Groups and Representations Conference; March 25-27, 2004; Eugene, OR

University Of Oregon Eugene, Eugene OR

Investigators

Abstract

Principal Investigator: Alexander Kleshchev Proposal Number: DMS- 0244651 Institution: University of Oregon, Eugene Title: Groups and Representations conference Abstract: The subject of this meeting will be recent developments in the structure theory of simple algebraic and finite groups, their representation theory, and the interplay between these theories. The meeting will cover three main, inter-related areas. The first is the representation theory of simple algebraic groups in defining characteristic. There is no known character formula for the irreducible modules, but some years ago, Lusztig proposed a conjectural formula, which has become the main focus of attention. While Lusztig's conjecture remains far from proved, attempts to prove it have led to spectacular progress in the area in recent years. The second area is modular representation theory in non-defining characteristic. Here one considers the finite groups of Lie type over a field of characteristic p, and studies representations over fields of characteristic different from p. One should also include in this area the representation theory of Coxeter groups (such as the symmetric groups), which is intimately related to that of groups of Lie type. Again there have been spectacular developments in recent years. Fundamental conjectures have been formulated by Alperin, Dade and Broue, and while these are again nowhere near proved, many special cases have been solved, leading to a much deeper general understanding of this field. The third area is the structure theory of simple algebraic and finite groups, particularly the subgroup structure, and its relationship with the representation theory discussed above. Powerful parallel theories for subgroups of both the finite and the algebraic simple groups have been developed, using representation theory as one of the main tools, via the actions of classical groups on their natural modules, and of exceptional groups on their adjoint modules. The symmetry of any system, physical or mathematical, abstract or concrete, is encapsulated in its symmetry group. Thus, the theory of groups finds many applications, both in mathematics and in the physical sciences. Much of group theory is concerned with the study of the actions of groups on spaces of various kinds. The study of group actions on vector spaces is known as representation theory, and that of group actions on sets as permutation group theory, and the focus for this meeting will be on these two areas and their applications. The building blocks of all finite groups are the so-called simple groups, and most of these arise in a natural way from simple algebraic groups (such as SL(n,K), the group of n x n determinant 1 matrices over an algebraically closed field K). Consequently, most attention is devoted to the representation and permutation group theory of these simple groups. These areas are alive with basic conjectures, such as those of Lusztig, Alperin, Broue and Dade. While these are all far from proved, attempts to prove them have led to spectacular progress in the subject in recent years. The meeting will focus on this progress and its applications. There is a healthy number of graduate students working in these areas, and one of the goals is to stimulate interaction between graduate students, young researchers and some of the established leaders in the field.