Modular Representation Theory of Finite and Algebraic Groups

Parshire/Scott，97-00965这个建议的目标是为李型有限群G创建一个可行的不可约模表示理论。有两种情况需要考虑：(1)描述特征理论，其中表示取在具有与G的定义特征p相同特征的域k上；(2)非描述特征理论，其中k具有不同于p的特征。在过去的十年中，研究人员发展了许多新的技术(通常与E.Cline合作)来攻击情形(1)。这些方法通常是通过将几何方法，如倒叶理论，应用于经典代数(特别是有限维代数的表示理论)而产生的。在拟议的工作中，他们将继续朝着这个方向前进。同时，研究人员将把他们的方法与高维变形技术相结合，这些技术也已应用于(1)。此外，研究人员最近开发了用于攻击的新的自同态代数技术(2)。特别是，他们(与杜杰一起)将Kazhdan-Lusztig胞格理论引入到Hecke自同态代数理论中，表明在发展非描述特征理论的一般理论(即对所有类型都有效)方面可以取得很大进展。该项目涉及或将应用于数学的许多其他领域，包括上同调、李理论、有限维代数和有限群的极大子群理论。数学群是对称性的抽象体现。它的具体实现可以为复杂的物理和通信系统的行为提供强有力的约束。这个项目集中于确定最重要的抽象有限群在有限数系统上找到作为方阵的具体表示的方法。研究人员研究的群和表示构成了创建所有有限群表示的一般理论的最重要的基本成分。在本世纪，关于常见连续数系统上的连续群的类似理论在量子理论和基本粒子理论中发挥了很大的作用。它们的有限类比在通信和数据存储设备的设计中已经被证明是有价值的，尽管这个有限理论仍然非常不完善。在下个世纪，人们有理由预计，计算机和通信的有限离散世界将变得更加重要。因此，创建一个可行的有限群表示的一般理论的任务--这是研究人员的长期目标--是未来的一个中心问题。

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.