Mathematical Sciences: Invariant Theory and Applications to Representation Theory

摘要Howe 9622916这个项目的主要方面是研究与无重数动作和旗形上的动作相关的不变量理论的各个方面。作者与J.Horvath一起描述了Gln在与驯服箭图相关的旗流形乘积上的各向同性作用群的工作，将被用来研究与野生问题有关的作用中密集轨道的出现。对于一些可用多重数自由作用分析的低维可约作用，将研究不变理论第一基本定理的实例。一个主要目标将是研究某些对理解张量积和相关问题很重要的环。在某些情况下，这些环可以作为某些自然多项式子环上的模是自由的，这在某些情况下已被证明，也在另一些情况下被怀疑。对这些自由模的NICE基的描述可以提供对Littlewood-Richardson规则和相关公式的深入了解。在时间允许的情况下，还可以研究几何不变理论的普通几何意义，以及通过Hecke代数研究p-adic群的Plcherel定理。说明性摘要：在数学中，人们经常对一些对象感兴趣，这些对象可能是不同的，但在某些目的上可能是等价的。因此，在初等几何中有同余的概念：两个三角形或其他图形，如果一个可以移动到恰好覆盖另一个的位置，那么它们是全等的。在工业环境中，一致性是可互换部件背后的理念。不变量理论的思想是为两个对象提供数字标准，使其相对于任何相关的等价概念都是等价的。因此，当两个三角形的边的长度完全相同时，它们是全等的。可以用来确定两个对象是否等价的数字称为*不变量(It)*。不变量理论试图找到几何构型的不变量。这个项目试图将不变量理论应用于新的方向。它将为一些新的构型寻找不变量系统，还将试图在一些最重要的不变量系统中找到一些新的结构。对称性将被大量利用，其结构由对称性非常强烈地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.