A Nekrasov-Okounkov type formula for type C

TL;DR

This paper extends Nekrasov-Okounkov type formulas for powers of the Dedekind eta function to type C, using Macdonald's identity and combinatorial bijections, revealing new formulas and relations among types.

Contribution

It introduces a new combinatorial expansion of eta powers for type C using Macdonald's identity and a novel bijection, expanding the understanding of these identities.

Findings

Derived a symplectic hook formula.

Established relations between Macdonald's identities in types C, B, and BC.

Provided new combinatorial expansions of eta powers.

Abstract

In 2008, Han rediscovered an expansion of powers of Dedekind $\eta$ function due to Nekrasov and Okounkov by using Macdonald's identity in type $\widetilde{A}$. In this paper, we obtain new combinatorial expansions of powers of $\eta$, in terms of partition hook lengths, by using Macdonald's identity in type $\widetilde{C}$ and a new bijection. As applications, we derive a symplectic hook formula and a relation between Macdonald's identities in types $\widetilde{C}$, $\widetilde{B}$, and $\widetilde{BC}$.