Representation of finite groups and association schemes

我们在研究项目上取得了以下成果。利用特征标理论，我们给出了具有一定不可约特征标的有限群的一个群理论刻画，目前正在研究具有上述条件的所有有限群的判定问题。(M.Kiyota)2.我们研究了自旋模型构造的结合方案的性质，对于每个自旋模型都是在一个结合方案(K.Nomura)上构造的。3.对于每个正定积分二次型，我们得到了一个关于块中的特征标数和Cartan整数的不等式。如果正定积分二次型为A型，则该不等式已被称为和田不等式。(T.Wada)4.我们有两个猜想：如果Cartan矩阵的最大特征值是整数，那么它是亏群的阶；如果Cartan矩阵的最大特征值是亏群的阶，那么特征值和初等因子是重合的。如果亏群满足特殊条件，我们证明了猜想。(M.Kiyota，M.Murai和T.Wada)5.如果素数p与3模4同余，且p-块的Brauer特征标数为2，则证明了上述猜想。我们研究了Cartan整数只有两个值的情况。(M，Kiyota)6.我们有一个更强的猜想，它包含了文4中的猜想。我们证明了如果一个块B与某种特殊的Brauer树是循环的，我们就证明了这个更强的猜想。我们正在研究一般的循环区块B(M.Kiyota和T.Wada)。

We obtained the following results on our research project.1. We had a group theoretical characterization on finite groups with a certain irreducibie characters by using the theory of sharp characters, We are now studying the determination of all finite groups with the above condition. (M. Kiyota)2. We studied the property of association schemes on which spin models constructed, for each spin models is constructed on an association scheme, (K. Nomura)3. For every positive definite integral quadratic form, we obtained an inequality on the number of characters in a block and the Cartan integers. If the positive definite integral quadratic form is type A, the inequality is already known as Wada's inequality. (T. Wada)4. We had two conjectures that if the maximal eigenvalue of Cartan matrix is an integer then it is the order of the defect group, and that if the maximal eigenvalue of Cartan matrix is the order of the defect group then the eigenvalues and the elementary divisors are coincide. We proved the conjectures if the defect group satisfies special conditions. (M. Kiyota, M. Murai and T. Wada)5. If a prime p is congruent to 3 modulo 4 and the number of Brauer characters in a p-block is 2, then we proved the above conjectures. We studied them if the Cartan integers have just two values. (M, Kiyota)6.We had a stronger conjecture which implies the conjectures in 4. We proved the stronger conjecture if a block B is cyclic with some special Brauer tree. We are studying it for general cyclic block B. (M. Kiyota and T. Wada)