Towards wreath Macdonald theory

Abstract: Wreath Macdonald polynomials are generalizations of Macdonald polynomials wherein the symmetric groups are replaced with their wreath products with a cyclic group of order $\ell$. They were defined by Haiman, and mirroring the usual Macdonald theory, it is not obvious that they exist. Haiman also conjectured for them a generalization of his celebrated proof of Macdonald positivity where the Hilbert scheme of points on the plane is replaced with certain cyclic Nakajima quiver varieties. This conjecture was proven by Bezrukavnikov and Finkelberg, which also implies the existence of the polynomials. Analogues of standard formulas and results of usual Macdonald theory remain to be explored. I will present an approach to the study of the wreath variants via the quantum toroidal algebra of $\mathfrak{sl}_\ell$, generalizing the fruitful interactions between the usual Macdonald theory and the quantum toroidal algebra of $\mathfrak{gl}_1$. As applications, I'll present an analogue of the norm formula and a conjectural path towards "wreath Macdonald operators" that makes contact with the spin Ruijsenaars-Schneider integrable system.

mathematical physicsalgebraic geometrydifferential geometrygeometric topologyoperator algebrasrepresentation theorysymplectic geometry

Audience: researchers in the topic

Geometry, Physics, and Representation Theory Seminar

Series comments: If you would like to receive announcements, please join our mailing list here: listserv.neu.edu/cgi-bin/wa?SUBED1=GPRT-SEMINAR&A=1