Geometric Methods in Representation Theory

这项研究是在接口的几何和表示理论。几何学是对空间的形状和性质的数学研究，表示论使用代数技术来研究对称性。几何和表示论的几个研究项目，通过连接代数和几何思想来增强数学的统一性。其中一个项目是利用一种新的表示论方法解决一个长期存在的关于计算拉丁方个数的猜想，具体来说，所提出的项目概括如下：第一个项目是回答Kostant的一个著名问题，即描述半单李代数的两个不可约表示的张量积的分解，其中两个不可约表示具有相同的最高权;第二个计划是将著名的Parthasarathy-Ranga Rao-Varadarajan定理推广到一个新的背景下，第三个计划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的张量积的Littlewood-Richardson系数一致;第四个项目是解决Alon和Tarsi关于计算某类拉丁方个数的组合猜想。

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.