Association schemes and characters of finite groups

我们在研究项目上取得了以下成果。给出了群结合方案为q-多项式的有限群的完全分类。我们注意到，所有这些群体都有鲜明的个性。(M.Kiyota和H.Suzuki)2.我们得到了距离正则图(K.Nomura)上自旋模型的一般公式(含两个参数)。3.我们得到了块中的特征标数与Cartan整数之间的关系。利用这个关系，我们得到了Cartan矩阵的最大特征值的下界。(T.Wada)4.对于一族p-可解群，我们完全确定了它们的Cartan矩阵。(T.Wada)5.在p-可解群中，我们得到了Cartan矩阵的特征值与初等因子重合的条件。(M.Kiyota和T.Wada)6.我们有了新的概念，p-良群和良模。我们有关于p-好群和好模的结构的猜想，我们研究(现在正在研究)它们。(M.Kiyota)

We obtained the following results on our research project.1. We had the complete classification of finite groups whose group association schemes are Q-polynomial. We note that all these groups have sharp characters. (M. Kiyota and H. Suzuki)2. We obtained a general formulas (with two parameters) for spin models on distance-regular graphs (K. Nomura)3. We obtained a relation between the number of characters in a block and the Cartan integers. Using the relation, we get the lower bound of the maximal eigenvalue of the Cartan matrix. (T. Wada)4. For a certain family of p-solvable groups, we had the complete determination of their Cartan matrices. (T. Wada)5. In p-solvable groups, we obtained the condition under which the eigenvalues and the elementary divisors of Cartan matrix are coincide. (M. Kiyota and T. Wada)6. We had a new concepts, p-good groups and good modules. We had conjectures on the structures of p-good groups and good modules, and we studied (are now studying) them. (M. Kiyota)