Modular Representations of Finite Groups

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The project is an investigation into the representation theory and cohomology of finite groups over fields of prime characteristic. The Principal Investigator is particularly interested in the homological properties of representations which underlie the basic module theory. He plans to consider a question open for more than 20 years on the classification of a specific type of modules that play an important role in the larger category theory of the modules, and also to look the structure of the cohomology ring of the group which acts on the fundamental homological constructions. Carlson and his collaborators have shown that many facets of the module category are controlled by the group cohomology of p-subgroups. The proposed work would build on this foundation. Other projects involve investigations of the structure of module categories of finite groups and the general theory of extensions of modules. Results from the project could be of interest in the area of algebraic topology as well as in representation theory. Professor Carlson plans to continue his development of computer algebra systems for experimentation with modules and homomorphisms. He intends to expand his collection of programs for the computation of group cohomology and other aspects of the module theory. The programs are also being rewritten for more general applications in the area of the representation theory of algebras. In basic terms the Principal Investigator will look at certain types of algebraic 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 like the geometric rotation of points on a space. The project will concentrate on the classification and properties of modules whose associated operators have a preset collection of interactions. 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. Some of the methods of the study are closely related to geometric techniques used in topology.