Modular Representations and Cohomology

本文研究有限Lie型群的模表示理论和上同调.一个主要的推动力涉及定义特征理论，在该理论中，底层模块是在一个与群体具有“相同”特征的领域上建立的。交叉特征理论是另一个主题;在这里，模块的特征不同于群体的特征。在这两种情况下，连续李群的（已知的）表示提供了表示的起点。这在定义特征群的情况下最为明显，在交叉特征群中，量子群起着类似的作用，但是，到目前为止，仅限于有限的一般线性群。主要的问题涉及到不可约模的分类、维数和特征标的确定以及同调性质的理解。涉及Weyl群（和Heckealgebras），有限维代数（例如，Parshall和Scott已经发展并贡献了许多不同的方法，目前的项目将继续在这个方向上，但也应用新的（和深入的结果），如最近解决的对称群的Broue猜想（由Chuang和Rouquier）。例如，作者将联合收割机这项工作与他们自己最近的工作有关的Kazhdan-Lusztig多项式特征公式的表示的特殊线性群的定义特征的表示对称群。群和表示在这里研究包括最重要的基本成分，为建立一般理论的所有有限群表示，在过去的世纪，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。创建一个可行的有限群表示的一般理论的任务-研究人员的长期目标-是，因此，未来的中心问题。最后，研究生和本科生将充分参与该项目。

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.