Monotone Hurwitz Numbers in Genus Zero

Goulden, I. P. ; Guay-Paquet, Mathieu ; Novak, Jonathan

Canadian journal of mathematics, Tome 65 (2013) no. 5, pp. 1020-1042

Résumé

Hurwitz numbers count branched covers of the Riemann sphere with specified ramification data, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers related to the expansion of complete symmetric functions in the Jucys–Murphy elements, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra–Itzykson–Zuber integral. In this paper we begin a detailed study of monotone Hurwitz numbers. We prove two results that are reminiscent of those for classical Hurwitz numbers. The first is the monotone join-cut equation, a partial differential equation with initial conditions that characterizes the generating function for monotone Hurwitz numbers in arbitrary genus. The second is our main result, in which we give an explicit formula for monotone Hurwitz numbers in genus zero.

Détail

DOI : 10.4153/CJM-2012-038-0

Mots-clés : 05A15, 14E20, 15B52, Hurwitz numbers, matrix models, enumerative geometry

Bibliographie

[1] [1] Bouchard, V. and Mariño, M., Hurwitz numbers, matrix models and enumerative geometry, In: From Hodge theory to integrability and TQFT tt*-geometry. Proc. Sympos. Pure Math., 78, American Mathematical Society, Providence, RI, 2008, pp. 263–283. Google Scholar

[2] [2] Bousquet-Mélou, M. and Schaeffer, G., Enumeration of planar constellations. Adv. in Appl. Math. 24(2000), no. 4, 337–368. Google Scholar | DOI

[3] [3] Ekedahl, T., Lando, S., Shapiro, M., and Vainshtein, A., Hurwitz numbers and intersections on moduli spaces of curves. Invent. Math. 146(2001), no. 2, 297–327, Google Scholar | DOI

[4] [4] Eynard, B., Mulase, M., and Safnuk, B., The Laplace transform of the cut-and-join equation and the Bouchard-Marino conjecture on Hurwitz numbers.arxiv:0907.5224. Google Scholar

[5] [5] Eynard, B. and Orantin, N., Invariants of algebraic curves and topological expansion. Commun. Number Theory Phys. 1(2007), no. 2, 347–452. Google Scholar

[6] [6] Féray, V., On complete functions in Jucys-Murphy elements. Ann. Comb., to appear. arxiv:1009.0144 Google Scholar

[7] [7] Gewurz, D. A. and Merola, F., Some factorisations counted by Catalan numbers. European J. Combin. 27(2006), no. 6, 990–994. Google Scholar | DOI

[8] [8] Goulden, I. P.,Guay-Paquet, M., and Novak, J., Polynomiality of monotone Hurwitz numbers in higher genera. arxiv:1210.3415 Google Scholar

[9] [9] Goulden, I. P., Monotone Hurwitz numbers and the HCIZ integral I.arxiv:1107.1015. Google Scholar

[10] [10] Goulden, I. P. and Jackson, D. M., Transitive factorisations into transpositions and holomorphic mappings on the sphere. Proc. Amer. Math. Soc. 125(1997), no. 1, 51–60. Google Scholar | DOI

[11] [11] Goulden, I. P., Jackson, D. M., and Vainshtein, A., The number of ramified coverings of the sphere by the torus and surfaces of higher genera. Ann. Comb. 4(2000), no. 1, 27–46. Google Scholar | DOI

[12] [12] Goulden, I. P. and Jackson, David M., Combinatorial enumeration. Dover Publications Inc., Mineola, NY, 2004, Reprint of the 1983 original. Google Scholar

[13] [13] Chandra, Harish, Differential operators on a semisimple Lie algebra. Amer. J. Math. 79(1957), 87–120. Google Scholar | DOI

[14] [14] Hurwitz, A., Über Riemann’sche Flächen mit gegebenen Verzweigungspunkten. Mathematische Annalen 39(1891), no. 1, 1–60. Google Scholar

[15] [15] Itzykson, C. and Zuber, J. B., The planar approximation. II. J. Math. Phys. 21(1980), no. 3, 411–421. Google Scholar | DOI

[16] [16] Jucys, A.-A. A., Symmetric polynomials and the center of the symmetric group ring Rep. Mathematical Phys. 5(1974), no. 1, 107–112. Google Scholar | DOI

[17] [17] Kazarian, M. E. and Lando, S. K., An algebro-geometric proof of Witten's conjecture. J. Amer. Math. Soc. 20(2007), no. 4, 1079–1089. Google Scholar | DOI

[18] [18] Kontsevich, M., Intersection theory on the moduli space of curves and the matrix Airy function. Comm. Math. Phys. 147(1992), no. 1, 1–23, Google Scholar | DOI

[19] [19] Lando, S. K. and Zvonkin, A. K., Graphs on surfaces and their applications. Encyclopaedia of Mathematical Sciences, 141, Low-Dimensional Topology, II, Springer-Verlag, Berlin, 2004. Google Scholar

[20] [20] Lassalle, M., Class expansion of some symmetric functions in Jucys-Murphy elements. arxiv:1005.2346 Google Scholar

[21] [21] Macdonald, I. G., Symmetric functions and Hall polynomials. Second ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. Google Scholar

[22] [22] Okounkov, A. and Pandharipande, R., Gromov-Witten theory, Hurwitz numbers, and matrix models. In: Algebraic geometry—Seattle 2005. Part 1, Proc. Sympos. Pure Math., 80, American Mathematicla Society, Providence, RI, 2009, pp. 325–414. Google Scholar

[23] [23] Okounkov, A. and Vershik, A., A new approach to representation theory of symmetric groups. Selecta Math. (N.S.) 2(1996), no. 4, 581–605. Google Scholar | DOI

[24] [24] The Sage-Combinat community, Sage-Combinat: enhancing Sage as a toolbox for computer exploration in algebraic combinatorics. 2008. http://combinat.sagemath.org. Google Scholar

[25] [25] Steinet et al., W. A., Sage mathematics software (version 4.6), 2010, http://www.sagemath.org Google Scholar

[26] [26] Strehl, V., Minimal transitive products of transpositions—the reconstruction of a proof of A. Hurwitz. Sém. Lothar. Combin. 37(1996), Art. S37c, 12 pp. (electronic). Google Scholar

[27] [27] Witten, E., Two-dimensional gravity and intersection theory on moduli space. In: Surveys in differential geometry (Cambridge, MA, 1990), Lehigh Univ., Bethlehem, PA, 1991, pp. 243–310. Google Scholar

[28] [28] Zinn-Justin, P., HCIZ integral and 2D Toda lattice hierarchy. Nuclear Phys. B 634(2002), no. 3, 417–432. Google Scholar | DOI

[29] [29] Zinn-Justin, P. and Zuber, J.-B., On some integrals over the U(N) unitary group and their large N limit. J. Phys. A 36(2003), no. 12, 3173–3193. Google Scholar | DOI

[30] [30] Zvonkin, A., Matrix integrals and map enumeration: an accessible introduction. Combinatorics and Physics (Marseilles, 1995). Math. Comput. Modelling 26(1997), no. 8–10, 281–304. Google Scholar | DOI