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

摘要：Macdonald多项式、对角调和函数和Hilbert格式的几何。Haiman教授正在完成一系列涉及Macdonald多项式、所谓的“n阶乘”猜想和对角调和函数的特征公式的证明。 该方法涉及一个详细的代数几何研究的希尔伯特计划的点在平面和相关的代数品种。 在早期的工作教授海曼表明，某些假设的奇异性和层上同调这些品种将意味着所需的代数结果（并与加西亚，表明后者意味着相关的组合结果）。 Macdonald多项式是一类新的对称函数族。 他们的发现麦克唐纳在1988年是一个令人惊讶的发展理论的对称函数，这是一个基本的和经典的一部分数学的根源工作欧拉，雅可比，柯西在世纪前。 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，表示论，甚至理论物理。 在他们发现的时候，麦克唐纳声称与他的多项式相关的某些系数应该是正整数，这一证明仍然是这一领域最重要的未解决的问题。 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”，沿着Garsia和研究人员的相关表示论猜想，称为“n阶乘”猜想，一些相关的组合结构，以及希尔伯特方案的强大的新几何性质，这些可能在几何和表示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.