1999 Fiscal Year Final Research Report Summary
A STUDY OF THE CARTAN MATRICES OF FINITE GROUPS
| Project/Area Number |
09640015
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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 University of Agriculture & Technology |
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Principal Investigator |
WADA Tomoyuki TOKYO UNIVERSITY OF AGRICULTURE AND TECHNOLOGY (TUAT), FACULTY OF TECHNOLOGY, PROFESSOR, 工学部, 教授 (30134795)
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| Co-Investigator(Kenkyū-buntansha) |
MAEDA Hironobu TUAT TECHNOLOGY, ASSOCIATE PROFESSOR, 工学部, 助教授 (50173711)
MASHIMO Katsuya TUAT TECHNOLOGY, ASSOCIATE PROFESSOR, 工学部, 助教授 (50157187)
YAMAGATA Kunio TUAT TECHNOLOGY, PROFESSOR, 工学部, 教授 (60015849)
KIYOTA Masao TOKYO MEDICAL AND DENTAL UNIVERSITY, GENERAL EDUCATION, PROFESSOR, 教養部, 教授 (50214911)
TASHIRO Yoshiaki TUAT TECHNOLOGY, PROFESSOR, 工学部, 教授 (00014928)
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| Project Period (FY) |
1997 – 1999
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| Keywords | FINITE GROUP / REPRESENTATION / CARTAN MATRIX / BLOCK / PERRON-FROBENIUS EIGENVALUE / IRREDUCIBLE CHARACTER / BRAUER CHARACTER / DEFECT GROUP |
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| Research Abstract |
Let G be a finite group, FG the group algebra of G over an algebraically closed field F of characteristic p > 0, and B a block of FG. Let CィイD2BィエD2 be the Cartan matrix of B and p(B) the Perron-Frobenius eigenvalue (i.e. the largest one) of CィイD2BィエD2. Let k(B), l(B) be the number of ordinary and modular (i.e. over F) irreducible characters, respectively.
We have found k(B) 【less than or equal】 p(B)l(B) holds for any block B of G in [1], further we posed a conjecture that k(B) 【less than or equal】 p(B) for any p-solvable groups. This is a stronger assertion than Brauer's famous k(B)-conjecture for p-solvable groups. In [2] we completed to calculate the Cartan matrix of a certain class of finite solvable groups, which have many 0 entries. We also verified that k(B) 【less than or equal】 p(B) in this case. In [3] we investigated on the fundamental question when eigenvalus and elementary divisors of CィイD2BィエD2 coincide. We have had a striking conjecture that p(B) is rational if and only if B is Morita equivalent to the Brauer correspondent b of NィイD2GィエD2(D). In [4] we newly obtained a definition of P-good module and p-good group which is related to eigenvalues of CィイD2BィエD2.
[1] T. Wada, A lower bound of the Perron-Frobenius eigenvalue of the Cartan matrix for finite groups. Arch. Math. 73 (1999), 407-413. [2] T. Wada, The Cartan matrix of a certain class of finite solvable groups. (accepted to Osaka Jour. Math.) [3] M. Kiyota, M. Murai and T. Wada, Rationality of eigenvalues of the Cartan matrices of finite groups. (in preperation) [4] A. Hanaki, M. kiyota, M. Murai and T. Wada, P-good modules and p-good groups. (in preperation)
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