Combinatorial Methods in Geometric Representation Theory

几何表示理论中的组合方法

• 批准号: 1248171
• 负责人: Julianna Tymoczko
• 金额: $ 13.97万
• 依托单位: Smith College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-01-01 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1248171&HistoricalAwards=false
• 关键词: Combinatorial; Methods; Geometric; Representation; Theory

Representation theorists seek to quantify, analyze, and predict how a group of motions---like rotations or reflections---act on physical and geometric objects. The fundamental problem in geometric representation theory is to construct representations using geometric tools. The fundamental problem in combinatorial representation theory is to understand a representation explicitly in terms of essential combinatorial parameters: for instance to decompose a given representation into the fundamental building blocks of irreducible representations. Both kinds of representation theorists seek to understand representations, but a problem that may seem answered to geometric representation theorists, such as `find a natural representation', is only partially answered to combinatorial representation theorists, who seek to `analyze explicitly a given representation'. This proposal addresses several problems in geometric and combinatorial representation theory, using tools from combinatorics, algebraic geometry, and topology. The broader impact includes work in many other areas of mathematics, including knot theory and commutative algebra, as well as fields like mathematical physics. Educationally, the PI will establish programs for graduate student retention and excellence, in both research and teaching.

代表理论家寻求量化，分析和预测一组动作（例如旋转或反射）如何对物理和几何对象作用。 几何表示理论中的基本问题是使用几何工具构建表示。组合表示理论中的基本问题是要从基本组合参数方面明确理解一种表示：例如，将给定表示形式分解为不可减至表示的基本构建块。两种代表理论家都试图理解表示形式，但是似乎回答了几何代表理论家（例如“找到自然代表”）的问题只是对组合代表理论家的部分答复，他们试图“明确地分析给定的代表”。该建议使用组合学，代数几何学和拓扑的工具解决了几何和组合表示理论中的几个问题。 更广泛的影响包括在许多其他数学领域的工作，包括结理论和交换代数，以及数学物理等领域。 在教育上，PI将在研究和教学中建立用于研究生保留和卓越的计划。