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

摘要：麦克唐纳多项式、对角谐波和Hilbert格式的几何。海曼教授正在努力完成一系列关于麦克唐纳多项式的猜想的证明，即所谓的“n阶乘”猜想，以及对角谐波的特征公式。这些方法包括对平面上点的希尔伯特格式和相关的代数变体进行详细的代数几何研究。在早期的工作中，海曼教授表明，关于这些变体的奇点和束上同调的某些假设将隐含期望的代数结果（并与Garsia一起表明后者隐含相关的组合结果）。目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。Macdonald在1988年的发现是对称函数理论的一个惊人发展，对称函数理论是数学的一个基础和经典部分，植根于欧拉、雅可比和柯西一个多世纪前的工作。麦克唐纳多项式已经被发现在广泛的领域有重要的应用，包括几何，表示理论，甚至理论物理。在他们发现的时候，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。这个项目的成功完成将解决这个问题，证明了“Macdonald正性猜想”，以及Garsia和研究者的一个相关的表示理论猜想，即“n阶乘”猜想，一些相关的组合猜想，以及Hilbert格式的强大的新几何性质，这些可能在几何和表示理论中有进一步的应用。

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.