RESEARCH ON THE MODULAR REPRESENTATIONS OF FINITE GROUPS

有限群模表示的研究

• 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

(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)有限群上置换模的自同态环的研究关于Iwahori-Hecke代数，Tsushima在对称群的模表示方面建立了一些结果。特别是，他使用Young图上称为分支的操作来构造Speht模块的一些简单组成部分。准确地说，卡特-佩恩定理被推广到柱枝型，该定理只适用于杆枝型。此外，还证明了每个Speht模块都有简单成分，其对应的Young图是原始Young图的分支。对有限群的不可分解模Watanabe的研究表明，如果群有素数阶的交换Sylow p-子群，则Alperin猜想对主p-块成立，从而证明了Broue猜想在完备等距上的正确性。(3)Lie型群的表示理论Kaneda建立了单连通单代数群上同调群的Andersen-Haboush定理的量子模拟它同时给出了Kempf消失定理的量子类比。

项目成果

Kaneda,Masahain: "Cohomology of in finitesimal quantum algebras"J.Algebra. 226. 250-282 (2000)

Kaneda，Masahain：“有限小量子代数的上同调”J.代数。

Kawata,Shigeto: "On Auslander-Reiten components and projective lattice of p-groups"Osaka J.of Math.. 38(印刷中). (2001)

Kawata, Shigeto：“论 Auslander-Reiten 分量和 p 群的射影格”Osaka J.of Math.. 38（出版中）。

Shigeo Koshitani and Katshusi Waki: "The Loewy structure of the projective modules of the Mathieu group M_<12> and its automorphism group in characteristic 3"Communications in Algebra. 27(1). 1-36 (1999)

Shigeo Koshitani 和 Katshusi Waki：“Mathieu 群 M_<12> 的射影模的 Loewy 结构及其特征 3 中的自同构群”《代数通讯》。

S.Koshitani: "The principal 3-blocks for four-and five-dimensional projective special linear groups in non-defining characteristic"J.Algebra. 226. 788-806 (2000)

S.Koshitani：“非定义特征中四维和五维射影特殊线性群的主要 3 块”J.代数。

Atumi Watanabe: "The Glauberman character correspondence and porbect isometries for blocks of finite groups"Journal of Algebra. 216. 548-565 (1999)

Atumi Watanabe：“有限群块的格劳伯曼特征对应和对象等距”代数杂志。