Double Hurwitz numbers via the infinite wedge

Abstract

We derive an algorithm to produce explicit formulas for certain generating functions of double Hurwitz numbers. These formulas generalize a formula of Goulden, Jackson and Vakil for one part double Hurwitz numbers. Immediate consequences include a new proof that double Hurwitz numbers are piecewise polynomial, an understanding of the chamber structure and wall crossing for these polynomials, and a proof of the Goulden, Jackson and Vakil's Strong Piecewise Polynomiality conjecture. The method is a straightforward application of Okounkov's expression for double Hurwitz numbers in terms of operators on the infinite wedge. We begin with a introduction to the infinite wedge tailored to our use.

25 pages, 2 figures. Comments welcome

References in corpus (3)

• Hurwitz numbers and intersections on moduli spaces of curves
• On double Hurwitz numbers in genus 0
• Lectures on the ELSV formula

Cited by in corpus (5)

• Polynomiality of Hurwitz numbers, Bouchard-Mariño conjecture, and a new proof of the ELSV formula
• Monotone Hurwitz numbers and the HCIZ integral II
• Bijections for simple and double Hurwitz numbers
• The combinatorics of real double Hurwitz numbers with real positive branch points
• Covers counting via Feynman Calculus