Tropical R maps and affine geometric crystals

Kashiwara, Masaki; Nakashima, Toshiki; Okado, Masato

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

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

DOI: https://doi.org/10.1090/S1088-4165-2010-00379-9

Published electronically: July 7, 2010

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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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