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

(1)几何晶体由Berenstein和Kazhdan公理化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。利用该几何晶体的矩阵实现，构造了与该几何晶体作用的乘积（热带R）交换的双分映射，并证明其满足Yang-Baxter方程。我们相信，一个几何晶体存在于对应于李代数的Dynkin图的每一个顶点上。对于D型，有n个顶点，因此预计有n个不同的几何晶体。在上一学期，我们通过数学计算了k=2的几何晶体与京都大学RIMS的Masaki Kashiwara合作。数据是巨大的，没有办法打印出来。然而，如何用一种有意义的方式写出它，除了扩展到k大于2的情况外，这是一个未来的问题。（ii）给出了异常仿射李代数D_4^(3)的一系列有限晶体的坐标表示，并明确地得到了零作用。此外，我们构造了一个与这一系列晶体相对应的元胞自动机，并确定了系统中出现的孤子的内部自由度和两个孤子的散射规律。我们将超离散可积系统的一个重要例子——盒球系统推广到只有一个反射端的情况。与通常的盒球系统类似，它具有无限的通勤时间演化族和与每次演化相关的守恒量。我们还定义了孤子态，并从晶体组合学的角度描述了单孤子的反射规律和双孤子的散射规律。

(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.