010
MathSciNet review: 2661518
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: By modifying an earlier method of the authors (2008), certain affine geometric crystals are realized in affinization of the fundamental representation $W(\varpi_1)_l$ , and the tropical R maps for the affine geometric crystals are described explicitly. We also define prehomogeneous geometric crystals and show that for a positive geometric crystal, the connectedness of the corresponding ultra-discretized crystal is the sufficient condition for prehomogeneity of the positive geometric crystal. Moreover, the uniqueness of tropical R maps is discussed.

[BK] Arkady Berenstein and David Kazhdan, Geometric and unipotent crystals, Geom. Funct. Anal. Special Volume (2000), 188–236. GAFA 2000 (Tel Aviv, 1999). MR 1826254 (2003b:17013), http://dx.doi.org/10.1007/978-3-0346-0422-2_8
[D1] V. G. Drinfel′d, Quantum groups, (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR 934283 (89f:17017)
[D2] V. G. Drinfel′d, On some unsolved problems in quantum group theory, Quantum groups (Leningrad, 1990) Lecture Notes in Math., vol. 1510, Springer, Berlin, 1992, pp. 1–8. MR 1183474 (94a:17006), http://dx.doi.org/10.1007/BFb0101175
[HKOTY] G. Hatayama, A. Kuniba, M. Okado, T. Takagi, and Y. Yamada, Scattering rules in soliton cellular automata associated with crystal bases, field theory (Charlottesville, VA, 2000) Contemp. Math., vol. 297, Amer. Math. Soc., Providence, RI, 2002, pp. 151–182. MR 1919817 (2003k:81104), http://dx.doi.org/10.1090/conm/297/05097
[J] Michio Jimbo, A 𝑞-difference analogue of 𝑈(𝔤) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), no. 1, 63–69. MR 797001 (86k:17008), http://dx.doi.org/10.1007/BF00704588
[Kac] Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219 (92k:17038)
[KKM] Seok-Jin Kang, Masaki Kashiwara, and Kailash C. Misra, Crystal bases of Verma modules for quantum affine Lie algebras, Compositio Math. 92 (1994), no. 3, 299–325. MR 1286129 (95h:17016)
[KMN1] Seok-Jin Kang, Masaki Kashiwara, Kailash C. Misra, Tetsuji Miwa, Toshiki Nakashima, and Atsushi Nakayashiki, Affine crystals and vertex models, Infinite analysis, Part A, B (Kyoto, 1991) Adv. Ser. Math. Phys., vol. 16, World Sci. Publ., River Edge, NJ, 1992, pp. 449–484. MR 1187560 (94a:17008)
[KMN2] Seok-Jin Kang, Masaki Kashiwara, Kailash C. Misra, Tetsuji Miwa, Toshiki Nakashima, and Atsushi Nakayashiki, Perfect crystals of quantum affine Lie algebras, Duke Math. J. 68 (1992), no. 3, 499–607. MR 1194953 (94j:17013), http://dx.doi.org/10.1215/S0012-7094-92-06821-9
[K1] Masaki Kashiwara, On level-zero representations of quantized affine algebras, Duke Math. J. 112 (2002), no. 1, 117–175. MR 1890649 (2002m:17013), http://dx.doi.org/10.1215/S0012-9074-02-11214-9
[K2] Masaki Kashiwara, Level zero fundamental representations over quantized affine algebras and Demazure modules, Publ. Res. Inst. Math. Sci. 41 (2005), no. 1, 223–250. MR 2115972 (2005i:17021)
[KOTUY] Atsuo Kuniba, Kailash C. Misra, Masato Okado, Taichiro Takagi, and Jun Uchiyama, Crystals for Demazure modules of classical affine Lie algebras, J. Algebra 208 (1998), no. 1, 185–215. MR 1643999 (99h:17008), http://dx.doi.org/10.1006/jabr.1998.7503
[KNO] Masaki Kashiwara, Toshiki Nakashima, and Masato Okado, Affine geometric crystals and limit of perfect crystals, Trans. Amer. Math. Soc. 360 (2008), no. 7, 3645–3686. MR 2386241 (2009e:17020), http://dx.doi.org/10.1090/S0002-9947-08-04341-9
[KOTY] A. Kuniba, M. Okado, T. Takagi, and Y. Yamada, Geometric crystal and tropical 𝑅 for 𝐷⁽¹⁾_{𝑛}, Int. Math. Res. Not. 48 (2003), 2565–2620. MR 2013509 (2004h:17016), http://dx.doi.org/10.1155/S1073792803209041
[N1] Toshiki Nakashima, Geometric crystals on Schubert varieties, J. Geom. Phys. 53 (2005), no. 2, 197–225. MR 2110832 (2006d:17016), http://dx.doi.org/10.1016/j.geomphys.2004.06.004
[N2] Toshiki Nakashima, Geometric crystals on unipotent groups and generalized Young tableaux, J. Algebra 293 (2005), no. 1, 65–88. MR 2173966 (2006j:20064), http://dx.doi.org/10.1016/j.jalgebra.2005.07.025
[PK] Dale H. Peterson and Victor G. Kac, Infinite flag varieties and conjugacy theorems, Proc. Nat. Acad. Sci. U.S.A. 80 (1983), no. 6 i., 1778–1782. MR 699439 (84g:17017)
[Y] Yasuhiko Yamada, A birational representation of Weyl group, combinatorial 𝑅-matrix and discrete Toda equation, Physics and combinatorics, 2000 (Nagoya), World Sci. Publ., River Edge, NJ, 2001, pp. 305–319. MR 1872262 (2002k:20021), http://dx.doi.org/10.1142/9789812810007_0014

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2010): 17B37, 17B67, 22E65, 14M15

Retrieve articles in all journals with MSC (2010): 17B37, 17B67, 22E65, 14M15

Additional Information

Masaki Kashiwara
Affiliation: Research Institute for Mathematical Sciences, Kyoto University, Kitashiwakawa, Sakyo-ku, Kyoto 606, Japan
Email: masaki@kurims.kyoto-u.ac.jp

Toshiki Nakashima
Affiliation: Department of Mathematics, Sophia University, Kioicho 7-1, Chiyoda-ku, Tokyo 102-8554, Japan
Email: toshiki@mm.sophia.ac.jp

Masato Okado
Affiliation: Department of Mathematical Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan
Email: okado@sigmath.es.osaka-u.ac.jp

DOI: http://dx.doi.org/10.1090/S1088-4165-2010-00379-9
PII: S 1088-4165(2010)00379-9
Keywords: Prehomogeneous geometric crystal, perfect crystal, folding, tropical $R$ map, ultra-discretization
Received by editor(s): September 2, 2008
Posted: July 7, 2010
Additional Notes: This work was supported in part by JSPS Grants in Aid for Scientific Research, numbers 18340007(M.K.), 19540050(T.N.), 20540016(M.O.)
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|