Modular Representations of Finite Groups

本课题是研究有限群和代数在素数特征域上的表示理论和上同调。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。在群表示方面，他将继续对模论与群上同的关系进行长期的研究。具体问题涉及有界支持变量模块的块分类及其与群的普通块理论的关系。卡尔森将研究与有限群和群方案表示的线性代数有关的其他基本问题。拟议的工作将建立在卡尔森教授多年来奠定的基础上。此外，首席研究员计划继续开发用于模块和同态实验的计算机代数系统。最近的工作导致了提取矩阵代数的生成和关系的系统的发展。该系统将扩展到研究有限维代数的一般同调性质以及对群表示的应用。其他项目涉及与代数群的表示理论和群扩展的一般理论的联系。首席研究员将研究代数系统和算子的动作。这样的系统可能是一个空间，其中的运算符是旋转或一些随时间进展的表示。系统被称为一个模块，它可能有很多维度，因为它依赖于很多变量。该项目将集中于对某些类型的模块进行分类，这些模块的相关操作符的交互满足预设条件。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。

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.