Integrable Systems and Combinatorial Representation Theory

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During the period of research project, we mainly obtained the following results.1. [Affine geometric crystal]In collaboration with M. Kashiwara and T. Nakashima, we constructed geometric crystals associated to nonexceptional affine Lie algebras. We confirmed that the ultra-discrete limit of these geometric crystals coincide with the limit of previously known perfect crystals. Moreover, except type C, we obtained explicit formulas for birational maps, called tropical R maps, that satisfy the Yang-Baxter equation.2. [Existence of crystal bases of the KR modules for nonexceptional types]There was a conjecture saying that any finite-dimensional representation of a quantum affirm algebra that has an integer multiple of a level 0 fundamental weight as highest weight (KR module) has a crystal base. We solved this conjecture for all affine Lie algebras of nonexceptional types. In collaboration with A. Schilling, we also proved that the crystals of type B^<(1)>_n, D^<(1)>_n, and A^<(2)>_<2n-1> are isomorphic to the combinatorial crystals recently constructed by Schilling.3. [Construction of the coherent family of perfect crystals for exceptional types]In collaboration with M. Kashiwara, K.C. Misra and D. Yamada, we revealed the crystal structure of the perfect crystals associated to the exceptional affine lie algebra D^<(3)>_4 at any level.