Ext operators for wreath Macdonald polynomials

TL;DR

This paper introduces a new wreath Macdonald polynomial analogue of a vertex operator and proves a modular Nekrasov--Okounkov formula for higher ranks, confirming a conjecture by Walsh and Warnaar.

Contribution

It develops a wreath Macdonald polynomial analogue of the Carlsson--Nekrasov--Okounkov vertex operator and proves a related modular formula for $r extgreater 3$, advancing the theory.

Findings

Established a wreath Macdonald polynomial analogue of the vertex operator.

Proved a modular $(q,t)$-Nekrasov--Okounkov formula for $r extgreater 3$.

Confirmed a conjecture by Walsh and Warnaar.

Abstract

We introduce a wreath Macdonald polynomial analogue of the Carlsson--Nekrasov--Okounkov vertex operator. As an application, we prove a modular $(q,t)$-Nekrasov--Okounkov formula for $r\ge 3$ originally conjectured by Walsh and Warnaar.