Modular Representations of Finite Groups

Jon F.Carlson DMS-0401431项目的获奖摘要：有限群的模表示。该项目是对素特征域上的有限群的表示理论和上同调的研究。作为基本模理论基础的表示的同调性质，是主要研究者特别感兴趣的问题。他将继续研究在更大的模范畴理论中扮演重要角色的某些特定类型的模的分类，并研究上同调环的一般结构。Carlson和他的合作者已经证明，群上同调控制着群代数的模范畴的许多方面。拟议的工作将建立在这一基础上。卡尔森教授计划继续开发用于模和同态实验的计算机代数系统。特别令人感兴趣的是研究有限维代数的同调性质的算法的发展。PI打算扩大他的程序集，用于计算群上同调和模理论的其他方面。其他项目涉及与代数群的表示理论和群扩张的一般理论的联系。在基本条件下，首席研究人员将研究某些类型的代数系统以及算子的作用。这样的系统被称为模块，在依赖于许多变量的意义上，它可能具有许多维度。这些运算可能代表了空间中点的几何旋转。该项目将集中于其相关算子来自群或代数的模的分类和性质。这意味着操作员具有彼此交互的预设集合。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。

Abstract for award of Jon F. Carlson DMS-0401431Title of Project: Modular Representations of Finite Groups.The project is an investigation into the representation theory and cohomology of finite groups over fields of prime characteristic. The PrincipalInvestigator is particularly interested in the homological properties of representations which underlie the basic module theory. He will continue working on the classification of certain specific types of modules that play an important role in the larger category theory of modules, and also to look at the general structure of the cohomology rings. Carlson and his collaborators have shown that many facets of the module category for groupalgebras are controlled by the group cohomology. The proposed work would build on this foundation. Professor Carlson plans to continue his development of computer algebra systems for experimentation with modules and homomorphisms. Of particular interest is the development of algorithms for studying homologicalproperties for finite dimensional algebras. The PI intends to expand his collection of programs for the computation of group cohomology and other aspects of the module theory. Other projects involve connections withthe representation theory of algebraic groups and the general theoryof group extensions.In basic terms the Principal Investigator will look at certain types ofalgebraic 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 likethe geometric rotation of points on a space. The project will concentrateon the classification and properties of modules whose associated operatorscome from a group or algebra. This means that the operators have a preset collection of interactions with each other. A significant part of the project is the development of computational techniques and software for analyzing the structure and properties of modules. Groups of transformationson modules and spaces are basic objects in modern mathematics and arise in many applications of the mathematics.