Association schemes and characters of finite groups

关于我们的研究项目，我们得到了以下结果。我们得到了尖锐字符类型变换的一般定理。即证明了对于具有一定条件的L型尖锐特征，通过对原尖锐特征的变形，可以构造出另一个不同L型尖锐特征。由于L'的基数是L的基数的约数，我们得到了一个新的类型更小的尖锐字符。因此，我们可以利用该定理将L型尖锐字符的确定简化为较小类型尖锐字符的确定。我们正在研究将上述结果应用于尖锐字的分类。(M.Kiyota) 2。在关联方案的表示理论中自然出现的三对角对是在一定条件下确定的。现在我们正在研究三对角线对的分类。(K.Nomura) 3。在有限群中块的Cartan矩阵C上，我们得到了一个更强的猜想，它蕴涵了原来的猜想。即，我们推测C的初等除数根据C的特征值的代数共轭类划分为类，使得相应的类具有(a)相等的基数，(b)相等的乘积，并且(C)最大初等除数的类对应于最大特征值的类。如果区块满足以下条件之一，我们就证明了这个猜想。(1)驯服块，(2)1(B)<=5的循环块，(3)具有某种特殊Brauer树的循环块。我们现在正在研究可解群的猜想。（清太和田）

We obtained the following results on our research project.1.We had a general theorem on the transformation for types of sharp characters. Namely we proved that for a sharp character of type L with a certain condition, we can construct another sharp one of different type L' by deforming the original one. Since the cardinality of L' is a divisor of that of L, we obtain a new sharp character with smaller type. Hence we can reduce the determination of sharp characters of type L to that of smaller type by using the theorem. We are now studying the application of the above result to the classification of sharp characters. (M.Kiyota)2.Tridiagonal pairs (two linear transformations each tridiagonal with respect to an eigenbasis of the other), which appeared naturally in the representation theory of association schemes, are determined under certain conditions. Now we are studying the tridiagonal pairs toward the classification. (K.Nomura)3.We have found a stronger conjecture, which implies the original ones, on the Cartan matrix C of a block in a finite group. Namely, we conjectured that the elementary divisors of C are partitioned into classes according to the algebraically conjugate classes of the eigenvalues of C such that the corresponding classes have (a) equal cardinality, (b) equal product, and moreover (c) the class of maximal elementary divisor corresponds to that of maximal eigenvalue. We proved this conjecture if the block satisfies the one of the following conditions. (1)tame blocks, (2)cyclic blocks with 1(B)<=5, (3)cyclic blocks with some special Brauer tree. We are now studying the conjecture for solvable groups. (M.Kiyota and T.Wada)