Maps in Locally Orientable Surfaces and Integrals Over Real Symmetric Surfaces

Goulden, I. P. ; Jackson, D. M.

Canadian journal of mathematics, Tome 49 (1997) no. 5, pp. 865-882

Résumé

The genus series for maps is the generating series for the number of rooted maps with a given number of vertices and faces of each degree, and a given number of edges. It captures topological information about surfaces, and appears in questions arising in statistical mechanics, topology, group rings, and certain aspects of free probability theory. An expression has been given previously for the genus series for maps in locally orientable surfaces in terms of zonal polynomials. The purpose of this paper is to derive an integral representation for the genus series. We then show how this can be used in conjunction with integration techniques to determine the genus series for monopoles in locally orientable surfaces. This complements the analogous result for monopoles in orientable surfaces previously obtained by Harer and Zagier. A conjecture, subsequently proved by Okounkov, is given for the evaluation of an expectation operator acting on the Jack symmetric function. It specialises to known results for Schur functions and zonal polynomials.

Détail

DOI : 10.4153/CJM-1997-045-9

Mots-clés : 05C30, 05A15, 05E05, 15A52

Bibliographie

[1] 1. Bessis, D., Itzykson, C. and Zuber, J.-B., Quantum field theory techniques in graphical enumeration. Adv. in Appl. Math. 1(1980), 109–157. Google Scholar

[2] 2. Farahat, H.K. and Higman, G., The centres of symmetric group rings. Proc. Roy. Soc. London Ser. A 250(1959), 212–221. Google Scholar

[3] 3. Goulden, I.P., Harer, J.L. and Jackson, D.M., The virtual Euler characteristic for the moduli spaces of real and complex algebraic curves. Preprint, September 1996. Google Scholar

[4] 4. Goulden, I.P. and Jackson, D.M., Symmetric functions and Macdonald's result for top connexion coefficients in the symmetric group. J.Algebra 166(1994), 364–378. Google Scholar

[5] 5. Goulden, I.P., Maps in locally orientable surfaces, the double coset algebra, and zonal polynomials. Canad. J. Math. 48(1996), 569–584. Google Scholar

[6] 6. Goulden, I.P., Connection coefficients, matchings, maps and combinatorial conjectures for Jack symmetric functions. Trans. Amer.Math. Soc. 348(1996), 873–892. Google Scholar

[7] 7. Hanlon, P.J., Stanley, R.P. and Stembridge, J.R., Some combinatorial aspects of the spectra of normally distributed random matrices. Contemp. Math. 138(1992), 151–174. Google Scholar

[8] 8. Harer, J. and Zagier, D., The Euler characteristic of the moduli space of curves. Invent. Math. 85(1986), 457–485. Google Scholar

[9] 9. Helgason, S., Differential geometry, Lie groups and symmetric spaces. Academic Press, NewYork, 1978. Google Scholar

[10] 10. Hooft, G.'t, A planar diagram theory for strong interactions. Nuclear Phys. B 72(1974), 461–473. Google Scholar

[11] 11. Itzykson, C. and Zuber, J.-B., Matrix integration and combinatorics of modular groups. Commun. Math. Phys. 134(1990), 197–207. Google Scholar

[12] 12. Jack, H., A class of symmetric functions with a parameter. Proc. Roy. Soc. Edinburgh Sect.A 69A(1970), 1–18. Google Scholar

[13] 13. Jackson, D.M., On an integral representation for the genus series for 2-cell embeddings. Trans. Amer. Math. Soc. 344(1994), 755–772. Google Scholar

[14] 14. Jackson, D.M., The genus series for maps. J. Pure and Appl. Algebra 105(1995), 293–297. Google Scholar

[15] 15. James, A.T., Zonal polynomials of the real positive definite matrices. Ann. Math. 74(1961), 475–501. Google Scholar

[16] 16. Kontsevich, M., Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys. 147(1992), 1–23. Google Scholar

[17] 17. Macdonald, I.G., Symmetric functions and Hall polynomials. Clarendon Press, Oxford, 1979. Google Scholar

[18] 18. Macdonald, I.G., Private communication, June 1995. Google Scholar

[19] 19. Mehta, M.L., Random matrices. Academic Press, New York, 1967. Google Scholar

[20] 20. Mehta, M.L., Statistical mechanics of membranes and surfaces. (Eds. Nelson, D., Piran, T. and Weinberg, S.) World Scientific, New Jersey, 1989. Google Scholar

[21] 21. Okounkov, A., Proof of a conjecture of Goulden and Jackson. Canad. J. Math (this issue). Google Scholar

[22] 22. Penner, R.C., Perturbative series and the moduli space of Riemann surfaces. J. Diff.Geometry 27(1988), 35–53. Google Scholar

[23] 23. Richmond, L.B. and Wormald, N. C., Almost allmaps are asymmetric. J.Combin. Theory Ser.B 63(1995), 1–7. Google Scholar

[24] 24. Stanley, R.P., Some combinatorial properties of Jack symmetric functions. Advances inMath. 77(1989), 76–115. Google Scholar

[25] 25. Szegö, G., Orthogonal polynomials. Amer. Math. Soc. Colloq. Publ. 23, Providence, RI, 1939. Google Scholar

[26] 26. Tutte, W.T., Graph theory. Encyclopedia of Math. and its Applications 21, Addison-Wesley, London, 1984. Google Scholar

[27] 27. Voiculescu, D.V., Dykema, K.J. and A.Nica, Free random variables. C.R.M.Monograph Series 1(1993). Published by A.M.S. Google Scholar