Seamus Albion: A modular $(q,t)$-Nekrasov–Okounkov formula

A modular $(q,t)$-Nekrasov–Okounkov formula

Seamus Albion (IMFM)

Abstract: The Nekrasov–Okounkov formula gives a beautiful expansion for an arbitrary complex power of the Dedekind eta function as a sum over integer partitions weighted by a certain product of hook-lengths. It has afforded many refinements and generalisations, including Han's modular analogue involving hook-lengths divisible by a fixed positive integer $r$, and a $(q,t)$-analogue conjectured by Hausel and Rodriguez Villegas and proved indepdendently by Rains and Warnaar and Carlsson and Rodriguez Villegas. I will discuss a combination of these two generalisations in the form of a modular $(q,t)$-Nekrasov–Okounkov formula, which was originally conjectured by Walsh and Warnaar. The proof of this identity is based on the theory of wreath Macdonald polynomials, a multisymmetric function generalisation of the ordinary Macdonald polynomials. This is joint work with Joshua Jeishing Wen (University of Vienna).

Please note that the seminar will be held in room 3.06 (Jadranska 21).