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：麦克唐纳多项式、对角调和和希尔伯特格式的几何。海曼教授正在努力完成一系列涉及麦克唐纳多项式猜想、所谓“n阶”猜想以及对角调和的特征公式的证明。 这些方法涉及平面上点的希尔伯特格式和相关代数簇的详细代数几何研究。 在早期的工作中，海曼教授表明，关于这些簇的奇点和束上同调的某些假设将暗示所需的代数结果（并且与加西亚一起，表明后者意味着相关的组合结果）。 当前的工作是建立这些几何假设。麦克唐纳多项式是一类新的对称函数。 麦克唐纳于 1988 年的发现是对称函数理论的惊人发展，对称函数理论是数学的基础和经典部分，根源于一个多世纪前欧拉、雅可比和柯西的工作。 此后人们发现麦克唐纳多项式在几何、表示论甚至理论物理等广泛领域具有重要的应用。 在他们发现时，麦克唐纳推测与他的多项式相关的某些系数应该是正整数，其证明仍然是该领域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“麦克唐纳正性猜想”，以及加西亚和研究者的相关表示论猜想，即“n阶乘”猜想，一些相关的组合猜想，以及强有力的新猜想。希尔伯特格式的几何性质，可能在几何和表示论中有进一步的应用。