RESEARCH ON THE MODULAR REPRESENTATIONS OF FINITE GROUPS

有限群模表示的研究

基本信息

批准号：
11640042
负责人：
TSUSHIMA Yukio
金额：
$ 2.24万
依托单位：
OSAKA CITY UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640042/
关键词：
Iwahori-Hecke algebra symmetric group Specht module q-Schur algebra Alperin conjecture Broue conjecture groups of Lie type cohomology 中山予想直既約加群 AR-quiver AR component Lie型群代数群モジュラー表現 Young diagram Mathieu群

项目摘要

(1) Study of endomorphism rings of permutatuion modules over finite groupsIn connection with the Iwahori-Hecke algebras, Tsushima has established some results on the modular representations of symmetric groups. In particular he constructs some simple constituents of the Specht modules using the operation on the Young diagram called branch. To be precise, Carter-Payne's theorem which is known to be true only for bar branch type is extended to pillar branch type. Also it is shown that each Specht module has simple constituents whose corresponding Young diagrams are branches of the original Young diagram. Moreover a complete proof has been given to the Nakayama conjecture for the q-Schur algebras, which is done because the original proof to the conjecture given by James and Mathas contains a gap.(2) Study of indecomposable modules of finite groupsWatanabe has shown that Alperin conjecture is true for the principal p-block if the group under consideration has an abelian Sylow p-subgroup with automizer of prime order, which induces the validity of Broue's conjecture on perfect isometry.(3) Representation theory of groups of Lie typeKaneda has established the quantum analogue of Andersen-Haboush's theorem on the cohomology group of the simply connected simple algebraic groups, which yields at the same time the quantum analogue of Kempfs vanishing theorem.

(1)有限群上置换模自同态环的研究Tsushima结合Iwahori-Hecke代数建立了对称群模表示的一些结果。特别是，他使用杨图上称为分支的操作构造了 Specht 模块的一些简单组成部分。准确地说，卡特-佩恩定理仅适用于杆状分支类型，现已扩展到柱状分支类型。还表明每个 Specht 模块都有简单的组成部分，其相应的杨图是原始杨图的分支。此外，还对 q-Schur 代数的中山猜想给出了完整的证明，这是因为 James 和 Mathas 给出的猜想的原始证明存在空白。 (2) 有限群不可分解模的研究渡边证明：如果所考虑的群具有带有素数阶自动子的阿贝尔 Sylow p 子群，则 Alperin 猜想对于主 p 块成立，这导致了 Broue 的有效性（3）李型群的表示论Kaneda建立了单连通简单代数群的上同调群上Andersen-Haboush定理的量子类比，同时给出了Kempfs消失定理的量子类比。