Integrable Systems and Combinatorial Representation Theory

在课题研究期间，主要取得了以下成果. [仿射几何晶体]与M. Kashiwara和T. Nakashima，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以前已知的完美晶体的极限相一致。此外，除了C型外，我们还得到了满足Yang-Baxter方程的双有理映射（称为热带R映射）的显式公式. [非例外类型的KR模块的晶体基的存在性]有一个猜想说，任何量子肯定代数的有限维表示，具有0级基本权重的整数倍作为最高权重（KR模块）具有晶体基。我们解决了这个猜想的所有仿射李代数的非例外类型。与A合作。Schilling等提出的方法，我们还证明了B^&lt;（1）&gt;_n，D^&lt;（1）&gt;_n和A^&lt;（2）&gt;_n型晶体<2n-1>与Schilling最近构造的组合晶体同构. [构建特殊类型的完美晶体的相干家族]与M. Kashiwara，K.C. Misra和D. Yamada等人的工作揭示了与特殊仿射李代数D^&lt;（3）&gt;_4在任何水平上相关联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.