Structures and Representations of Association Schemes

a.代数与几何的联系：距离正则图的基子集的Terwilliger代数被定义并得到了广泛的研究。利用这一代数可以很容易地研究任意端点的Terwilliger代数的模，并给出了Delsarte理论与Terwilliger代数及其模之间的联系。平衡条件：对Q-多项式群结合方案进行分类。弱距离正则有向图：给出了弱距离正则有向图的定义和各种结构。部分作为一个联合工作与王凯顺，分类的情况下，程度2，并定期组的情况下，这是所谓的薄，完成。度3例，围度2例。II型基质：作为与细谷理惠的联合工作，它表明，如果一个II型矩阵是非本原的，它可以被分解为一个广义张量积的II型矩阵的较小的大小。这一结果有望应用于非本原自旋模型的分类。距离正则图的绝对有界猜想及相关主题：完成并发表了（s，2）阶距离正则图的分类。使用Terwilliger代数被证明是有用的。特别地，如果图是3-瘦的，则几何围长以11为界。

a. Connection between Algebra and Geometry: The Terwilliger algebra with respect to a base subset of a distance-regular grasph was defined and studied extensively. The use of this algebra enables to investigate modules of Terwilliger algebras of any endpoint much easily and gives a connection between the theory of Delsarte and Terwilliger algebra and its modules.b. Balanced Conditions: Q-polynomial group association schemes were classified.c. Weakly Distance-Regular Digraphs: The definition of weakly distance-regular digraphs together with various constructions were given. Partly as a joint work with Kaishun Wang, classifications of the case with degree 2, and regular group cases, which are called thin, were completed. Degree 3 cases and girth 2 cases are inprogress.d. Type II Matrices: As a joint work with Rie Hosoya, it was shown that if a type II matrix is imprimitive, it can be decomposed as a generalized tensor product of type II matrices of smaller size. The application of this result to a classification of imprimitive spin models is expected.e. Absolute Bound Conjecture of Distance-Regular Graphs and Related Topics: A classification of distance-regular graphs of order (s, 2) was completed and published. The use of Terwilliger algebra was proved to be useful. In particular, if the graph is 3-thin, geometric girths are bounded by 11.