Modular Representation Theory of Finite and Algebraic Groups

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PARSHALL/SCOTT, 97-00965 The goals of this proposal center on creating a viable theory of irreducible modular representations for finite groups G of Lie type. There are two cases to consider: (1) the describing characteristic theory in which the representations are taken over a field k having the same characteristic as the defining characteristic p of G; and (2) the non-describing characteristic theory in which k has characteristic different from p. Over the past decade the investigators have developed many new techniques (often in collaboration with E. Cline) for attacking case (1). These methods have often come about by adapting geometric methods, such as the theory of perverse sheaves, to classical algebra (especially the representation theory of finite dimensional algebras). In the proposed work they will continue in that direction. At the same time, the investigators will integrate their approach with high dimensional deformation techniques which have also been applied to (1). In addition, the investigators have recently developed new endomorphism algebra techniques for attacking (2). In particular, their introduction (with Jie Du) of Kazhdan-Lusztig cell theory into the theory of Hecke endomorphism algebras suggests that much progress can be made on developing a general theory (i.e., valid for all types) for the non-describing characteristic theory. The project involves, and/or will have applications to, many other areas of mathematics, including cohomology, Lie theory, finite dimensional algebras, and the maximal subgroup theory of finite groups. A mathematical group is an abstract embodiment of symmetry. Its concrete realizations can provide powerful constraints on the behavior of complicated physical and communications systems. This project concentrates on determining the ways in which the most important abstract finite groups find concrete representations as square matrices over finite number systems. The groups and representati ons the investigators study comprise the most important basic ingredients for creating a general theory of all finite group representations. In this century, similar theories for continuous groups over familiar continuous number systems have played a large role in quantum theory and the theory of elementary particles. Their finite analogs have already proved valuable in the design of communications and data storage devices, though this finite theory remains very incomplete. In the next century, one reasonably expects that the finite discrete worlds of computers and communications will become even more important. The task of creating a viable general theory of finite group representations -- as is the investigators' long-term goal -- is, thus, a central problem for the future.