Representation of finite groups and association schemes

有限群和关联方案的表示

• 批准号: 12640012
• 负责人: KIYOTA Masao
• 金额: $ 1.22万
• 依托单位: Tokyo Medical and Dental University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640012/
• 关键词: finite group; Cartan matrix; association scheme; sharp character

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)

本课题取得了以下成果：1.利用sharp特征标理论，我们对具有某种不可约特征标的有限群进行了群论刻画，我们现在研究的是所有具有上述条件的有限群的判定问题。（M. Kiyota）2.本文研究了自旋模型建立在其上的缔合方案的性质，对于每个自旋模型都建立在一个缔合方案上，（K。野村证券（Nomura）3.对于正定整二次型，得到了一个关于块特征标个数与Cartan整数的不等式。如果正定积分二次型是A型，则该不等式已经被称为Wada不等式。（T. Wada）4.给出了Cartan矩阵的最大特征值为整数时为亏群的阶和最大特征值为亏群的阶时特征值与初等因子一致的两个定理。证明了亏群满足特殊条件时的性质。（M. Kiyota，M. Murai和T. Wada）5.若素数p与3模4全等，且p-块中Brauer特征标的个数为2，则证明了上述定理。我们研究了当嘉当整数只有两个值时的情形。（M，Kiyota）6.我们有一个更强的猜想，它包含了4中的猜想。我们证明了一个更强的猜想，如果一个块B是循环的，有一些特殊的Brauer树。我们正在研究一般的环状嵌段B。（M. Kiyota和T.和田）

项目成果

清田正夫, 村井正文, 和田倶幸: "Rationdity of eigenvalues of Cartan matrces in finite groups"Journal of Algebra. (to appear).

Masao Kiyota、Masafumi Murai、Yasuyuki Wada：《有限群中嘉当矩阵特征值的合理性》《代数杂志》（待发表）。

Masao Kiyota, Masafumi Murai and Tomoyuki Wada: "Rationality of eigenvalues of Cartan matrices in finite groups"J. Algebra. (to appear).

Masao Kiyota、Masafumi Murai 和 Tomoyuki Wada：“有限群中嘉当矩阵特征值的有理性”J.

和田倶幸: "The Cartan Matrix of a certain class of finite solvable groups"Osaka Journal of Mathematics. 37. 1-14 (2000)

和田 Yoshiyuki Wada：“某类有限可解群的嘉当矩阵”《大阪数学杂志》37. 1-14 (2000)。

Kazumasa Nomura and Brian Curtin: "Spin models and strongly hyper-self-dual Bose-Mesner algebras"J. Algebraic Combinatorics. 13. 173-186 (2000)

Kazumasa Nomura 和 Brian Curtin：“自旋模型和强超自对偶 Bose-Mesner 代数”J.

清田正夫, 鈴木寛: "Character products and Q-pdynomial group association schemes"Journal of Algebra. 226. 533-546 (2000)

Masao Kiyota，Hiroshi Suzuki：“字符乘积和 Q-pdynomial 群关联方案”代数杂志 226. 533-546 (2000)。