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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
Published electronically: July 7, 2010
MathSciNet review: 2661518
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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.

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Additional Information

Masaki Kashiwara
Affiliation: Research Institute for Mathematical Sciences, Kyoto University, Kitashiwakawa, Sakyo-ku, Kyoto 606, Japan

Toshiki Nakashima
Affiliation: Department of Mathematics, Sophia University, Kioicho 7-1, Chiyoda-ku, Tokyo 102-8554, Japan

Masato Okado
Affiliation: Department of Mathematical Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan

Keywords: Prehomogeneous geometric crystal, perfect crystal, folding, tropical $R$ map, ultra-discretization
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.)
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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