A Study on Glauberman-Watanabe Correspondence and Derived Equivalence of Blocks

格劳伯曼-渡边对应及块的导出等价性研究

• 批准号: 18540004
• 负责人: OKUYAMA Tetsuro
• 金额: $ 2.4万
• 依托单位: Hokkaido University of Education
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2006
• 资助国家: 日本
• 起止时间: 2006 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18540004/
• 关键词: Representation of Finite Group; Block Algebra; Derived Equivalence; Glauberman Correspondence; Complex of Modules; Cohomology; 加群の複体とコホモロジー

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).

在本课题的研究中，我们研究了有限群表示理论中的主要问题之一--“阿贝尔亏格猜想”。我们主要关注Glauberman-Watanabe的通信1。我们在群U(4，q^2)的主块上构造了一个双边复形，它给出了一个稳定的等价。在我们的调查中。我们解决了有限约化群中与Alvis-Curtis-Kawanaka对偶有关的某些模范畴的复形和有限Coxeter群的Hecke代数中的自等价性的猜想。利用域自同构群研究了Holloway-Koshitani-Kuugi提出的关于SL(2，q)的块与其扩张群之间的关系的问题。我们为群Sz(Q)构造了一些单边复形，我们可能会认为它是一个倾斜复形。图中显示的区块具有非阿贝尔缺陷群和循环焦点亚群。我们还研究了一个与拟完全等距有关的问题，它是Narasaki-Uno最近研究的完全等距概念的推广。解决了他们关于具有定义特征的域上的群U(4，q^2)和群Sz(Q)的猜想。我们研究了群^2F_4(2)和^2F_4(Q)的块之间的完美等比关系。

项目成果

Brauer対応とGreen対応

布劳尔兼容和绿色兼容

• 发表时间: 2007
• 作者: M.Homma;S.J.Kim;佐々木 洋城
• 通讯作者: 佐々木 洋城

On a certain simple modules and cohomology of the symmetric group over GF(2)

关于GF(2)上对称群的某个简单模和上同调

• 发表时间: 2008
• 期刊: RIMS Kokyuroku(数理解析研究所講究録) 1581
• 作者: H.Nishimura;S.Kuroda;T. Okuyama
• 通讯作者: T. Okuyama

Parabolic geometries associated with differential equations of finite type

与有限型微分方程相关的抛物线几何

• 发表时间: 2007
• 期刊: Progress in Mathematics 252
• 作者: H.Nishimura;S.Kuroda;T. Okuyama;T. Okuyama;T. Okuyama;H. Sasaki;K. Yamaguchi and T. Yatsui
• 通讯作者: K. Yamaguchi and T. Yatsui

Cohomology algebras of blocks of finite groups and Brauer correspondence

有限群块的上同调代数和布劳尔对应

• 发表时间: 2006
• 期刊: Algebras and Representation Theory 9
• 作者: K.Yamaguchi;T.Yatsui;H. Kawai and H. Sasaki
• 通讯作者: H. Kawai and H. Sasaki

Brauer correspondence and Green correspondence

布劳尔通信和格林通信

• 发表时间: 2007
• 作者: Hiroki;Sasaki
• 通讯作者: Sasaki