| Project/Area Number |
15540011
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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) |
NOMURA Kazumasa Tokyo Medical and Dental University, College of Liberal Arts and Sciences, Professor, 教養部, 教授 (40111645)
WADA Tomoyuki Tokyo University of Agriculture and Technology, Faculty of Engineering, Professor, 工学部, 教授 (30134795)
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| Project Period (FY) |
2003 – 2004
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| Project Status |
Completed (Fiscal Year 2004)
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| Budget Amount *help |
¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2004: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2003: ¥900,000 (Direct Cost: ¥900,000)
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| Keywords | Association schemes / Finite groups / Cartan matrices / Sharp characters |
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| Research Abstract |
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)
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