Mathematical Sciences: Invariant Theory and Representation Theory

该奖项支持不变量理论和表示理论的六个项目。其中四个项目集中在不变量理论中的问题，与对偶对和“多重性自由作用法”有关。他们都关心的是扩大例子的范围，在这些例子中，人们可以有效地计算感兴趣的量。具体地说，它们涉及张量积、权空间、多面体、计算某些多面体中的格点、广义球谐和表示限制中的多重性-1现象。第五个项目涉及实约化群主级数表示的显式结构的计算。第六个项目是研究p-进群上同余Hecke代数的结构及其对p-进表示理论中极小K型存在的启示。研究的支持涉及有限群的表示理论。群是用于研究变换的代数对象。正因为如此，群体是物理、化学和计算机科学以及数学中的基本工具。表象理论是确定群结构的重要方法。

This award supports six projects in invariant theory and representation theory. Four of these projects are focussed on questions in invariant theory, related to the dual pairs and the "method of multiplicity-free actions". They all are concerned with extending the range of examples in which one can effectively compute quantities of interest. Specifically, they are concerned with tensor products, weight spaces, plethysms, counting lattice points in certain polyhedra, generalized spherical harmonics, and multiplicity-one phenomena in restrictions of representations. A fifth project concerns the computation of explicit structure of principal series representations of real reductive groups. A sixth project involves the study of structure in congruence Hecke algebras on p-adic groups and its implications for the existence of minimal K-types in p-adic representation theory. The research support concerns the representation theory of finite groups. A group is an algebraic object used to study transformations. Because of this, group are a fundamental tool in physics, chemistry and computer science as well as mathematics. Representation theory is an important method foe determining the structure of groups.