Combinatorics of Koornwinder polynomials and stable double affine Hecke algebras

这个项目的研究将集中在表象理论中出现的结构的组合方面。组合学的力量在于它能够阐明和澄清表征的结构。这一现象的一个基本但具有指导性的例子是著名的公式，该公式将舒尔多项式表示为半标准Young图面上的加权和。在这个项目中，PI将研究Koornwinder多项式、双仿射Hecke代数(DAHAS)及其高阶稳定化。该项目的一个重要主题是Koornwinder多项式的普适性。这有两个方面。首先，通过参数的特殊化，将Koornwinder多项式投影到所有经典和混合仿射类型的Macdonald多项式。这项拟议的研究旨在证明在Koornwinder案例中关于A型Macdonald多项式的著名结果的相似性；然后将这些结果专门化，将这些结果投影到经典类型和混合类型上。A类Macdonald多项式的积分性质就是一个例子，它与组合学和几何有很深的联系。普适性的第二个方面是大阶稳定化，这涉及到Koornwinder多项式的生成算子的应用，变形泛特征标的特化，环面纽结多项式的对偶和稳定化，稳定Daha的标准基，以及经典类型的椭圆Kostka多项式。通过作为研究助理直接参与对研究生的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。所有得到的软件将在开源软件系统SAGE中公开可用。PI将开发出Koornwinder型球面Daha的稳定(大秩数)极限，并利用这项研究Koornwinder多项式的组合学及其对对称函数的提升。其目的是证明关于稳定的Koornwinder Daha的基本结果，并将这些结果应用于解决下列领域的问题和突出猜想：-Koornwinder多项式的生成算子，并应用于Koornwinder对称函数的RAINS积分猜想--非约化仿射类型的Daha环面结点多项式的对偶猜想和A型以外有理Shuffle猜想的推广--Koornwinder多项式的组合公式在精神上的推广和经典仿射根系的特化-Koornwinder对称函数到q-变形泛特征标的特化-由稳定Daha得到的对称函数的显式公式

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