Wreath Macdonald polynomials as eigenstates

Joshua Jeishing Wen (University of Illinois at Urbana-Champaign)

Wednesday 2nd October 16:00-17:00 Maths 311B

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. The result is 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'.

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