Combinatorics of Koornwinder polynomials and stable double affine Hecke algebras

这个项目的研究将集中在表征理论中出现的结构的组合方面。组合数学的力量在于它能够阐明和澄清表示的结构。这种现象的一个基本但指导性的例子是著名的公式表示舒尔多项式作为加权和半标准杨tableaux。在这个项目中，PI将研究Koornwinder多项式，双仿射Hecke代数（DAHA）及其大秩稳定化。该项目的一个重要主题是Koornwinder多项式的普遍性。这有两个方面。首先，通过参数的特殊化，Koornwinder多项式投影到所有经典和混合仿射类型的Macdonald多项式。拟议的研究的目的是证明类似物在Koornwinder的情况下，著名的A型麦克唐纳多项式的结果;专业化，然后将这些结果投影到经典和混合类型。一个例子是A型麦克唐纳多项式的积分性质，它与组合数学和几何学有着深刻的联系。普适性的第二个方面是大秩稳定化，这涉及到Koornwinder多项式的创建算子的应用，变形普适特征的专门化，环面结多项式的对偶和稳定化，稳定DAHA的规范基，以及经典类型的椭圆Kostka多项式。通过研究生作为研究助理的直接参与对他们进行培训是该项目的一个重要组成部分。该项目还将涉及大量的符号计算。所有由此产生的软件将在开源软件系统Sage中公开提供。PI将开发Koornwinder类型的球形DAHA的稳定（大秩）极限，并使用这项研究Koornwinder多项式的组合学及其对称函数的提升。目的是证明稳定Koornwinder DAHA的基础成果，并将这些成果应用于解决以下领域的问题和突出问题：- 用于Koornwinder多项式的创建运算符，应用于Rains对Koornwinder对称函数的完整性猜想-非约化仿射型DAHA环面结多项式的对偶性和超越A型的有理洗牌猜想的推广-Koornwinder多项式的组合公式的精神和这种公式对经典仿射根系统的专门化-Koornwinder对称函数对q的专门化-变形泛特征标.由稳定DAHA产生的对称函数算子的显式公式

The research of this project will focus on combinatorial aspects of structures arising in representation theory. The power of combinatorics lies in its ability to illuminate and clarify the structure of representations. A basic yet guiding example of this phenomenon is the famous formula expressing a Schur polynomial as a weighted sum over semi-standard Young tableaux. In this project, the PIs will study Koornwinder polynomials, double affine Hecke algebras (DAHAs), and their large-rank stabilizations. An important theme of the project is the universality of Koornwinder polynomials. This has two facets. First, by specialization of parameters, the Koornwinder polynomials project to the Macdonald polynomials of all classical and mixed affine types. The proposed research aims to prove analogues in the Koornwinder case of celebrated results on type A Macdonald polynomials; specialization then projects such results onto classical and mixed types. An example is the integrality property of type A Macdonald polynomials, which has deep connections to combinatorics and geometry. The second facet of universality is that of large-rank stabilization.This involves applications to creation operators for the Koornwinder polynomials, specializations to deformed universal characters, duality and stabilization of torus knot polynomials, canonical bases of stable DAHA, and elliptic Kostka polynomials for classical types. The training of graduate students, through their direct involvement as research assistants, is an important component of the project. The project will also involve substantial symbolic computation. All resulting software will be made publicly available in the open-source software system Sage.The PIs will develop a stable (large rank) limit of the spherical DAHA of Koornwinder type and use this study the combinatorics of Koornwinder polynomials and their lifts to symmetric functions. The aim is to prove foundational results on the stable Koornwinder DAHA and apply these results to solve problems and outstanding conjectures in the following areas:- Creation operators for Koornwinder polynomials, with applications to Rains' integrality conjecture for Koornwinder symmetric functions- Duality conjectures for DAHA torus knot polynomials of non-reduced affine type and generalizations of the rational shuffle conjecture beyond type A- Combinatorial formulas for Koornwinder polynomials in the spirit and specializations of such formulas to classical affine root systems- Specializations of Koornwinder symmetric functions to q-deformed universal characters- Explicit formulas for symmetric function operators arising from the stable DAHA