Tesler identities for wreath Macdonald polynomials

TL;DR

This paper introduces explicit operators for wreath Macdonald polynomials, enabling new results on their duality, evaluation, and interpolation, and explores their connections to bispectral problems and algebraic structures.

Contribution

It provides explicit formulas for operators on wreath Macdonald polynomials and initiates the study of wreath interpolation Macdonald polynomials, advancing understanding of their algebraic properties.

Findings

Derived a plethystic formula for wreath (q,t)-Kostka coefficients

Proved new reciprocity results including Macdonald--Koornwinder duality

Presented series solutions to the wreath Macdonald bispectral problem

Abstract

We give an explicit formula for an operator that sends a wreath Macdonald polynomial to the delta function at a character associated to its partition. This allows us to prove many new results for wreath Macdonald polynomials, especially pertaining to reciprocity: Macdonald--Koornwinder duality, evaluation formulas, etc. Additionally, we initiate the study of wreath interpolation Macdonald polynomials, derive a plethystic formula for wreath $(q,t)$-Kostka coefficients, and present series solutions to the bispectral problem involving wreath Macdonald operators. Our approach is to use the eigenoperators for wreath Macdonald polynomials that have been produced from quantum toroidal and shuffle algebras.