Modular Representation Theory of Finite and Algebraic Groups

本文的目标是建立一类有限群G的不可约模表示的可行理论。有两种情况需要考虑：(1)描述特征理论，在该理论中，表征被置于具有与G的定义特征p相同特征的域k上；(2)非描述性特征理论，其中k具有与p不同的特征。在过去的十年中，研究人员开发了许多新技术（通常与E. Cline合作）来攻击案例(1)。这些方法通常是通过将几何方法（如反常束理论）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，他们将继续朝这个方向努力。同时，研究人员将把他们的方法与高维变形技术相结合，这些技术也应用于(1)。此外，研究人员最近开发了新的自同态代数攻击技术(2)。特别是，他们（与杜杰）将Kazhdan-Lusztig细胞理论引入Hecke自同态代数理论，这表明在发展非描述特征理论的一般理论（即对所有类型有效）方面可以取得很大进展。该项目涉及和/或将应用于许多其他数学领域，包括上同调、李论、有限维代数和有限群的极大子群理论。数学群是对称的抽象体现。它的具体实现可以对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以方阵的形式找到具体表示的方式。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，在我们熟悉的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重要作用。它们的有限类似物已经在通信和数据存储设备的设计中被证明是有价值的，尽管这种有限理论仍然非常不完整。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——这是研究者的长期目标——是未来的一个核心问题。

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.