Hecke Algebras, MacDonald Polynomials, and Applications

赫克代数、麦克唐纳多项式及其应用

• 批准号: 9622829
• 负责人: Ivan Cherednik
• 金额: $ 6.6万
• 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: 1996
• 资助国家: 美国
• 起止时间: 1996-05-15 至 1999-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622829&HistoricalAwards=false
• 关键词: Hecke; Algebras; MacDonald; Polynomials; Applications

This award provides funds for a project to study affine and double affine Hecke algebras. The latter algebras (recently introduced by the principal investigator) lead to a new approach to various problems in representation theory, special functions, harmonic analysis, conformal field theory, combinatorics, topology, and number theory. The main objectives are (1) the theory of differential and difference operators acting on polynomials, theta functions and various generalizations, (2) representations of double affine Hecke algebras at roots of unity, connections with the monodromy of the double affine KZ equations and elliptic braid groups, (3) the action of the modular group on the Macdonal polynomials, (4) relations to quantum groups of elliptic type, Kac-Moody algebras, matrix models, and W-algebras, and (5) applications to harmonic analysis. This research is in the general area of Combinatorics. Combinatorics attempts to find efficient methods to study how discrete collections of objects can be arranged. The behavior of discrete systems is extremely important to modern communications. For example, the design of large networks, such as those occurring in telephone systems, and the design of algorithms in computer science deal with discrete sets of objects, and this makes use of combinatorial research. This research also deals with orthogonal polynomials, which are useful in designing optical transmission lines.

该奖项为研究仿射和双仿射Hecke代数的项目提供资金。后者代数（最近推出的主要调查员）导致一个新的方法来解决各种问题的代表性理论，特殊功能，谐波分析，共形场理论，组合学，拓扑学和数论。主要目的是（1）微分和差分算子作用于多项式、θ函数和各种推广的理论，（2）双仿射Hecke代数在单位根上的表示，与双仿射KZ方程和椭圆辫群的单值性的联系，（3）模群对Macdonal多项式的作用，（4）与椭圆型量子群、Kac-Moody代数、矩阵模型和W-代数的关系;（5）在调和分析中的应用。 这项研究是在组合数学的一般领域。组合数学试图找到有效的方法来研究如何安排离散的对象集合。 离散系统的行为对现代通信极为重要。例如，设计大型 网络，如电话系统中的网络，以及计算机科学中的算法设计，处理的是离散的对象集，这就利用了组合研究。本研究亦探讨正交多项式，其可用于设计光传输线。