Piecewise Linear Representation Theory of Quantum Groups and Geometric Crystals

几何晶体理论是通过考虑某些群作用而得到的关于晶基的代数簇的类比理论，它实际上是晶体算符的类比算符。众所周知，通过热带化/超离散过程，我们可以从几何晶体中获得晶体，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，我已经成功地在与Kac-Moody群相关的Schubert簇上构造了几何晶体结构。此外，我还构造了由扩展Weyl群的平移得到的仿射Schubert簇上的几何晶体结构。这种几何晶体具有以下显着的性质：它具有自然的正结构，缔合的晶体与所谓的完美晶体的极限同构。得到了…乘积上的热带R图更多的是这些几何晶体，它是R-矩阵的类似对象。理想晶体是仿射型晶体研究中的重要对象，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。对于1的仿射量子群t根的表示，我们得到了两个赋值表示彼此同构的充要条件。马尔可夫迹是重要的拓扑不变量之一。从拓扑学和表示论的角度来看，马尔可夫迹是我们应该研究的一个相当有趣的对象。为了在一般框架下处理马氏迹，Gomi定义了马氏迹，并给出了用统一方法构造马氏迹的方法。Shinoda通过显式计算成功地得到了有限约群的Zeta函数与Gel‘fand Graev表示之间关系的一些有趣的结果。较少

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