Modular Representations and Cohomology

本文研究了李型有限群的模表示理论和上同调。一个主要的推力涉及定义特征理论，在该理论中，底层模块占据了与组具有“相同”特征的字段。交叉特征理论是另一个主题；这里，模块的特性不同于组的特性。在这两种情况下，连续李群的（已知的）表示为表示提供了一个起点。这是最明显的定义特征的情况下，其中一个环境代数群是可用的。在交叉特性中，量子群也起着类似的作用，但到目前为止，只适用于有限的一般线性群。未解决的主要问题包括不可约模的分类、模的维数和特征的确定以及模的同调性质的理解。涉及Weyl群（和hecke代数），有限维代数（例如，自同态和准遗传代数），逆凸几何理论等的问题在许多对这些问题的攻击中自然出现。Parshall和Scott已经开发并贡献了许多不同的方法，当前的项目将继续朝着这个方向发展，但也会应用新的（和深入的结果），例如最近对对称群的Broue猜想的解决方案（由Chuang和rouquier）。例如，作者将把这项工作与他们自己最近的工作结合起来，这些工作是关于定义特征中特殊线性群的表示的Kazhdan-Lusztig多项式特征公式到对称群的表示。这里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，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，创造一个可行的有限群表示的一般理论的任务——研究者的长期目标——是未来的一个中心问题。最后，研究生和本科生将全面参与项目。

This research concerns the modular representation theory andcohomology of finite groups of Lie type. A major thrust involves thedefining characteristic theory, in which the underlying module istaken over a field having the "same" characteristic as the group.The cross-characteristic theory is another theme; here, thecharacteristic of the module differs from that of the group. In bothcases, the (already known) representations of continuous Lie groups provide a starting point for representations. This is most apparent in the definingcharacterstic case, where an ambient algebraic group is available.In cross characteristic, quantum groups plays a similar role, but,so far, only for the finite general linear groups. The majorunsolved problems involve classifying the irreducible modules,determining their dimensions and characters, and understanding theirhomological properties. Issues involving Weyl groups (and Heckealgebras), finite dimensional algebras (e.g., endomorphism andquasi-hereditary algebras), the geometric theory of perversesheaves, etc. arise naturally in many attacks on these problems.Parshall and Scott have developed and contributed to many of thevarious approaches, and the current project will continue in thatdirection, but also apply new (and deep results), such as the recentsolution of the Broue conjecture for symmetric groups (by Chuang andRouquier). For example, the authors will combine this work withtheir own recent work relating Kazhdan-Lusztig polynomial characterformulas for representations of the special linear groups in the defining characteristic to the representations of symmetric groups.The groups and representations studied here comprise the mostimportant basic ingredients for creating a general theory of allfinite group representations, Over the past century, similartheories for continuous groups played a large role in quantum theoryand the theory of elementary particles. Their finite analogs havealready proved valuable in the design of communications and datastorage devices, though this finite theory remains very incomplete. In the future, one expects that the finite discrete worlds of computers andcommunications will become even more important. The task of creatinga viable general theory of finite group representations--theinvestigators' long-term goal--is, thus, a central problemfor the future. Finally, graduate and undergraduate students willbe fully involved in the project.