Macdonald Polynomials, Diagonal Harmonics, and the Geometry of Hilbert Schemes

麦克唐纳多项式、对角调和和希尔伯特方案的几何

• 批准号: 0070772
• 负责人: Mark Haiman
• 金额: $ 12.3万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2002-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070772&HistoricalAwards=false
• 关键词: Macdonald; Polynomials; Diagonal; Harmonics; Geometry

Abstract: Macdonald polynomials, diagonal harmonics, and the geometryof Hilbert schemes.Professor Haiman is working to complete the proof of a series of conjectures involving Macdonald polynomials, the so-called "n-factorial" conjecture, and the character formula for diagonal harmonics. The methods involve a detailed algebraic geometrical study of the Hilbert scheme of points in the plane and related algebraic varieties. In earlier work Professor Haiman showed that certain hypotheses on the singularities and sheaf cohomology of these varieties would imply the desired algebraic results (and with Garsia, showed that the latter imply related combinatorial results). The current work is to establish these geometric hypotheses.Macdonald polynomials are a new family of symmetric functions. Their discovery by Macdonald in 1988 was a surprising development in the theory of symmetric functions, which is a fundamental and classical part of mathematics with roots in the work of Euler, Jacobi, and Cauchy over a century ago. Macdonald polynomials have since been found to have important applications in a wide range of areas including geometry, representation theory, and even theoretical physics. At the time of their discovery, Macdonald conjectured that certain coefficients associated with his polynomials should be positive integers, the proof of which remains the most important unsolved problem in this area. The successful completion of this project will solve this problem, proving the "Macdonald positivity conjecture," along with a related representation-theoretic conjecture of Garsia and the investigator known as the "n-factorial" conjecture, some related combinatorial conjectures, and strong new geometric properties of Hilbert schemes, which are likely to have further applications in geometry and representation theory.

摘要：MacDonald多项式，对角线谐波和Hilbert Schemes.Professor Haiman正在努力完成一系列涉及MacDonald多项式的猜想的证明，涉及MacDonald多项式，即所谓的“ N因素”的猜想，以及对角色谐波的特征。 该方法涉及对平面中点及相关代数品种的希尔伯特方案的详细代数几何研究。 在较早的工作中，海曼教授表明，这些品种的奇异性和捆绑共同体学的某些假设将暗示所需的代数结果（并且与Garsia一起表明，后者暗示着相关的组合结果）。 当前的工作是建立这些几何假设。麦克唐纳德多项式是对称函数的新家族。 他们在1988年在麦克唐纳（MacDonald）在对称函数理论中发现了一个令人惊讶的发展，这是数学的基本和古典部分，在一个世纪前的欧拉（Euler），雅各比（Jacobi）和凯奇（Cauchy）的工作中起源于一世纪。 此后，麦当劳多项式在包括几何形状，表示理论甚至理论物理学在内的广泛领域具有重要应用。 在发现时，麦克唐纳猜想与他的多项式相关的某些系数应该是积极的整数，其证明仍然是该领域中最重要的未解决问题。 该项目的成功完成将解决这个问题，证明了“麦克唐纳的积极性猜想”，以及相关的GARSIA理论猜想以及被称为“ N因素”猜想的研究者，一些相关的组合构想，以及Hilbert Schemes的强大新几何特性，这可能进一步构成GEOMENT和GEOMENTINCTACTIADS CONDICTACTIONS PEROMENTINCTACTIANS CONDITS CONDITS CONDATINCTIADS和代表。