BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Joshua Wen (Northeastern University) DTSTART:20210128T195000Z DTEND:20210128T205000Z DTSTAMP:20260423T005854Z UID:GPRTatNU/14 DESCRIPTION:Title: Towards wreath Macdonald theory\nby Joshua Wen (Northeastern Univers ity) as part of Geometry\, Physics\, and Representation Theory Seminar\n\n \nAbstract\nWreath Macdonald polynomials are generalizations of Macdonald polynomials wherein the symmetric groups are replaced with their wreath pr oducts with a cyclic group of order $\\ell$. They were defined by Haiman\, and mirroring the usual Macdonald theory\, it is not obvious that they ex ist. Haiman also conjectured for them a generalization of his celebrated p roof of Macdonald positivity where the Hilbert scheme of points on the pla ne is replaced with certain cyclic Nakajima quiver varieties. This conject ure was proven by Bezrukavnikov and Finkelberg\, which also implies the ex istence of the polynomials. Analogues of standard formulas and results of usual Macdonald theory remain to be explored. I will present an approach t o the study of the wreath variants via the quantum toroidal algebra of $\\ mathfrak{sl}_\\ell$\, generalizing the fruitful interactions between the u sual Macdonald theory and the quantum toroidal algebra of $\\mathfrak{gl}_ 1$. As applications\, I'll present an analogue of the norm formula and a c onjectural path towards "wreath Macdonald operators" that makes contact wi th the spin Ruijsenaars-Schneider integrable system.\n LOCATION:https://researchseminars.org/talk/GPRTatNU/14/ END:VEVENT END:VCALENDAR