The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications

TL;DR

This paper refines and extends the Nekrasov-Okounkov hook length formula, providing an elementary proof and new applications, including a marked hook formula, by exploring properties of t-cores and using Macdonald identities.

Contribution

It introduces a refined and extended version of the hook length formula with a new proof method and applications, connecting Macdonald identities and t-core generating functions.

Findings

Derived an expansion formula for powers of the Euler Product in terms of hook lengths.

Provided a new property of t-cores leading to a refinement.

Established applications including the marked hook formula.

Abstract

The paper is devoted to the derivation of the expansion formula for the powers of the Euler Product in terms of partition hook lengths, discovered by Nekrasov and Okounkov in their study of the Seiberg-Witten Theory. We provide a refinement based on a new property of t-cores, and give an elementary proof by using the Macdonald identities. We also obtain an extension by adding two more parameters, which appears to be a discrete interpolation between the Macdonald identities and the generating function for t-cores. Several applications are derived, including the "marked hook formula".