2001 Fiscal Year Final Research Report Summary
Application of Shintani descent to the perfect isometry problem and Dade's conjecture
| Project/Area Number |
11440008
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | Osaka University |
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Principal Investigator |
UNO Katsuhiro Osaka Univ., Dept.of Math., Asso. Prof., 大学院・理学研究科, 助教授 (70176717)
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| Co-Investigator(Kenkyū-buntansha) |
WATANABE Atumi Kumamoto Univ., Dept.of Math., Asso. Prof., 理学部, 助教授 (90040120)
NAGATOMO Kiyokazu Osaka Univ., Dept.of Math., Asso. Prof., 大学院・理学研究科, 助教授 (90172543)
KAWANAKA Noriaki Osaka Univ., Dept.of Math., Prof., 大学院・理学研究科, 教授 (10028219)
OKUYAMA Tetsuro Hokkaido Univ. of Edu., Dept. of Math., Prof., 教育学部・旭川校, 教授 (60128733)
KOSHITANI Shigeo Chiba Univ., Dept.of Math., Prof., 理学部, 教授 (30125926)
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| Project Period (FY) |
1999 – 2001
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| Keywords | Dade's conjecture / perfect isometry / Broue conjecture / shintani descent / modular representation |
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| Research Abstract |
1. For block algebras of Chevalley groups of type G, having an extra special group of order 27 as defect groups, or of projective special linear groups or special unitary groups, having an.elementary abelian group of order 9 as defect groups, we have shown that Broue conjecture holds, namely, that there exists a derived equivalence between the block algebra and its Brauer correspondent. This result implies that there exists a perfect isometry between them, and moreover that Dade's conjecture is true.
In the situations above, the stabilizers of an automorphism induced from those of the denning fields are again Chevalley groups of the same type. The derived equivalences, and thus perfect isometries are shown to be compatible with taking the stabilizers. In other words, they are compatible with the Shintani descent. Moreover, we have proved that for general linear groups, local blocks are Morita equivalent and it is also compatible with Shintani descent.
2. For syinplectic groups of small rank, we have proved that the invariant form of Dade's conjecture is true. However, we have not checked that it is compatible with Shintani descent.
3. Meanwhile, Isaacs and Navarro proposed a conjecture. We have discussed the relationship between it and perfect isometries and Dade's conjecture, and finally proposed a new conjecture which includes all of them. For the sporadic simple groups of Lyons and Thompson, it is shown that the new conjecture is true. It is certainly necessary to study the new conjecture more.
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