Combinatorics of Young Tableaux and their Generalizations

Young Tableaux 的组合学及其概括

基本信息

批准号：
0405948
负责人：
Peter Magyar
金额：
$ 8.75万
依托单位：
Michigan State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2004
资助国家：
美国
起止时间：
2004-06-15 至 2007-12-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405948&HistoricalAwards=false
关键词：
Combinatorics Young Tableaux their Generalizations

项目摘要

DMS-0405948Peter M. MagyarThe investigator will examine combinatorial problems related to representations of Lie groups, both finite and infinite-dimensional. The aim is to gain an explicit combinatorial description of the representations, and conversely to use representation theory to answer combinatorial questions. First he will work in Sperner Theory, which examines methods to organize sets and numbers in ``non-interfering'' ways. The investigator will use a new connection with crystal bases of quantum group representations to address combinatorial problems involving Symmetric Chain Decomposition of posets. Next, he aims to apply combinatorics, specifically the Littelmann and Kyoto path models, to control and clarify the algebraic structure of loop group representations. Finally, the investigator will explore generalizations of combinatorial and geometric representation theory to configuration varieties, multiple flag varieties, and quiver representations.Representation theory is the theory of symmetric objects. Since symmetry is usually the key to obtaining exact solutions in complicated situations, representation theory is one of the most powerful tools in mathematics, physics, and chemistry. The investigator's work seeks to describe various kinds of symmetry in explicit combinatorial terms. Put simply, the aim is to understand the objects in n-dimensional (and infinite-dimensional) space, which, like the circle or sphere, have a large, continuous family of symmetries, and to encode this symmetry in discrete combinatorics.

DMS-0405948Peter M. Magyar 研究人员将检查与有限维和无限维李群表示相关的组合问题。目的是获得表示的明确组合描述，并反过来使用表示理论来回答组合问题。首先，他将研究斯佩纳理论，该理论研究以“互不干扰”的方式组织集合和数字的方法。研究人员将使用与量子群表示的晶体基的新连接来解决涉及偏序集对称链分解的组合问题。接下来，他的目标是应用组合学，特别是 Littelmann 和京都路径模型，来控制和阐明环群表示的代数结构。最后，研究者将探索组合和几何表示理论对配置变量、多旗变量和箭袋表示的推广。表示理论是对称对象的理论。由于对称性通常是在复杂情况下获得精确解的关键，因此表示论是数学、物理和化学领域最强大的工具之一。研究人员的工作旨在以明确的组合术语描述各种对称性。简而言之，目标是理解 n 维（和无限维）空间中的对象，这些对象与圆或球体一样，具有大量连续的对称族，并将这种对称性编码为离散组合。