Modular Representations of Finite Groups

该项目是一个调查的表示理论和有限群和代数的上同调域的素特征。主要研究者特别感兴趣的是代表性的基础基本模块理论的同调属性。在该地区的团体表示，他将继续他的长期调查之间的联系模块理论和组上同调。具体问题涉及到分类模块有界支持品种成块和关系与普通块理论的群体。卡尔森将研究其他基本问题有关的线性代数表示有限群和组计划。拟议的工作将建立在Carlson教授多年来奠定的基础上。此外，首席研究员计划继续开发计算机代数系统，用于模块和同态的实验。最近的工作导致了系统的发展，提取生成元和矩阵代数的关系。该系统将扩大到调查一般同调性质有限维代数以及应用群表示。其他项目涉及代数群的表示理论和群扩展的一般理论。首席研究员将研究代数系统以及算子的作用。这样的系统可能是一个空间，其算子是旋转或随时间推移的某种表示。这个系统被称为一个模块，它可能有许多维度，因为它依赖于许多变量。该项目将集中在某些类型的模块的分类，其相关的运营商，其相互作用满足预设条件。该项目的一个重要组成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，并在数学的许多应用中出现。

The project is an investigation into the representation theory and cohomology of finite groups and algebras over fields of prime characteristic. The Principal Investigator is particularly interested in the homological properties of representations which underlie the basic module theory. In the area of group representations he will continue his long investigation into the connections between module theory and group cohomology. Specific problems are concerned with the classification of modules with bounded support varieties into blocks and the relations with the ordinary block theory of the group. Carlson will study other basic issues concerned with the linear algebra of representations of finite groups and group schemes. The proposed work would build on the foundation laid by Professor Carlson over many years. In addition, the Principal Investigator plans to continue his development of computer algebra systems for experimentation with modules and homomorphisms. Recent work has led to the development of systems for extracting generators and relations for matrix algebras. The system will be expanded to investigate general homological properties for finite dimensional algebras as well as application to group representations. Other projects involve connections with the representation theory of algebraic groups and the general theory of group extensions.The Principal Investigator will look at algebraic systems together with the actions of operators. Such a system might be a space with the operators being rotations or some representation of progression over time. The system is called a module and it might have many dimensions in the sense of depending on many variable. The project will concentrate on the classification of certain types of modules whose associated operators whose interactions satisfy preset conditions. 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.