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Piecewise Linear Representation Theory of Quantum Groups and Geometric Crystals

量子群和几何晶体的分段线性表示理论

基本信息

批准号：
16540039
负责人：
NAKASHIMA Toshiki
金额：
$ 1.79万
依托单位：
Sophia University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540039/
关键词：
Quantum Grouos Crystal Bses Geometric Crystals Tropicalization Ultra-Discretizations Algebraic Groups Hecke Akgebras Markov Trace 完全結晶 tropical R Drinfeld 多項式マルコフトレース 1の冪根 Markov trace Hecke環

项目摘要

The theory of geometric crystal is obtained as an analogus theory on algebraic varieties to crystal bases by considering certain group actions on the vatirety, which turns out to be an analogus operator to the crystal operators. It is well-known that by the tropicalization / ultra-discretization procedure we obtain crystals from geometric crystals.The purpose of the research is to construct a geometric crystal structure on various algebraic varieties. In fact, I have succeeded to construct the geometric crystal structure on the Schubert varieties associated with Kac-Moody groups. Furthermore, I have constructed the geometric crystal structures on the affine Schubert varieties which is obtained from the translations of the extended Weyl groups. This geometric crystal possesses the following remarkable properties : it has a natural positive structure and the associated crystal is isomorphic to the so-called the limit of perfect crystals. We obtained the tropical R maps on the product of … More these geometric crystals, which is an analogus object to R-matrices. Perfect crystals are crucial objects in the study of affine type crystals and they play a central role in the theory of solvable lattice models in mathematical physics and the theory of Kirillov-Reshetikhin modules.As for the representation of affine quantum groups t roots of 1, we obtain the sufficient and necessary condition for that two evaluation representations are isomorphic to each other.The Markov trace is one of important topological invariants. From the view point of topology and representation theory, the Markov trace turns out to be an rather interesting object which we should study. In order to treat the Markov trace in the general framework, Gomi defined the Markov property and present the way to construct the Markov trace by the unified method.Shinoda succeeded in obtaining certain interesting results on relations between zeta functions and the Gel'fand Graev representations of finite reductive groups by performing the explicit calculations. Less

几何晶体理论是代数簇与晶体基的类比理论，通过考虑代数簇上的某些群作用，得到代数簇与晶体基的类比算子。众所周知，通过热带化/超离散化的方法，我们可以从几何晶体中得到晶体，本研究的目的是在各种代数簇上构造几何晶体结构。事实上，我已经成功地构造了与Kac-Moody群相关的Schubert簇的几何晶体结构。此外，我构造了仿射Schubert簇上的几何晶体结构，这些簇是由扩展Weyl群的平移得到的。这种几何晶体具有以下显着的性质：它具有自然的正结构，并且相关的晶体同构于所谓的完美晶体的极限。我们得到的热带R地图的产品 ...更多信息这些几何晶体，这是一个类似的对象R-矩阵。理想晶体是仿射型晶体研究的重要对象，在数学物理中的可解格模型理论和Kirillov-Reshetikhin模理论中起着核心作用。我们得到了两个赋值表示同构的充要条件，马尔可夫迹是重要的拓扑表示之一，不变量从拓扑学和表示论的观点来看，马尔可夫迹是一个值得研究的有趣对象。为了处理马尔可夫迹的一般框架，五味定义的马尔可夫性质，并提出了如何构建马尔可夫迹的统一方法。筱田成功地获得了一些有趣的结果之间的关系zeta函数和Gelfand Graev表示有限约化群进行明确的计算。少

项目成果

期刊论文数量（30）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

The Markov traces and the Fourier transforms

马尔可夫迹和傅里叶变换

DOI：
发表时间：
2006
期刊：
Journal of Algebra vol.303
影响因子：
0
作者：
末信郁也;久保富士男;Fujio Kubo;Fujio Kubo;Fujio Kubo;Masao Tsuzuki;Ken-ichi Shinoda;Toshiki Nakashima;T. Nakashima;K. Shinoda;Y. Gomi;Yasushi Gomi
通讯作者：
Yasushi Gomi

Geometric Crystals on Schubert Varieties

舒伯特品种中的几何晶体