Tesler identities for wreath Macdonald polynomials

TL;DR

The paper extends the Tesler identity and related reciprocity phenomena from classical Macdonald theory to wreath Macdonald polynomials indexed by r-colors. It constructs an explicit operator $\\mathsf V$, built from wreath nabla operators, the Omega kernel, and a translation, that converts a wreath Macdonald polynomial into a delta-function delta at the corresponding core-parameter, via $\\mathbb E_\\lambda$. Central to the approach are Delta and $\\Delta^\\dagger$ operators and the series $\\mathbb D$ (and its adjoint), realized through quantum toroidal and shuffle algebra frameworks, which yield eigenoperators for $H_\\lambda$ and its dual. The work proves wreath analogues of Macdonald–Koornwinder duality, derives evaluation formulas, and furnishes a plethystic formula for wreath $(q,t)$-Kostka coefficients, alongside shifted and star variants, all anchored by a robust delta-function calculus. The results illuminate reciprocity and duality in the wreath setting, broadening the reach of Macdonald theory to the finite group wreath products via core–quotient combinatorics and color-graded operator algebras, with potential links to interpolation Macdonald polynomials and bispectral problems. The methods harmonize eigenoperator techniques from quantum toroidal algebras with wreath-specific combinatorics, yielding structured, explicit constructions that generalize foundational type A results to the wreath context.

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.

Figures (3)

• Figure 1: The Maya diagram for the partition $(4,2,2)$.
• Figure 2: The quotient decomposition for $\lambda = (4,2,2)$ when $r=3$.
• Figure 3: Two origin=c]180L-shaped segments that avoid the addable boxes of $\alpha$. The black squares are the $\chi$-poles while the chains of $t^{-1}$- and $q^{-1}$-poles are indicated by the gray squares. If a segment starting at $z_i$ has $\chi$-pole northwest of the segment ending at $z_{i+1}$, then $\alpha$ has a removable ribbon of length $nr$ for some $n$, as illustrated by the dashed line.

Theorems & Definitions (57)

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• Remark 2.10
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• Remark 2.13
• Remark 3.4
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• Remark 3.12
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