Seamus Albion Ferlinc

A modular (q,t)-Nekrasov–Okounkov formula and wreath Macdonald polynomials

The Nekrasov--Okounkov formula gives a beautiful power series expansion for an arbitrary complex power of the Dedekind eta function in terms of hook-lengths of integer partitions. It has found many refinements and generalisations including a modular refinement, in which the hook-lengths are replaced by those divisible by a fixed positive integer $r$, and a $(q,t)$-analogue coming from the theory of Macdonald polynomials. I will discuss a proof of a unified modular $(q,t)$-Nekrasov--Okounkov formula originally conjectured by Walsh and Warnaar. This hinges on a new vertex operator acting on wreath Macdonald polynomials, themselves a deep generalisation of the Macdonald polynomials intimately related to modular hook lengths.
This is joint work with Joshua Jeishing Wen (University of Vienna).