Association schemes and characters of finite groups

在我们的研究项目中，我们得到了以下结果：1.我们得到了一个关于尖锐字符类型转换的一般定理。即证明了对于一个具有一定条件的L型尖锐字，通过对原有的尖锐字进行变形，可以构造出另一个不同类型的L尖锐字。由于L的基数是L基数的除数，因此我们得到了一个新的较小类型的尖锐特征。因此，利用该定理可以将L型锐字的判定简化为小型字的判定。我们正在研究上述结果在尖锐字符分类中的应用。(M.Kiyota)2.在一定条件下确定了结合式表示理论中自然出现的三对角线对(两个线性变换，每个线性变换相对于另一个特征基三对角线)。现在我们正在研究分类的三对角线对。(K.Nomura)3.我们在有限群中块的Cartan矩阵C上发现了一个更强的猜想，它蕴含了原来的猜想。也就是说，我们猜想，根据C的特征值的代数共轭类，C的初等因子被划分成类，使得相应的类具有(A)相等的基数，(B)相等的乘积，并且(C)极大初等因子的类对应于最大特征值的类。如果块满足下列条件之一，我们证明了这一猜想。(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。(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)