2007 Fiscal Year Final Research Report Summary
Study on eigenvalues and elementary divisors of Cartan matrices in finite groups
| Project/Area Number |
17540014
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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 and Technology |
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Principal Investigator |
WADA Tomoyuki Tokyo University of Agriculture and Technology, Institute of Symbiotic Science and Technology, Professor (30134795)
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| Co-Investigator(Kenkyū-buntansha) |
YAMAGATA Kunio Tokyo University of Agricultute and Technology, Institute of Symbiotic Science and Technology, Professor (60015849)
KIYOTA Masao Tokyo Medical and Dental University, General Education, Professor (50214911)
ENOMOTO Yoko (ENOMOTO, Usami) Ochanomizu Woman's University, Science, Professor (90151993)
FUKUSHIMA Hiroshi Gunma University, Education, Professor (30125869)
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| Project Period (FY) |
2005 – 2007
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| Keywords | finite group / modular representation / block / Cartan matrix / Frobenius-Perron eigenvalue / elementary divisor / Morita equivalent / defect group |
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| Research Abstract |
Let G be a finite group and F be an algebraically closed field of characteristic p>0. Let B be a block of the group algebra FG with defect group D. Let C be the Cartan matrix of B and let ρ be the Frobenius-Perron eigenvalue of C. It is the purpose of this research to investigate the structure of G or B when ρ is an integer. We found from many examples that ρ is an integer if and only if B and its Brauer correspondent b are Morita equivalent. But there is a counter example to this statement when D is not abelian. So, if D is abelian, we conjectured that ρ is an integer if and only if B and b are Morita equivalent. We proved that the conjecture is true in the following.
1 When D is cyclic, or p=2 and D is the four group, the conjecture is true.
2 When D is elementary abelian 3-group of order 9 and B is the principal 3-block, the conjecture is true.
3 When G has an abelian 2-Sylow subgroup and B is the principal 2-block, the conjecture is true.
Since Morita equivalence means derived equivalence, our concern that ρ is an integer is in turn deeply related to Broue's abelian defect group conjecture.
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