Mathematical Sciences: Invariant Theory and Representation Theory

该奖项支持不变理论和表征理论领域的6个项目。其中四个项目集中在不变量理论中的问题，与对偶和“无多重行为方法”有关。它们都是关于扩展例子的范围，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、体积、某些多面体中的点阵点计数、广义球谐和表示限制中的多重一现象。第五个项目涉及实约化群的主级数表示的显式结构的计算。第六个项目涉及研究p进群上的同余Hecke代数的结构及其对p进表示理论中最小k型存在的意义。研究支持有限群的表示理论。群是用来研究变换的代数对象。正因为如此，群是物理、化学、计算机科学以及数学的基本工具。表征理论是确定群体结构的重要方法。

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.