1998 Fiscal Year Final Research Report Summary
Research on blocks of finite groups
| Project/Area Number |
09640048
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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 | Kumamoto University |
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Principal Investigator |
WATANABE Atumi Faculty of Science, Associate Professor, 理学部, 助教授 (90040120)
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| Co-Investigator(Kenkyū-buntansha) |
HIRAMINE Yutaka Department of Education, Professor, 教育学部, 教授 (30116173)
YAMAKI Hiroyoshi Department of Science, Professor, 理学部, 教授 (60028199)
UNO Katsuhiro Department of Mathematics, Graduate School of Science, Osaka University, Associa, 大学院・理学研究科, 助教授 (70176717)
OKUYAMA Tetsuro Laboratory of Mathematics, Asahikawa Campus, Hokkaido University of Education, P, 教育学部・旭川校, 教授 (60128733)
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| Project Period (FY) |
1997 – 1998
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| Keywords | block / perfect isometry / isotypy / Glauberman correspondence / Shintani descent / Isaacs correspondence / アルペリン予想 / イソタイピー |
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| Research Abstract |
1. We obtained some interesting examples of perfect isometries and isotypies between blocks of finite groups. Let S and C be finite groups such that S acts on G via automorphism and (|S|, |G|) 1. In this situation there is a natural bijection, what we call, Glauberman-Isaacs correspondence pi(G.S) from the set Irr_s (G) of S-invariant irreducible characters of G onto the set Irr(C_G(S)) of irreducible characters of C_G(S). We showed that pi(G.S) gives isotypies between blocks of G and C_G(S) under some assumptions. We conjecture that a perfect isometry obtained from the Isaacs ocrrespondence is induced by a Morita equivalence. We also proved that the Shintani descent for irreducible characters of finite general linear groups gives perfect isometries between the principal blocks of those groups. We showed that the naturally Morita equivalence between blocks of finite groups in normal subgroups gives isotypies.
2. We studied blocks of finite groups with abelian defect groups. We obtained results on generalized decomposition numbers and isotypies of blocks of p-solvable groups with abelian defect groups. And we applied them to a problem on decompositions into tensor products of block algebras with abelian defect groups.
3. We proved that Broue's conjecture is true for some blocks of finite groups and we obtained results on the Auslander- Reiten components of blocks of finite groups.
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