A Nekrasov-Okounkov Type formula for $\widetilde{C}$

TL;DR

This paper extends combinatorial formulas for powers of the Dedekind eta function using type c4; Macdonald identities, revealing new relations and generating functions for partitions.

Contribution

It introduces new combinatorial expansions of eta powers via type c4; Macdonald identities and a novel bijection, expanding understanding of partition hook lengths.

Findings

Derived a symplectic hook formula

Established relations between Macdonald identities in types c4, b2, and bc

Produced new weighted generating functions for partitions

Abstract

In 2008, Han rediscovered an expansion of powers of Dedekind $\eta$ function attributed to Nekrasov and Okounkov (which was actually first proved the same year by Westbury) by using a famous identity of Macdonald in the framework of type $\widetilde{A}$ affine root systems. In this paper, we obtain new combinatorial expansions of powers of $\eta$, in terms of partition hook lengths, by using the Macdonald identity in type $\widetilde{C}$ and a new bijection between vectors with integral coordinates and a subset of $t$-cores for integer partitions. As applications, we derive a symplectic hook formula and an unexpected relation between the Macdonald identities in types $\widetilde{C}$, $\widetilde{B}$, and $\widetilde{BC}$. We also generalize these expansions through the Littlewood decomposition and deduce in particular many new weighted generating functions for subsets of integer partitions and refinements of hook formulas.