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.

在研究项目期间，我们主要获得以下结果1。 [仿射几何晶体]与M. Kashiwara和T. Nakashima合作，我们构建了与非感受的仿射相关的几何晶体。我们证实了这些几何晶体的超滴限与先前已知的完美晶体的极限。此外，除了C型外，我们还获得了满足Yang-Baxter方程的Birational图的明确公式，称为热带R地图。2。 [对于非敏感类型的KR模块的晶体碱基的存在]有一个猜想说，量子肯定的代数的任何有限维度表示，该代数具有0级基本权重的整数倍数作为最高权重（KR模块）具有晶体碱基。我们解决了所有非见面类型的仿射代数的猜想。与A. Schilling合作，我们还证明了B^<（1）> _ n，d^<（1）> _ n的晶体和A^<（2）> _ <2N-1>是与Schilling最近构建的组合晶体的同构。3。 [为特殊类型的完美晶体的连贯家族的建设]与K.C. M. Kashiwara合作Misra和D. Yamada，我们揭示了与特殊的Offine Like Like d^<（3）> _ 4相关的完美晶体的晶体结构。

项目成果

Combinatorial aspect of integrable systems

可积系统的组合方面

• 发表时间: 2007
• 作者: A.Kuniba;M.Okado
• 通讯作者: M.Okado

On the crystal bases of the KR module of type D^<(1)>_n

在 D^<(1)>_n 类型 KR 模块的晶体底座上

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

Affine Geometric Crystals and Limit of Perfect Crystals

仿射几何晶体与完美晶体的极限

• 发表时间: 2008
• 期刊: Transactions in American Mathematical Society 30
• 作者: 中島俊樹;(M. Kashiwara;M. Okado);Takumi Noda;河田成人;中島俊樹(with M. Kashiwara M. Okado)
• 通讯作者: 中島俊樹(with M. Kashiwara M. Okado)

Crystal interpretation of Kerov-Kirillov-Reshetikhin bijection

Kerov-Kirillov-Reshetikhin 双射的晶体解释

• 发表时间: 2006
• 期刊: Nucl. Phys. B 740[PM]
• 作者: A.Kuniba;M.Okado;R.Sakamoto;T.Takagi;Y.Yamada
• 通讯作者: Y.Yamada