Combinatorial Study of Crystal Bases and its Application to Discrete Integrable Systems

(I)由Berenstein和Kazhdan公理地引入的几何晶体，对于Kac-Moody李代数的D^(1)_n型是显式构造的。利用几何晶体的矩阵实现，我们构造了在几何晶体作用下的乘积(热带R)上的二元映射，并证明了它满足Yang-Baxter方程。人们认为，在对应于李代数的动态图的每个顶点上都存在一个几何晶体。对于类型D，有n个顶点，因此预计会有n个不同的几何晶体。在上一节课中，我们和京都大学Rims的Masaki Kashiwara合作，用数学计算了k=2的几何晶体。数据非常庞大，无法打印出来。(Ii)给出了一个例外的痤疮李代数的晶体和孤子元胞自动机，给出了一个例外仿射李代数D_4^(3)的一系列有限晶体的坐标表示，并明确地得到了它的零作用量。此外，我们还构造了与这一系列晶体相对应的元胞自动机，并确定了系统中孤子的内部自由度和两个孤子的散射规则。(Iii)带反射端的Box-ball系统我们将超离散可积系统的一个重要例子Box-ball系统推广到有一个反射端的情况。类似于通常的箱球系统，它有一个通勤时间演化的无穷族，以及与每个时间演化相关的守恒量。我们还定义了孤子态，并用晶体组合学描述了一个孤子的反射规律和两个孤子的散射规律。

(i) Geometric crystalGeometric crystal, indtoduced axiomatically by Berenstein and Kazhdan, was consttructed explicitly for type D^(1)_n of Kac-Moody Lie algebras. By using a matrix realization of this geometric crystal, we constructed a birational mapping on the product (tropical R) commuting with the action of the geometric crystal and also,showed that it satisfies the Yang-Baxter equation. It is believed that a geometric crystal exists arrogated to each vertex of the Dynkin diagram corresponding to the Lie algebra. For type D, there are n vertices, so it is expected that there are n distinct geometric crystals. During the last yeah we calculated the geometric crystal for k=2 by Mathematics in collaboration with Masaki Kashiwara at RIMS, Kyoto University. Data is huge and no way to print it out. However to write it down in a meaningful manner is, besides with the extension to the case when k is greater than 2, becomes a future problem.(ii)Crystal and soliton cellular automaton associated to an exceptional acne Lie algebraCoordinate representation of a series of finite crystals for an exceptional affine Lie algebra D_4^(3) is given and the zero action is explicitly obtained. Moreover we constructed a cell automaton corresponding to this series of crystals and determined the internal degree of freedom of the solitons appearing in the system and the scattering rule of two solitons.(iii)Box-ball system with reflecting endWe have extended the box-ball system, an important example of ultra discrete integrable systems, to the case with one reflecting end. Similar to the usual box-ball system, it has an infinite family of commuting time evolutions and conserved quantities associated to each time evolution. We also defined soliton states and described the reflection rule of one soliton and the scattering rule of two solitons in terms of combinatorics of crystals.