2007 Fiscal Year Final Research Report Summary
A Study on Glauberman-Watanabe Correspondence and Derived Equivalence of Blocks
| Project/Area Number |
18540004
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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 | Hokkaido University of Education |
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Principal Investigator |
OKUYAMA Tetsuro Hokkaido University of Education, Faculty of Education, Professor (60128733)
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| Co-Investigator(Kenkyū-buntansha) |
HASEGAWA Izumi Hokkaido University of Education, Faculty of Education, Professor (50002473)
KITAYAMA Masashi Hokkaido University of Education, Faculty of Education, Professor (80169888)
OKUBO Kazuyoshi Hokkaido University of Education, Faculty of Education, Professor (80113661)
YATSUI Tomoaki Hokkaido University of Education, Faculty of Education, Associate Professor (00261371)
IAI Shin-Ichiro Hokkaido University of Education, Faculty of Education, Associate Professor (50333125)
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| Project Period (FY) |
2006 – 2007
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| Keywords | Representation of Finite Group / Block Algebra / Derived Equivalence / Glauberman Correspondence / Complex of Modules / Cohomology |
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| Research Abstract |
In the research of this project, we studied "the abelian defect conjecture" which is one of main problems in representation theory of finite groups. We were mainly concerned with Glauberman-Watanabe correspondence.
1. We have constructed a two sided complex in the principal blocks of the group U(4,q^2) which gives a stable equivalence. In our investigation. we have solved conjectures on complexes of some module categories related to Alvis-Curtis-Kawanaka duality in finite reductive groups and to self-equivalences in Hecke algebras of finite Coxeter groups.
2. We studied the problem raised by Holloway-Koshitani-Kunugi which concerns with relations between blocks of SL(2,q) and its extension group by a field automorphism group. And we constructed some one sided complex for the group Sz(q) which we might expect to be a tilting complex. Blocks appearing here have non-abelian defect groups with cyclic focal subgroup.
3. We also worked on a problem concerning with quasi-perfect isometries which is a generalization of notion of perfect isometries recently studied by Narasaki-Uno. And we solved their conjecture for the group U(4,q^2) and Sz(q) over the field with defining characteristic. We studied a perfect isometriy between the blocks of the groups ^2F_4(2) and ^2F_4(q).
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