Modular Representations of Finite Groups

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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.