2001 Fiscal Year Final Research Report Summary
Representation of finite groups and association schemes
| Project/Area Number |
12640012
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Tokyo Medical and Dental University |
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Principal Investigator |
KIYOTA Masao Tokyo Medical and Dental University, College of Liberal Arts and Sciences,Professor, 教養部, 教授 (50214911)
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| Co-Investigator(Kenkyū-buntansha) |
WADA Tomoyuki Tokyo University of Agriculture and Technology Department of Mathematics, Professor, 工学部, 教授 (30134795)
NOMURA Kazumasa Tokyo Medical and Dental University, College of Liberal Arts and Sciences,Professor, 教養部, 教授 (40111645)
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| Project Period (FY) |
2000 – 2001
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| Keywords | finite group / Cartan matrix / association scheme / sharp character |
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| Research Abstract |
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)
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