结合方案和距离正则图是代数组合论的重要研究课题。Terwilliger代数是上述课题研究所发展起来的一种代数表示理论，是代数组合领域非常重要的研究工具之一。它包含了Bose-Mesner代数，蕴含更多的代数和结构信息，在对称结合方案研究中发挥强有力的作用。非对称的结合方案是比对称结合方案更一般的情形，近期受到越来越多研究者的关注。已有的工作主要从Bose-Mesner代数或者图结构角度对其进行刻画。.为了推动非对称结合方案的Terwilliger代数研究，本项目拟从非对称P-多项式结合方案（其组合对象是距离正则有向图）入手，研究距离正则有向图的Terwilliger代数表示，包括距离正则有向图的直径、局部图的特征值，进而刻画其不可约T-模的结构。申请者期望通过研究，得到处理非对称结合方案等此类问题的一般性方法，并应用于更一般的组合对象中，推动该领域的进一步发展。

The association schemes and the distance-regular graphs are important research topics in algebraic combinatorics. Terwilliger algebra is an algebraic representation theory developed by the above research topic and is one of the main research tools in the field of algebraic combinatorics. It contains Bose-Mesner algebra, and contains more algebra and structural information, plays a powerful role in the study of symmetric association schemes. The nonsymmetric association scheme is a more general case of the symmetric association scheme, and has attracted more and more attention recently. The existing work is mainly portrayed from the perspective of Bose-Mesner algebra or graph's structure..In order to promote the research of Terwilliger algebras of nonsymmetric association scheme, this project intends to start from the nonsymmetric P-polynomial association scheme, whose combinatorial object is the distance-regular digraph, study the representation theory of Terwilliger algebras of distance-regular digraph, including the diameter of distance-regular digraph, the eigenvalues of the local graphs, and then the structure of its irreducible T-module. It is expected that through our research, a general approach for dealing with such problems such as nonsymmetric association scheme can be obtained and applied to more general combinatorial objects, then contribute to the further development of algebraic combinatorics.