亏群为亚循环群的块代数研究

项目名称： 亏群为亚循环群的块代数研究

项目编号： No.11301393

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 杨胜

作者单位： 温州大学

项目金额： 22万元

中文摘要： 有限群模表示理论中，一个很重要目的是弄清楚有限群的一个p-块结构多大程度上依赖于他的亏群结构。通过R.Brauer, E.C.Dade, G.J.Janusz 和 J.B.Olsson等学者的一系列出色工作，亏群为循环群的p-块和亏群为二面体群、广义四元数群、拟二面体群的2-块结构得到清楚的刻画。但是，关于亏群为其他类型的p-块很少有比较整齐的结果。 张继平教授提出：今后国际上将有更多人结合现有的几个重要猜想研究亚循环亏数群的问题。受此启发，本项目将在前人工作基础上研究亏群为亚循环群的p-块（p是一个奇素数）结构，计算出一些重要的块不变量。 此外，我们还将对这类块验证Brauer的K(B) 猜想、 Olsson 猜想和Alperin猜想。 本项目的预期研究成果将在一定程度上回答张老师的问题，对进一步发展和完善局部表示论有关方法有比较重要的意义。

中文关键词： 块；亏群；亚循环群；惯性指数；下亏群

英文摘要： One of the aims of modular representation theory is to show how the structure of a block of a finite group depends on the structure of its defect group. Here the structure of a block is expressed in certain numerical invariants, such as the numbers of the irreducible ordinary and modular characters. By the outstangding work of R.Brauer, E.C.Dade, G.J.Janusz and J.B.Olsson, we have gained thorough knowledge of the structure of the bloaks with cyclic, dihedral, generalized quaternion or quasidihedral defect group. However, it is very difficult to get conplete results for the blocks with other types of defect groups. Professor Zhang Jiping:" There will be more people to investigate the structure of matacyclic block combined with several open conjectures in the future." In this project, we carry out extensive research into the blocks with odd metacyclic defect groups and calculate some important invariants of the blocks. In addition, we verify Brauer's K(B) conjecture, Olsson's conjecture and Alperin's conjecture in the case that D is metacyclic and p is odd. As one will see, our research results will to some extent answer Professor Zhang Jiping's problem and help to further develop methods of local representation theory.

英文关键词： Block；Defect Group；Metacyclic Group；Inertial Index；Lower Defect Group