The Combinatorics of Macdonald Polynomials and Symmetric Function Operators

对称函数在代数学中一直很重要，近年来在代数几何和数学物理、代数组合学、统计力学和表示论等许多数学领域中发挥着越来越重要的作用。在数学和科学中有许多多项式的例子，它们依赖于多个变量，并且具有重要的应用。在这个项目中，将研究与这些多项式相关的组合数学。麦克唐纳多项式，是满足正交关系的多变量对称函数，在代数组合学中起着核心作用，并应用于特殊函数，代数几何和统计力学。最近，麦克唐纳多项式出现在数学物理学和与弦理论相关的纽结不变量的研究中。最近在这些物体的组合学上取得了很多进展，特别是对于被称为环面结的特殊结，但许多重要的问题仍然悬而未决。这些问题涉及到麦克唐纳多项式和作用于它们的各种算子，本项目将进一步发展麦克唐纳多项式及其算子的理论。在未来的几年里，Macdonald多项式在数学物理、表示论和组合数学中的作用似乎越来越大，因此，从这个项目中产生的关于Macdonald多项式及其算子的结果可能会应用于数学的许多领域。Macdonald多项式依赖于一组变量、一个分区和两个参数，与希尔伯特方案的研究密切相关，在纽结不变量和特征标公式的研究中产生。这个项目旨在建立在以前的结果上的组合的麦克唐纳多项式和字符公式描述通过运营商适用于麦克唐纳多项式。一个特别重要的例子是对角谐波的性质。最近证明的“洗牌定理”给出了这个特征的一个很好的组合表达式。这个项目的一个中心焦点是研究最近的一个推广的洗牌猜想称为德尔塔猜想。这个猜想在组合数学中的一些有趣的应用已经被发现，证明将大大扩展我们对麦克唐纳多项式的理解，以及与希尔伯特方案和结不变量相关的特征公式。另外两个项目涉及与对角谐波连接的组合对象，如Tesler矩阵和LLT多项式（由Alain Lascoux，Bernard Leclerc和Jean-Yves Thibon引入的对称函数家族），以及Demazure字符和原子的研究。这最后三个项目将涉及更直接的组合分析和经典的对称函数理论。

Symmetric functions, long important in algebra, are playing an increasing role in recent years in many other areas of mathematics including algebraic geometry and mathematical physics, algebraic combinatorics, statistical mechanics and representation theory. There are many examples in mathematics and science of polynomials which depend on several variables, and which have important applications. In this project the combinatorics associated with these polynomials will be investigated. Macdonald polynomials, which are multi-variate symmetric functions which satisfy an orthogonality relation and play a central role in algebraic combinatorics with applications to special functions, algebraic geometry, and statistical mechanics. Recently, Macdonald polynomials have arisen in mathematical physics and the study of knot invariants connected to string theory. There has been a lot of recent progress on the combinatorics of these objects, especially for special knots known as torus knots, but many important questions remain open. These questions involve both Macdonald polynomials and various operators which act on them, and this project will further develop the theory of Macdonald polynomials and their operators. It seems likely that Macdonald polynomials will play an increasing role in mathematical physics, representation theory, and combinatorics over the coming years, so results on Macdonald polynomials and their operators arising from this project will likely have applications to many areas of mathematics.Macdonald polynomials, which depend on a set of variables, a partition, and two parameters, are closely connected to the study of the Hilbert scheme, and arise in the study of knot invariants and character formulas. This project seeks to build on previous results on the combinatorics of Macdonald polynomials and character formulas described via operators applied to a Macdonald polynomial. A particularly important example is the character of diagonal harmonics. The recently proved ''shuffle conjecture'' gives a nice combinatorial expression for this character. A central focus of this project is to study a recent generalization of the shuffle conjecture called the Delta conjecture. A number of interesting applications of this conjecture to combinatorics have already been found, and a proof would significantly expand our understanding of Macdonald polynomials, and character formulas connected to the Hilbert scheme and knot invariants. Two other projects involve combinatorial objects connected to diagonal harmonics such as Tesler matrices and LLT polynomials (a family of symmetric functions introduced by Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon), as well as the study of Demazure characters and atoms. These last three projects will involve more direct combinatorial analysis and classical symmetric function theory.