Wreath Macdonald polynomials as eigenstates

Abstract

We show that the wreath Macdonald polynomials for $\mathbb{Z}/\ell \mathbb{Z}\wr \hat{I}{\pounds }_n$, 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.

v8, 72 pages. Fixed an error concerning the coproduct. The arguments in Section 5 were flawed and thus the section has been completely rewritten. Our new arguments now give a new proof for the existence of wreath Macdonald polynomials