Geometric Methods in Representation Theory

本研究是几何与表征理论的结合。几何学是对空间形状和性质的数学研究，表征理论使用代数技术来研究对称性。完成几何学和表示理论方面的若干研究项目，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数拉丁平方数的猜想。更详细地说，提出的方案总结如下：第一个方案是回答Kostant的一个著名问题，描述具有相同最高权值的半简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parthasarthy - ranga Rao-Varadarajan定理扩展到一个新的环境；第三个项目是寻找关于一般线性群的经典事实的更一般群的类比，即grassmannian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的组合猜想，计算某种类型的拉丁平方的数量。

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.