Wreath Macdonald polynomials as eigenstates

TL;DR

This paper demonstrates that wreath Macdonald polynomials serve as eigenstates of a certain algebraic structure within quantum algebra, providing new proofs and insights into their properties using shuffle algebra techniques.

Contribution

It establishes wreath Macdonald polynomials as eigenstates in the quantum toroidal algebra context and offers a novel proof of their existence.

Findings

Wreath Macdonald polynomials diagonalize the horizontal Heisenberg subalgebra.

The proof utilizes shuffle algebra methods.

A new proof of the existence of wreath Macdonald polynomials is provided.

Abstract

We show that the wreath Macdonald polynomials for $\mathbb{Z}/\ell\mathbb{Z}\wr\Sigma_n$, when naturally viewed as elements in the vertex representation of the quantum toroidal algebra $U_{\mathfrak{q},\mathfrak{d}}(\ddot{\mathfrak{sl}}_\ell)$, diagonalize its horizontal Heisenberg subalgebra. Our proof makes heavy use of shuffle algebra methods, and we also obtain a new proof of existence of wreath Macdonald polynomials.