Mathematical Sciences: Invariant Theory and Applications to Representation Theory

摘要豪9622916 该项目的主要方面是 研究与多重性相关的不变量理论- 自由作用和旗流形上的作用。提议人的工作 与J. Horvath描述各向同性组的作用GLn上 将使用与驯服颤动相关的旗流形的乘积 来研究在与 一些低维可约作用的不变论第一基本定理的证明 的多重性自由行动将被调查。一个主要的目标将是调查某些环的重要了解- 张量积及相关问题。这在一些国家已经得到证明。 在某些情况下，并怀疑在其他情况下，这些环可能是免费的， 自然多项式子环上的模的描述 这些免费模块的良好基础可以提供对 Littlewood-Richardson规则及相关公式如果时间允许， 几何不变量的普通几何输入的研究 理论，并通过Hecke的p进群的Plancherel定理 代数，也可以进行。 摘要：在数学中，人们通常对以下内容感兴趣： 对象可能不同，但对于某些 目的因此，在初等几何中， 全等：两个三角形，或其他图形，是全等的，如果一个 可以移动以便正好适合另一个。在工业 在上下文中，一致性是可互换部件的基础思想。 不变量理论的思想是为下列问题提供数值判据： 两个对象是等价的，相对于什么是 等价的相关概念。因此，两个三角形全等 当它们的边长是三倍时， 号码可用于确定是否 两个等价的对象称为 *invariants（it）*。不变 理论试图找到几何构型的不变量。这 该项目旨在将不变理论应用于新的方向。它将寻找一些新的配置类的不变量系统， 它还将尝试在一些最重要的 重要的不变量系统。对称性将被大量利用， 而那些结构强烈地由对称性决定的系统将成为研究的中心对象。

Abstract Howe 9622916 The primary aspect of this project is to investigate aspects of invariant theory related to multiplicity- free actions, and actions on flag manifolds. Work of the proposer with J. Horvath describing isotropy groups of actions of GLn on products of flag manifolds associated to tame quivers will be used to study the occurrence of dense orbits in actions associated to wild quivers.Instances of the First Fundamental Theorem of Invariant Theory for some low dimensional reducible actions analyzable by means of multiplicity-free actions will be investigated. A major goal will be an investigation of certain rings important for understan- ding tensor products and related questions. It has been proved in some cases, and is suspected in others, that these rings may be free as modules over certain natural polynomial subrings. The description of nice bases for these free modules may provide insight into the Littlewood-Richardson Rule and related formulas. As time permits, investigations of the ordinary geometric import of geometric invariant theory, and of the Plancherel Theorem of p-adic groups via Hecke algebras, may also be undertaken. Expository Abstract: In mathematics, one is often interested in objects which may be different, but may be equivalent for some purposes. Thus, in elementary geometry there is the notion of congruence: two triangles, or other figures, are congruent if one may be moved so as to fit exactly over the other. In an industrial context, congruence is the idea underlying interchangeable parts. The idea of invariant theory is to provide numerical criteria for two objects to be equivalent, with respect to whatever is the relevant notion of equivalence. Thus, two triangles are congruent exactly when the lengths of their sides are the same triple of numbers. The numbers which may be used to determine whether or not two objects are equivalent are called *invariants(it)*. Invariant theory seeks to f ind invariants for geometric configurations. This project seeks to apply invariant theory in new directions. It will look for systems of invariants for some new classes of configurations, and it will also attempt to find some new structure in some of the most important systems of invariants. Symmetry will be heavily exploited, and systems whose structure is very strongly determined by symmetries will form the central objects of investigation.