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.

该奖项为研究仿期和双播hecke代数的项目提供了资金。后一个代数（主要研究者最近引入）为代表理论，特殊功能，谐波分析，保形场理论，组合学，拓扑和数字理论的各种问题的新方法提供了新的方法。 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椭圆类型，kac-moody代数，基质模型和W-代数，以及（5）谐波分析的应用。 这项研究位于组合学的一般领域。组合学试图找到有效的方法来研究如何安排对象的离散收集。 离散系统的行为对于现代通信非常重要。例如，大型网络的设计，例如在电话系统中发生的网络，以及计算机科学中算法的设计与离散对象集有关，这利用了组合研究。这项研究还涉及正交多项式，这对于设计光传输线很有用。