Modular Representations

李氏有限群的模表示理论是数学研究的一个重要而活跃的课题。该领域受到了其他领域的发展的启发和强烈影响，例如群和李代数的上同调、量子群、特征零表示理论、反常束和有限维代数理论。该项目处理描述和非描述理论。描述（说明）：（非描述）理论关注的是模块化表示，在这种表示中，底层模块被接管具有相同属性的字段。（不同的）特征作为群体的定义特征。在描述理论中，研究人员寻找相对未探索的仿射李代数表示在正特征中的作用增加。在非描述理论中，他们计划攻击几个重要的猜想，旨在找到一种适用于所有类型的统一方法。他们还继续探索这两种理论之间可能的相似之处。最后，Parshall继续他最近在上同调计算方面的工作，而Scott则扩展了他最近的计算工作。在数学中，“群”是对称的抽象体现。除了应用于数学的其他部分之外，群的具体实现可以为复杂的物理和通信系统的行为提供强大的约束。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。研究者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。在过去的一个世纪里，类似的连续群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。在未来，人们可以合理地预期，由计算机和通信组成的有限的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——这是研究者的长期目标——是未来的中心问题。

The modular representation theory of finite groups of Lietype is a major, active topic of mathematical research. The area has both inspired and been stronglyinfluenced 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 thedescribing 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 relativelyunexplored area of affine Lie algebra representations inpositive 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 cohomologycomputations, 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 ofcomplicated physical and communications systems. This project concentrateson determining the ways in which the most important abstract finite groupsfind concrete representations as square matrices over finite number systems.The groups and representations the investigators study comprise the mostimportant basic ingredients for creating a general theory of all finitegroup representations. Over the past century, similar theories forcontinuous groups have played a large role in quantum theory and the theoryof elementary particles. Their finite analogs have already proved valuablein the design of communications and data storage devices, though this finitetheory remains very incomplete. In the future, one reasonably expects thatthe finite discrete worlds of computers and communications will become evenmore important. The task of creating a viable general theory of finite grouprepresentations -- as is the investigators' long-term goal -- is, thus, acentral problem for the future.