Tropical R maps and Affine Geometric Crystals

Details Tropical R maps and Affine Geometric Crystals Tropical R maps and Affine Geometric Crystals

2008-08-18

arxiv.org/abs/0808.2411v1

Mathematics

Quantum Algebra

53 pages

Scientific paper

By modifying the method in [KNO], 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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