Double Hurwitz numbers via the infinite wedge
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- by Paul Johnson PDF
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Abstract:
We derive an algorithm to produce explicit formulas for certain generating functions of double Hurwitz numbers. These formulas generalize a formula in the work of Goulden, Jackson, and Vakil for one part double Hurwitz numbers. 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 Strong Piecewise Polynomiality Conjecture of their work.
The proof is an 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.
References
- Renzo Cavalieri, Paul Johnson, and Hannah Markwig, Chamber structure for double Hurwitz numbers, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), Discrete Math. Theor. Comput. Sci. Proc., AN, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2010, pp. 227–238 (English, with English and French summaries). MR 2673838
- Renzo Cavalieri, Paul Johnson, and Hannah Markwig, Tropical Hurwitz numbers, J. Algebraic Combin. 32 (2010), no. 2, 241–265. MR 2661417, DOI 10.1007/s10801-009-0213-0
- Robbert Dijkgraaf, Mirror symmetry and elliptic curves, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 149–163. MR 1363055, DOI 10.1007/978-1-4612-4264-2_{5}
- Torsten Ekedahl, Sergei Lando, Michael Shapiro, and Alek Vainshtein, Hurwitz numbers and intersections on moduli spaces of curves, Invent. Math. 146 (2001), no. 2, 297–327. MR 1864018, DOI 10.1007/s002220100164
- I. P. Goulden, D. M. Jackson, and R. Vakil, Towards the geometry of double Hurwitz numbers, Adv. Math. 198 (2005), no. 1, 43–92. MR 2183250, DOI 10.1016/j.aim.2005.01.008
- Tom Graber and Ravi Vakil, Hodge integrals and Hurwitz numbers via virtual localization, Compositio Math. 135 (2003), no. 1, 25–36. MR 1955162, DOI 10.1023/A:1021791611677
- V. G. Kac and A. K. Raina, Bombay lectures on highest weight representations of infinite-dimensional Lie algebras, Advanced Series in Mathematical Physics, vol. 2, World Scientific Publishing Co., Inc., Teaneck, NJ, 1987. MR 1021978
- Sergei K. Lando and Alexander K. Zvonkin, Graphs on surfaces and their applications, Encyclopaedia of Mathematical Sciences, vol. 141, Springer-Verlag, Berlin, 2004. With an appendix by Don B. Zagier; Low-Dimensional Topology, II. MR 2036721, DOI 10.1007/978-3-540-38361-1
- Chiu-Chu Melissa Liu, Lectures on the ELSV formula, Transformation groups and moduli spaces of curves, Adv. Lect. Math. (ALM), vol. 16, Int. Press, Somerville, MA, 2011, pp. 195–216. MR 2883688
- T. Miwa, M. Jimbo, and E. Date, Solitons, Cambridge Tracts in Mathematics, vol. 135, Cambridge University Press, Cambridge, 2000. Differential equations, symmetries and infinite-dimensional algebras; Translated from the 1993 Japanese original by Miles Reid. MR 1736222
- Andrei Okounkov, Toda equations for Hurwitz numbers, Math. Res. Lett. 7 (2000), no. 4, 447–453. MR 1783622, DOI 10.4310/MRL.2000.v7.n4.a10
- Andrei Okounkov, Infinite wedge and random partitions, Selecta Math. (N.S.) 7 (2001), no. 1, 57–81. MR 1856553, DOI 10.1007/PL00001398
- A. Okounkov and R. Pandharipande, The equivariant Gromov-Witten theory of $\textbf {P}^1$, Ann. of Math. (2) 163 (2006), no. 2, 561–605. MR 2199226, DOI 10.4007/annals.2006.163.561
- A. Okounkov and R. Pandharipande, Gromov-Witten theory, Hurwitz theory, and completed cycles, Ann. of Math. (2) 163 (2006), no. 2, 517–560. MR 2199225, DOI 10.4007/annals.2006.163.517
- Mike Roth, Counting covers of an elliptic curve, www.mast.queens.ca/~mikeroth/notes/covers.pdf.
- S. Shadrin, M. Shapiro, and A. Vainshtein, Chamber behavior of double Hurwitz numbers in genus 0, Adv. Math. 217 (2008), no. 1, 79–96. MR 2357323, DOI 10.1016/j.aim.2007.06.016
Additional Information
- Paul Johnson
- Affiliation: Department of Mathematics, Weber Building, Colorado State University, Fort Collins, Colorado 80523-1874
- Address at time of publication: School of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, United Kingdom
- MR Author ID: 905435
- Email: johnson@math.colostate.edu, paul.johnson@shef.ac.uk
- Received by editor(s): March 6, 2013
- Received by editor(s) in revised form: July 9, 2013
- Published electronically: April 1, 2015
- Additional Notes: This research was supported in part by NSF grants DMS-0602191 and DMS-0902754.
- © Copyright 2015
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 367 (2015), 6415-6440
- MSC (2010): Primary 14N10
- DOI: https://doi.org/10.1090/S0002-9947-2015-06238-2
- MathSciNet review: 3356942