Modular representations and cohomology

Parshall和scott的DMS-0400966的奖励摘要Lie型有限群的表示有三大类。首先，在定义特征理论中，底层模块被接管为与组的定义特征具有相同特征的字段。其次，在交叉特征理论中，模块被接管了与定义特征不同的特征领域。如果想要了解一个有限李群可能嵌入另一个有限李群的所有方式，交叉特征研究是不可避免的。第三类涉及到底层Weyl群的所有特征的表征理论。在所有情况下，连续李群的表示都为表示提供了起点。这在定义特征的情况下是最明显的，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。可以说，存在第四类“李型有限群”，即与Weyl群——甚至Coxeter群——相关的Hecke代数。这些代数常常把这三类联系起来；例如，在交叉特征理论中，它们提供了有限一般线性群和量子群之间的联系。虽然量子群在其他类型中的类似作用尚未出现，但赫克代数的“联系”仍然存在，通常在连续几何中有解释。这个项目将推进这些领域的研究，无论是在当前的调查中，还是在寻找有限李群表示理论的统一方法上。例如，一切都可以在适当的丰富意义上简化为赫克代数（允许同调或几何结构）吗？最近，作者将有限一般线性群的定义特征Lusztig猜想“约简”为对称群上同调性质。对于其他类型或类似的交叉特征猜想是否有类似的减少？这些问题的进展将涉及作者最近在赫克代数上同调方面的工作的进一步发展，以及对Parshall的零锥研究的继续追求。其他正在进行的项目，包括斯科特对本科生的计算机研究，都很符合这些主题。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，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，人们预计计算机和通信的有限离散世界将变得更加重要。因此，研究人员的长期目标是创造一个可行的有限群表示的一般理论，这是未来的核心问题。

Abstract for award DMS-0400966 of Parshall and ScottThere are three great classes of representations of the finite groups of Lie type. First, in the defining characteristic theory, the underlying module is taken over a field with the same characteristic as the defining characteristic of the group. Second, in the cross-characteristic theory, modules are taken over fields of characteristic different from the defining characteristic. Cross-characteristic studies are unavoidable if one wants to understand all ways that one finite Lie group might embed in another. The third class involves the representation theory in all characteristics of the underlying Weyl group. In all cases, representations of continuous Lie groups provide starting point for representations. This is most apparent in the defining characterstic 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. Arguably, there is fourth class of "finite group of Lie type," viz., the Hecke algebras associated to Weyl---or even Coxeter---groups. These algebras often connect the three classes; e.g., in the cross-characteristic theory, they provide a link between the finite general linear groups and quantum groups. Though an analogous role for quantum groups in other types has yet to appear, the Hecke algebra "link" remains, often with interpretations in continuous geometry. This project will advance research in these areas, both in current investigations and in focusing anew on finding a unified approach to the representation theory of the finite Lie groups. For example, can everything be reduced to Hecke algebras in a suitably rich sense (allowing homological or geometric structure)? Recently, the authors "reduced" the defining characteristic Lusztig conjecture for finite general linear groups to proposed symmetric group cohomology properties. Is there a similar reduction for other types or analogous cross-characteristic conjectures? Progress on these questions will involve much further development of the authors' recent work on Hecke algebra cohomology, as well as continued pursuit of Parshall's nullcone research. Other continuing projects, including Scott's computer work with undergraduates, fit in well with these themes.The groups and representations studied here comprise the most important basic ingredients for creating a general theory of all finite group representations. Over the past century, similar theories for continuous groups 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 future, one 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-the investigators' long-term goal-is, thus, a central problem for the future.