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Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

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Tropical R maps and affine geometric crystals
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by Masaki Kashiwara, Toshiki Nakashima and Masato Okado
Represent. Theory 14 (2010), 446-509
Published electronically: July 7, 2010


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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Bibliographic Information
  • Masaki Kashiwara
  • Affiliation: Research Institute for Mathematical Sciences, Kyoto University, Kitashiwakawa, Sakyo-ku, Kyoto 606, Japan
  • MR Author ID: 98845
  • Email:
  • Toshiki Nakashima
  • Affiliation: Department of Mathematics, Sophia University, Kioicho 7-1, Chiyoda-ku, Tokyo 102-8554, Japan
  • Email:
  • Masato Okado
  • Affiliation: Department of Mathematical Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan
  • Email:
  • 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:
  • MathSciNet review: 2661518