A Nekrasov-Okounkov type formula for type C

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 $\tilde{A}$. 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 $\tilde{C}$ and a new bijection. As applications, we derive a symplectic hook formula and a relation between Macdonald's identities in types $\tilde{C}$, $\tilde{B}$, and $\widetilde{BC}$.

12 pages, 5 figures. This is an extended abstract, to appear in DMTCS proceedings