Structures and Representations of Association Schemes

a.代数与几何的联系：定义了关于距离正则图基子集的Terwilliger代数，并对其进行了广泛的研究。该代数的使用使研究任意端点的Terwilliger代数的模变得容易，并给出了Delsarte代数理论与Terwilliger代数及其模之间的联系。平衡条件：分类了q -多项式群关联方案。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构造。部分作为与王开顺的合作，完成了二级案例的分类，以及被称为薄的规则群案例。3度病例和2周围病例进展中。II型矩阵：作为与Rie Hosoya的联合工作，证明了如果II型矩阵是非基元的，它可以分解为较小尺寸的II型矩阵的广义张量积。期望将这一结果应用于非基元自旋模型的分类。距离正则图的绝对界猜想及相关主题：完成并发表了（s, 2）阶距离正则图的分类。特威利格代数的使用被证明是有用的。特别是，如果图形是3-thin，则几何周长以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.