Association schemes and characters of finite groups

在课题研究中，我们得到了如下结果：1.得到了一类尖锐特征标的变换的一般定理。即证明了对于一个L型的尖特征标，在一定条件下，通过变形原L型的尖特征标，可以构造出另一个不同L'型的尖特征标。由于L'的基数是L的基数的一个约数，我们得到了一个新的具有较小类型的尖锐特征标。因此，利用该定理可以将L型尖特征标的判定归结为较小型尖特征标的判定。我们现在正在研究将上述结果应用于尖字符的分类。（M.Kiyota）2.三对角对（两个线性变换，每一个关于另一个的特征基都是三对角的）在一定条件下被确定，它自然地出现在结合概型的表示论中。现在我们正在研究三对角对的分类。（K.Nomura）3.我们发现了关于有限群中块的Cartan矩阵C的一个更强的猜想，它蕴涵了原来的猜想。也就是说，我们证明了C的初等因子按C的特征值的代数共轭类划分成类，使得相应的类具有（a）相等的基数，（B）相等的乘积，并且（c）最大初等因子类对应于最大特征值类.我们证明了这个猜想，如果块满足以下条件之一。（2）1（B）≤ 5的循环块，（3）具有特殊Brauer树的循环块。我们现在正在研究可解群的猜想。（M.Kiyota和T.Wada）

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)