## Tropical R maps and affine geometric crystals

HTML articles powered by AMS MathViewer

- by Masaki Kashiwara, Toshiki Nakashima and Masato Okado
- Represent. Theory
**14**(2010), 446-509 - DOI: https://doi.org/10.1090/S1088-4165-2010-00379-9
- Published electronically: July 7, 2010
- PDF | Request permission

## 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.## References

- Arkady Berenstein and David Kazhdan,
*Geometric and unipotent crystals*, Geom. Funct. Anal.**Special Volume**(2000), 188–236. GAFA 2000 (Tel Aviv, 1999). MR**1826254**, DOI 10.1007/978-3-0346-0422-2_{8} - V. G. Drinfel′d,
*Quantum groups*, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR**934283** - 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**, DOI 10.1007/BFb0101175 - G. Hatayama, A. Kuniba, M. Okado, T. Takagi, and Y. Yamada,
*Scattering rules in soliton cellular automata associated with crystal bases*, Recent developments in infinite-dimensional Lie algebras and conformal field theory (Charlottesville, VA, 2000) Contemp. Math., vol. 297, Amer. Math. Soc., Providence, RI, 2002, pp. 151–182. MR**1919817**, DOI 10.1090/conm/297/05097 - Michio Jimbo,
*A $q$-difference analogue of $U({\mathfrak {g}})$ and the Yang-Baxter equation*, Lett. Math. Phys.**10**(1985), no. 1, 63–69. MR**797001**, DOI 10.1007/BF00704588 - Victor G. Kac,
*Infinite-dimensional Lie algebras*, 3rd ed., Cambridge University Press, Cambridge, 1990. MR**1104219**, DOI 10.1017/CBO9780511626234 - 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** - 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**, DOI 10.1142/s0217751x92003896 - 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**, DOI 10.1215/S0012-7094-92-06821-9 - Masaki Kashiwara,
*On level-zero representations of quantized affine algebras*, Duke Math. J.**112**(2002), no. 1, 117–175. MR**1890649**, DOI 10.1215/S0012-9074-02-11214-9 - 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**, DOI 10.2977/prims/1145475409 - 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**, DOI 10.1006/jabr.1998.7503 - 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**, DOI 10.1090/S0002-9947-08-04341-9 - 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**, DOI 10.1155/S1073792803209041 - Toshiki Nakashima,
*Geometric crystals on Schubert varieties*, J. Geom. Phys.**53**(2005), no. 2, 197–225. MR**2110832**, DOI 10.1016/j.geomphys.2004.06.004 - Toshiki Nakashima,
*Geometric crystals on unipotent groups and generalized Young tableaux*, J. Algebra**293**(2005), no. 1, 65–88. MR**2173966**, DOI 10.1016/j.jalgebra.2005.07.025 - 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**, DOI 10.1073/pnas.80.6.1778 - 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**, DOI 10.1142/9789812810007_{0}014

## Bibliographic Information

**Masaki Kashiwara**- Affiliation: Research Institute for Mathematical Sciences, Kyoto University, Kitashiwakawa, Sakyo-ku, Kyoto 606, Japan
- MR Author ID: 98845
- 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
- Received by editor(s): September 2, 2008
- Published electronically: 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.)
- © Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory
**14**(2010), 446-509 - MSC (2010): Primary 17B37, 17B67; Secondary 22E65, 14M15
- DOI: https://doi.org/10.1090/S1088-4165-2010-00379-9
- MathSciNet review: 2661518