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.
