Cohomology and Modular Representation Theory of Finite and Algebraic Groups

CLINE.TEXT 该基金的主要研究者将与弗吉尼亚大学的Brian Parshall和伦纳德斯科特合作，参与该基金的大部分研究活动。该小组将研究李型代数和有限群的模表示理论。在这种情况下，（素）的特点，基本领域是自然的特点，周围的代数群，主要问题的中心一个著名的猜想Lusztig的字符某些不可约表示。 最近的进展表明，这个猜想是正确的，但有限数量的素数。然而，目前还没有关于素数大小的有效界限。这个项目的一个目标是继续工作，以完成这个猜想的证明。在特性不是自然特性的情况下，理论就不那么完善了。 最近的工作克莱因，在合作与Parshall和斯科特，产量技术适用，一般来说，模块表示理论的有限群的李型的所有特点。因此，本项目的第二个目标是将这些新技术发展成为研究非自然特征情况下这些表征的有效工具。特别是，人们希望最终能对这些情况下的不可约模的性质有重要的了解。 群论及其表示是数学研究的一个中心领域，因为它的方法普遍适用于研究任何对象，特别是许多具有高度对称性的数学结构。对于一个给定的群，确定它的不可约表示是一个中心问题。 该研究的主要研究者在这个补助金将作为一个重点的确定这些不可约表示有限群的李型，及其相关的代数群。这个问题可分为两大类。第一种情况下，其中取得了重大进展的Lusztig猜想，是更好地发展比在第二个很少有人知道一般。 因此，工作 在第二种情况下，主要研究者在该补助金上的身份将比第一种情况更具初步性质。在任何一种情况下的进展将有应用在许多其他领域的数学，包括上同调理论，李理论，理论的有限维代数和研究的子群结构的有限和代数群的李型。

CLINE.TEXT The principal investigator of this grant will work in collaboration with Brian Parshall and Leonard Scott of the University of Virginia for a significant portion of the research activity on this grant. The group will study the modular representation theory of algebraic and finite groups of Lie type. In the case where the (prime) characteristic of the base field is the natural characteristic of the ambient algebraic group, the main problem centers upon a famous conjecture by Lusztig for the characters of certain irreducible representations. Recent progress shows that this conjecture is true for all but a finite number of primes. However, at present, there is no effective bound on the size of the prime. One goal of this project is to continue work toward a complete proof of this conjecture. In the case where the characteristic is not the natural characteristic, the theory is much less well developed. Recent work of Cline, in collaboration with Parshall and Scott, yields techniques which apply, generally, to the modular representation theory of the finite groups of Lie type in all characteristics. A second goal of this project therefore is to develop these new techniques into an effective tool for the study of these representations in the non-natural characteristic cases. In particular, one hopes eventually to obtain significant insight into the nature of the irreducible modules in these cases. The theory of groups and their representations is a central area of research in mathematics because its methods apply generally to the study of any object, in particular to many mathematical structures, with a high degree of symmetry. For a given group, the determination of its irreducible representations is a central problem. The research of the principal investigator in this grant will have as one focus the determination of these irreducible representations for the finite groups of Lie type, and their associated algebraic groups. The problem divid es into two major cases. The first case, in which major progress has been made on the Lusztig conjecture, is much more well developed than in the second where little is known in general. Therefore the work of the principal investigator on this grant will be of a more preliminary nature in the second case than in the first. Progress in either case would have applications in many other areas of mathematics including cohomology theory, Lie theory, the theory of finite dimensional algebras and the study of the subgroup structure of the finite and algebraic groups of Lie type.