2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/167238By 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.53 pagesQuantum AlgebraRepresentation Theory17B37; 17B67text