Title: Wreath-Macdonald polynomials as eigenstates
Speaker: Joshua Wen - UIUC
Abstract: Wreath Macdonald polynomials were defined by Haiman as generalizations of transformed Macdonald polynomials from the symmetric groups to their wreath products with cyclic groups of order m. In a sense, their definition was given in the hope that they would correspond to K-theoretic fixed point classes of cyclic quiver varieties, much like how Haiman's proof of Macdonald positivity assigns Macdonald polynomials to fixed points of Hilbert schemes of points on the plane. This hope was realized by Bezrukavnikov and Finkelberg, and the subject has been relatively untouched until now. I will present a first result exploring possible ties to integrable systems. Using work of Frenkel, Jing, and Wang, we can situate the wreath Macdonald polynomials in the vertex representation of the quantum toroidal algebra of sl_m. I will present the result that, in this setting, the wreath Macdonald polynomials diagonalize the horizontal Heisenberg subalgebra of the quantum toroidal algebra---a first step towards developing a notion of 'wreath Macdonald operators'.
