The modular representation theory of finite groups of Lie type is a major, active topic of mathematical research. The area has both inspired and been strongly influenced by developments in other areas, e.g., the cohomology of groups and Lie algebras, quantum groups, characteristic zero representation theory, perverse sheaves, and the theory of finite dimensional algebras. The project treats both the describing and non-describing theories. The describing (resp., nondescribing) theory concerns the modular representations in which the underlying module is taken over a field having the same (resp., different) characteristic as the defining characteristic of the group. In the describing theory, the investigators look for an increased role of the relatively unexplored area of affine Lie algebra representations in positive characteristic. In the non-describing theory, they plan to attack several important conjectures aimed at finding a unified approach valid for all types. They also continue to explore possible analogies between the two theories. Finally, Parshall continues his recent work on cohomology computations, while Scott extends his recent computational work.
In mathematics, a "group" is an abstract embodiment of symmetry. In addition to applications to other parts of mathematics, concrete realizations of groups 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 representations the investigators study comprise the most important basic ingredients for creating a general theory of all finite group representations. Over the past century, similar theories for continuous groups 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 future, 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.