Integrable Systems and Combinatorial Representation Theory

可积系统和组合表示理论

• 批准号: 18540030
• 负责人: OKADO Masato
• 金额: $ 2.46万
• 依托单位: Osaka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2006
• 资助国家: 日本
• 起止时间: 2006 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18540030/
• 关键词: Integrable system; Quantum group; Yang-Baxter eouation; 量了群; セルオートマトン; 可積文系

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.

在课题研究期间，我们主要取得了以下成果： 1． [仿射几何晶体]与M. Kashiwara和T. Nakashima合作，我们构建了与非例外仿射李代数相关的几何晶体。我们确认这些几何晶体的超离散极限与先前已知的完美晶体的极限一致。此外，除C型外，我们还得到了满足Yang-Baxter方程的双有理图的显式公式，称为热带R图。 2． [非例外类型 KR 模块的晶体基的存在]有一个猜想说，任何以 0 级基本权重作为最高权重（KR 模块）的整数倍的量子肯定代数的有限维表示都具有晶体根据。我们针对所有非例外类型的仿射李代数解决了这个猜想。与 A. Schilling 合作，我们还证明了 B^<(1)>_n、D^<(1)>_n 和 A^<(2)>_<2n-1> 类型的晶体同构于Schilling最近构建的组合晶体。3． [为特殊类型构建连贯的完美晶体家族]与 M. Kashiwara, K.C. 合作Misra 和 D. Yamada，我们揭示了与任何级别的特殊仿射李代数 D^<(3)>_4 相关的完美晶体的晶体结构。

项目成果

Crystal interpretation of Kerov-Kirillov-Reshetikhin bijection

Kerov-Kirillov-Reshetikhin 双射的晶体解释

• 发表时间: 2006
• 作者: M.Okado; et al.
• 通讯作者: et al.

Existence of Kirillov-Reshetikhin crystals for nonexceptional types

非特殊类型的基里洛夫-列谢蒂欣晶体的存在

• 发表时间: 2008
• 作者: M.; Okado; et. al.
• 通讯作者: et. al.

Combinatorial Aspect of Integrable Systems

可积系统的组合方面

• 发表时间: 2007
• 作者: M.Okado; et al.
• 通讯作者: et al.

Perfect crystals for U_q (D^(3)_4)

U_q (D^(3)_4) 的完美晶体

• 发表时间: 2007
• 作者: M. Okado; et. al.
• 通讯作者: et. al.

Crystal interpretation of Kerov-Kirillov-Reshetikhin bijection

Kerov-Kirillov-Reshetikhin 双射的晶体解释

• 发表时间: 2006
• 作者: A.; Kuniba; et. al.
• 通讯作者: et. al.