Wreath Macdonald polynomials as eigenstates

Geometry, Symmetry and Physics
Event time: 
Thursday, May 9, 2019 - 11:00am
LOM 215
Joshua Wen
Speaker affiliation: 
University of Illinois Urbana-Champaign
Event description: 

Wreath Macdonald polynomials were defined by Haiman as generalizations of transformed Macdonald polynomials from the symmetric groups to 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 torus-fixed points of cyclic quiver varieties, much like how Haiman’s proof of Macdonald positivity assigns Macdonald polynomials to torus-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 a proof that in this setting, the wreath Macdonald polynomials diagonalize the horizontal Heisenberg subalgebra of the quantum toroidal algebra.