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A Study on Broue's Conjecture in Representation Theory of Finite Groups

有限群表示论中布劳猜想的研究

基本信息

批准号：
10640001
负责人：
OKUYAMA Tetsuro
金额：
$ 2.05万
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：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640001/
关键词：
FINITE GROUPS MODULAR REPRESENTATION BLOCK THEORY BROUES CONJECTURE DERIVED EQUIVALENCES

项目摘要

In the research of this project, we studied a conjecture by Broue which says that a block algebra with abelian defect group of a finite group is derived equivalent to its Brauer correspondent. And we obtained good results as follows :1. We could give some methods to construct tilting complexes for two symmetric algebras which are stably equivalent of Morita type. As applications of our methods, we could show that the conjectures are true for principal 3-blocks of some finite groups with elementary abelian Sylow 3-subgroup of order 9, for example, Matieu groups M11, M21(=SL(3,4)), M22,M23 and Hignan-Sims group HS. We could also checked the conjecture for the principal blocks of the groups SL(2,q) over defining characteristic.2. There are many examples of blocks which are not known to be stably equivalent to its Brauer correspondent. We studied the principal 2-block of the smallest group of Janko. Using the splendid tilting complex for SL(2,4) given by Rickard , we could show that the conjecture is true for this block. We could also decide unknown parameters in the decomposition numbers of the group U(3,q) . We believe that our result can be used to study the conjecture for this group.3. Related to Broue's conjecture, there have been studied on tensor decompositions of blocks as algebras. We obtained some result on this problem by making use of theory of separable algebras and Puigis theory of source algebras of blocks.

在本项目的研究中，我们研究了 Broue 的一个猜想，该猜想表示有限群的具有阿贝尔缺陷群的分块代数等价于其 Brauer 对应推导。我们取得了良好的效果： 1．我们可以给出一些构造两个稳定等价于 Morita 型的对称代数的倾斜复形的方法。作为我们方法的应用，我们可以证明这些猜想对于一些具有 9 阶基本阿贝尔 Sylow 3 子群的有限群的主 3 块是正确的，例如，Matieu 群 M11、M21(=SL(3,4))、M22、M23 和 Hignan-Sims 群 HS。我们还可以通过定义特征来检查群 SL(2,q) 的主要块的猜想。2。有许多块的示例尚不知道是否与其布劳尔对应项稳定等效。我们研究了 Janko 最小群的主 2 块。使用 Rickard 给出的 SL(2,4) 的出色倾斜复形，我们可以证明该猜想对于该块是正确的。我们还可以决定群 U(3,q) 的分解数中的未知参数。我们相信我们的结果可以用来研究这一组的猜想。 3．与布劳猜想相关，人们对块的张量分解作为代数进行了研究。我们利用可分代数理论和块源代数Puigis理论在这个问题上取得了一些成果。