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

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 $\tilde{A}$ 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 $\tilde{C}$ 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 $\tilde{C}$, $\tilde{B}$, 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.

33 pages, 13 figures. This is the long version of the extended abstract accepted for FPSAC conference

References in corpus (1)

• Hall-Littlewood polynomials and characters of affine Lie algebras

Cited by in corpus (2)

• Polynomiality of some hook-content summations for doubled distinct and self-conjugate partitions
• Weyl-Kac character formula for affine Lie algebra in Deligne's category