Integrable Systems and Combinatorial Representation Theory

在课题研究期间，我们主要取得了以下成果。[仿射几何晶体]与M.Kashiwara和T.Nakashima合作，我们构造了与非例外仿射李代数相关的几何晶体。我们证实了这些几何晶体的超离散极限与已知的完美晶体的极限一致。此外，除了C型外，我们还得到了满足Yang-Baxter方程的双星映射的显式公式，称为热带R映射。[非例外类型的KR模的晶基的存在性]有一个猜想说，一个量子肯定代数的任何有限维表示，如果它的最高权为0级基权的整数倍(KR模)，则它有一个晶基。我们对所有非例外类型的仿射李代数解决了这个猜想。与A·席林合作，我们还证明了B^&lt；(1)&gt；_n，D^&lt；(1)&gt；_n，和A^&lt；(2)&gt；_&lt；2n-1&gt；与Schilling最近构造的组合晶体同构。与M.Kashiwara，K.C.Misra和D.Yamada合作，我们揭示了与例外仿射李代数D^&lt；(3)&gt；_4在任何水平上的完美晶体的晶体结构。

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.