Modular Algebraic Representation Theory

McNINCH，9970301该项目将研究群的模表示理论的各个方面，特别强调在p元素的素子域上定义和分裂的半单代数F群的有理表示。该项目的一个组成部分涉及考虑FG的表示的半简单性，其中G是一个任意群。 在适当的约束下的arepresentation，（例如，要求的尺寸不太大，关于p）最近的结果允许一个推导半简单性。沿着这些路线，该项目将研究应用于适当约束的半单表示的某些张量运算（推广张量幂，外幂和对称幂）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，该项目将寻求获得H的上同调群的消失结果，其系数在不可约FH-模中，其维数适当地以p为界;这种第一上同调的消失结果遵循一些已知的半单性结果。 本项目将介绍作者对H. H. Andersen的stilting模滤子;目的之一是在“Howe对偶对”的情况下得到所有素数p的Andersen求和公式。“类比之间的各种和公式-经典的结果jantzen，最近的工作安德森，和一些最近的工作Jantzenon非限制表示的李代数的g-将被考虑。将寻求对称群的双重覆盖群的不可约表示的一些显式构造;通过类比一般线性群和对称群之间熟悉的相互作用，希望可以利用简单的代数Spin群来获得对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.