Modular Representations of Finite Groups

本课题是对素数特征域上有限群的表示理论和上同调的研究。首席研究员对构成基本模块理论基础的表征的同调性质特别感兴趣。他计划考虑一个开放了20多年的问题，即在模的大范畴论中起重要作用的特定类型模的分类，并研究作用于基本同调结构的群的上同环的结构。Carlson和他的合作者已经证明了模范畴的许多方面是由p-子群的群上同调控制的。拟议的工作将建立在这个基础上。其他项目包括研究有限群的模类结构和模的可拓的一般理论。该项目的结果可能对代数拓扑和表示理论领域感兴趣。卡尔森教授计划继续开发计算机代数系统，用于模态和同态的实验。他打算扩充他的程序集，用于群上同调的计算和模论的其他方面。这些程序也正在被重写，以便在代数表示理论领域得到更广泛的应用。在基本术语中，首席研究员将研究某些类型的代数系统以及算子的动作。这样的系统被称为模块，它可能有许多维度，因为它依赖于许多变量。这些操作可以表示空间上点的几何旋转。该项目将集中于模块的分类和属性，其相关操作符具有预设的交互集合。该项目的一个重要部分是开发用于分析模块结构和性能的计算技术和软件。模和空间上的变换群是现代数学中的基本对象，在数学的许多应用中都有出现。本研究的一些方法与拓扑学中使用的几何技术密切相关。

The project is an investigation into the representation theory and cohomology of finite groups over fields of prime characteristic. The Principal Investigator is particularly interested in the homological properties of representations which underlie the basic module theory. He plans to consider a question open for more than 20 years on the classification of a specific type of modules that play an important role in the larger category theory of the modules, and also to look the structure of the cohomology ring of the group which acts on the fundamental homological constructions. Carlson and his collaborators have shown that many facets of the module category are controlled by the group cohomology of p-subgroups. The proposed work would build on this foundation. Other projects involve investigations of the structure of module categories of finite groups and the general theory of extensions of modules. Results from the project could be of interest in the area of algebraic topology as well as in representation theory. Professor Carlson plans to continue his development of computer algebra systems for experimentation with modules and homomorphisms. He intends to expand his collection of programs for the computation of group cohomology and other aspects of the module theory. The programs are also being rewritten for more general applications in the area of the representation theory of algebras. In basic terms the Principal Investigator will look at certain types of algebraic systems together with the actions of operators. Such a system is called a module and it might have many dimensions in the sense of depending on many variable. The operations may represent something like the geometric rotation of points on a space. The project will concentrate on the classification and properties of modules whose associated operators have a preset collection of interactions. A significant part of the project is the development of computational techniques and software for analyzing the structure and properties of modules. Groups of transformations on modules and spaces are basic objects in modern mathematics and arise in many applications of the mathematics. Some of the methods of the study are closely related to geometric techniques used in topology.