Modular Algebraic Representation Theory

这个项目将研究群的模表示理论的各个方面，特别强调定义的半单代数F-群的有理表示以及在p元素子域上的分裂。这个项目的一个组成部分涉及到FG的表示的半单性的考虑，其中G是任意群。在对表示的适当约束下(例如，要求尺寸相对于p不太大)，最近的结果允许人们推导出半简单性。沿着这些思路，该项目将研究某些张量运算(推广张量幂、外幂和对称幂)应用于适当约束的半单表示的半单性，以及某些“逆定理”。当H是有限域上半单代数群G的有理点的有限群时，本课题将求出不可约FH-模中系数为p的H的上同调群的消元结果，这种第一上同调的消元结果是从一些已知的半单结果得到的。本文将推广作者关于H·H·Andersen的Stilting模滤波的一些研究，目的之一是得到“Howe对偶”背景下所有素数p的Andersen求和公式，并考虑各种求和公式之间的相似之处--Jantzen的经典结果、Andersen最近的工作以及G的Lie代数的Jantzenon无限制表示的一些最近工作.我们将寻求对称群的双重覆盖群的一些不可约表示的显式构造；通过类比一般线性群和对称群之间的常见相互作用，希望可以利用简单的代数自旋群来获得所需构造的一些见解。

McNINCH, 9970301The project will investigate various aspects of the modularrepresentation theory of groups, with special emphasis on the rationalrepresentations of semisimple algebraic F-groups defined and splitover the prime subfield with p elements. One component of the projectinvolves consideration of the semisimplicity of representations of FG,where G is an arbitrary group. Under suitable constraints on arepresentation, (e.g. requirements that the dimension not be too largewith respect to p) recent results permit one to deducesemisimplicity. Along these lines, the project will study thesemisimplicity of certain tensor operations (generalizing the tensorpower, exterior power, and symmetric power) applied to appropriatelyconstrained semisimple representations, as well as certain "conversetheorems." When H is the finite group of rational points of asemisimple algebraic group G over a finite field, the project will seekto obtain vanishing results for cohomology groups of H withcoefficients in irreducible FH-modules whose dimension is suitablybounded in terms of p; such vanishing results for first cohomologyfollow from some known semisimplicity results. The project willextend some investigations of the author concerning H. H. Andersen'stilting module filtrations; one aim is to obtain Andersen's sumformula for all primes p in the setting of "Howe dual pairs."Analogies between various sum formulae -- the classical result ofJantzen, the recent work of Andersen, and some recent work of Jantzenon non-restricted representations of the Lie algebra of G -- will beconsidered. Some explicit constructions of irreducible representationsfor double covering groups of symmetric groups will be sought; byanalogy to the familiar interplay between the general linear group andthe symmetric group, it is hoped that the simple algebraic Spin groupcan be exploited to obtain some insight for the desired constructions.