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
DOI:
https://doi.org/10.1090/S1088-4165-2010-00379-9
Published electronically:
July 7, 2010
MathSciNet review:
2661518
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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.
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Additional 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
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.
