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
