Tropical R maps and Affine Geometric Crystals
Masaki Kashiwara, Toshiki Nakashima, Masato Okado
TL;DR
This paper constructs affine geometric crystals and explicit tropical R maps, introduces prehomogeneous crystals, and discusses the conditions for their connectedness and uniqueness.
Contribution
It presents a method to realize affine geometric crystals in fundamental representations and characterizes tropical R maps and prehomogeneity conditions.
Findings
Explicit description of tropical R maps for affine geometric crystals
Introduction of prehomogeneous geometric crystals and their properties
Connection between connectedness of ultra-discretized crystals and prehomogeneity
Abstract
By modifying the method in [KNO], certain affine geometric crystals are realized in affinization of the fundamental representation 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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Taxonomy
TopicsNonlinear Waves and Solitons · Algebraic structures and combinatorial models · Black Holes and Theoretical Physics