Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 040, 10 pages      arXiv:1307.2137
Contribution to the Special Issue on Asymptotics and Universality in Random Matrices, Random Growth Processes, Integrable Systems and Statistical Physics in honor of Percy Deift and Craig Tracy

Toda Equations and Piecewise Polynomiality for Mixed Double Hurwitz Numbers

I.P. Goulden a, Mathieu Guay-Paquet b and Jonathan Novak c
a) Department of Combinatorics and Optimization, University of Waterloo, 200 University Ave. W., Waterloo, ON, N2L 3G1 Canada
b) Département de mathématiques, Université du Québec à Montréal, C.P. 8888, succ. Centre-ville, Montréal, Québec, H3C 3P8 Canada
c) Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0404 USA

Received February 02, 2016, in final form April 13, 2016; Published online April 20, 2016

This article introduces mixed double Hurwitz numbers, which interpolate combinatorially between the classical double Hurwitz numbers studied by Okounkov and the monotone double Hurwitz numbers introduced recently by Goulden, Guay-Paquet and Novak. Generalizing a result of Okounkov, we prove that a certain generating series for the mixed double Hurwitz numbers solves the 2-Toda hierarchy of partial differential equations. We also prove that the mixed double Hurwitz numbers are piecewise polynomial, thereby generalizing a result of Goulden, Jackson and Vakil.

Key words: Hurwitz numbers; Toda lattice.

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  1. Biane P., Parking functions of types A and B, Electron. J. Combin. 9 (2002), 7, 5 pages.
  2. Carrell S.R., Diagonal solutions to the 2-Toda hierarchy, Math. Res. Lett. 22 (2015), 439-465, arXiv:1109.1451.
  3. Diaconis P., Greene C., Applications of Murphy's elements, Stanford University Technical Report no. 335, 1989.
  4. Do N., Dyer A., Mathews D.V., Topological recursion and a quantum curve for monotone Hurwitz numbers, arXiv:1408.3992.
  5. Ekedahl T., Lando S., Shapiro M., Vainshtein A., Hurwitz numbers and intersections on moduli spaces of curves, Invent. Math. 146 (2001), 297-327, math.AG/0004096.
  6. Farahat H.K., Higman G., The centres of symmetric group rings, Proc. Roy. Soc. London Ser. A 250 (1959), 212-221.
  7. Goulden I.P., Guay-Paquet M., Novak J., Monotone Hurwitz numbers in genus zero, Canad. J. Math. 65 (2013), 1020-1042, arXiv:1204.2618.
  8. Goulden I.P., Guay-Paquet M., Novak J., Polynomiality of monotone Hurwitz numbers in higher genera, Adv. Math. 238 (2013), 1-23, arXiv:1210.3415.
  9. Goulden I.P., Guay-Paquet M., Novak J., Monotone Hurwitz numbers and the HCIZ integral, Ann. Math. Blaise Pascal 21 (2014), 71-89, arXiv:1107.1015.
  10. Goulden I.P., Jackson D.M., Vakil R., Towards the geometry of double Hurwitz numbers, Adv. Math. 198 (2005), 43-92, math.AG/0309440.
  11. Johnson P., Double Hurwitz numbers via the infinite wedge, Trans. Amer. Math. Soc. 367 (2015), 6415-6440, arXiv:1008.3266.
  12. Kazarian M.E., Lando S.K., An algebro-geometric proof of Witten's conjecture, J. Amer. Math. Soc. 20 (2007), 1079-1089, math.AG/0601760.
  13. Matsumoto S., Novak J., Jucys-Murphy elements and unitary matrix integrals, Int. Math. Res. Not. 2013 (2013), 362-397, arXiv:0905.1992.
  14. Novak J., Vicious walkers and random contraction matrices, Int. Math. Res. Not. 2009 (2009), 3310-3327, arXiv:0705.0984.
  15. Okounkov A., Toda equations for Hurwitz numbers, Math. Res. Lett. 7 (2000), 447-453, math.AG/0004128.
  16. Okounkov A., Pandharipande R., Gromov-Witten theory, Hurwitz theory, and completed cycles, Ann. of Math. 163 (2006), 517-560, math.AG/0204305.
  17. Okounkov A., Pandharipande R., The equivariant Gromov-Witten theory of ${\bf P}^1$, Ann. of Math. 163 (2006), 561-605, math.AG/0207233.
  18. Olshanski G., Plancherel averages: remarks on a paper by Stanley, Electron. J. Combin. 17 (2010), 43, 16 pages, arXiv:0905.1304.
  19. Orlov A.Yu., Shcherbin D.M., Hypergeometric solutions of soliton equations, Theoret. and Math. Phys. 128 (2001), 906-926.
  20. Pandharipande R., The Toda equations and the Gromov-Witten theory of the Riemann sphere, Lett. Math. Phys. 53 (2000), 59-74, math.AG/9912166.
  21. Shadrin S., Spitz L., Zvonkine D., On double Hurwitz numbers with completed cycles, J. London Math. Soc. 86 (2012), 407-432, arXiv:1103.3120.
  22. Stanley R.P., Parking functions and noncrossing partitions, Electron. J. Combin. 4 (1997), 20, 14 pages.

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