Modular representations of finite groups

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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 endotrivial modules which define the stable equivalences of Morita type for the group algebra. The Principal Investigator will study other basic issues concerned with the linear algebra of representations of finite groups and group schemes and connections with the theory of bundles. The proposed work will build on the foundation laid by the Principal investigator over many years. In addition, he 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 analyzing and condensing matrix algebras down to their basic 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 collections 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.