A Nekrasov–Okounkov formula for Macdonald polynomials

Creators: Rains, Eric M.; Warnaar, S. Ole

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Abstract

We prove a Macdonald polynomial analogue of the celebrated Nekrasov–Okounkov hook-length formula from the theory of random partitions. As an application we obtain a proof of one of the main conjectures of Hausel and Rodriguez-Villegas from their work on mixed Hodge polynomials of the moduli space of stable Higgs bundles on Riemann surfaces.

Additional Information

© 2017 Springer Science+Business Media, LLC. Received: 18 December 2016; Accepted: 09 September 2017; First Online: 22 September 2017. Work supported by the National Science Foundation (Grant Number DMS-1001645) and the Australian Research Council. The second author is grateful to Masoud Kamgarpour for pointing out the papers of Hausel and Rodriguez-Villegas [22], and Hausel, Letellier and Rodriguez-Villegas [20, 21] on mixed Hodge polynomials, and to Dennis Stanton for helpful discussions on p-core partitions. We thank Fernando Rodriguez-Villegas for sending us a preliminary version of his paper [6] with Carlsson, which contains a different proof of Conjecture 1.1 based on the Carlsson–Nekrasov–Okounkov vertex operator [5]. We also thank Amer Iqbal for alerting us to the connection between our work and [24] and Jim Bryan for explaining the work of Waelder, which implies the elliptic Nekrasov–Okounkov formula described in the appendix. We thank the two referees for their helpful comments and corrections.

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