Modular Nekrasov-Okounkov formulas

TL;DR

This paper introduces a new variant of Littlewood's decomposition to develop modular analogues of hook-length formulas, leading to conjectures about modular versions of the Nekrasov-Okounkov formula.

Contribution

It presents a novel analogue of Littlewood's decomposition and a variant of the Han-Ji multiplication theorem for hook-length formulas.

Findings

New analogue of Littlewood's decomposition introduced

Modular analogues of hook-length formulas developed

Conjecture on a modular $q,t$-Nekrasov-Okounkov formula proposed

Abstract

Using Littlewood's map, which decomposes a partition into its $r$-core and $r$-quotient, Han and Ji have shown that many well-known hook-length formulas admit modular analogues. In this paper we present a variant of the Han-Ji `multiplication theorem' based on a new analogue of Littlewood's decomposition. We discuss several applications to hook-length formulas, one of which leads us to conjecture a modular analogue of the $q,t$-Nekrasov-Okounkov formula.