Modular Representations of Finite Groups

摘要奖的乔恩F。卡尔森DMS-0401431项目名称：有限群的模表示。该项目是对素特征域上有限群的表示理论和上同调的研究。首席研究员特别感兴趣的是代表性的基础基本模块理论的同调属性。他将继续工作的分类某些特定类型的模块，发挥了重要作用，在更大的范畴理论的模块，也看看一般结构的上同调环。卡尔森和他的合作者已经证明了群代数的模范畴的许多方面是由群上同调控制的。 拟议的工作将在此基础上开展。 卡尔森教授计划继续他的计算机代数系统的发展与模块和同态实验。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合，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展的一般理论的连接.在基本条款的主要研究者将着眼于某些类型的代数系统连同操作.这样的系统被称为模块，它可能有许多维度，取决于许多变量。这些操作可以表示类似于空间上点的几何旋转。该项目将集中在模块的分类和属性，其相关的运营商来自一个组或代数。这意味着操作员之间有一个预设的交互集合。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的基本对象，在数学的许多应用中出现。

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.