Geometric Methods in Representation Theory

表示论中的几何方法

• 批准号: 1501094
• 负责人: Shrawan Kumar
• 金额: $ 17万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-06-15 至 2018-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1501094&HistoricalAwards=false
• 关键词: Geometric; Methods; Representation; Theory

This research is at the interface of geometry and representation theory. Geometry is the mathematical study of shapes and properties of space, and representation theory uses algebraic techniques to study symmetries. Several research projects in geometry and representation theory that enhance the unity of mathematics by connecting algebraic and geometric ideas will be completed. Among them is a project to use a novel representation-theoretic approach to solve a long-standing conjecture about counting the number of Latin squares.In more detail, the proposed projects are summarized as follows: The first project is to answer of a well-known question of Kostant to describe the decomposition of the tensor product of two irreducible representations of a semisimple Lie algebra with the same highest weight; the second project is to extend the celebrated Parthasarathy-Ranga Rao-Varadarajan theorem to a new setting; the third project is to find an analog for more general groups of the classical fact about the general linear group that the cup product structure coefficients in the cohomology of Grassmannians coincide with the Littlewood-Richardson coefficients for tensor products of representations; and the fourth project is to solve a combinatorial conjecture of Alon and Tarsi counting the number of Latin squares of a certain type.

这项研究处于几何学和表示论的交叉领域。几何是对空间形状和属性的数学研究，表示论使用代数技术来研究对称性。将完成几何和表示论方面的几个研究项目，通过连接代数和几何思想来增强数学的统一性。其中一个项目是使用一种新颖的表示理论方法来解决一个长期存在的关于计算拉丁方数的猜想。更详细地说，所提出的项目总结如下：第一个项目是回答一个很好的问题Kostant 的已知问题，描述具有相同最高权重的半单李代数的两个不可约表示的张量积的分解；第二个项目是将著名的 Parthasarathy-Ranga Rao-Varadarajan 定理扩展到新的环境；第三个项目是找到关于一般线性群的经典事实的更一般群的类比，即格拉斯曼上同调中的杯积结构系数与表示张量积的利特伍德-理查森系数一致；第四个项目是解决Alon和Tarsi计算某种类型拉丁方数的组合猜想。