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Maps in Locally Orientable Surfaces and Integrals Over Real Symmetric Surfaces

Published online by Cambridge University Press:  20 November 2018

I. P. Goulden  and
D. M. Jackson
I. P. Goulden
Affiliation:
I. P. Goulden and D. M. Jackson, Faculty of Mathematics, University of Waterloo, Waterloo, ON, Canada
D. M. Jackson
Affiliation:
I. P. Goulden and D. M. Jackson, Faculty of Mathematics, University of Waterloo, Waterloo, ON, Canada
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Abstract

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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.

Keywords

05C3005A1505E0515A52
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 5 , 01 October 1997 , pp. 865 - 882
Copyright
Copyright © Canadian Mathematical Society 1997

References

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. 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. 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. 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. 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. 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. 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. Harer, J. and Zagier, D., The Euler characteristic of the moduli space of curves. Invent. Math. 85(1986), 457–485.Google Scholar
9. Helgason, S., Differential geometry, Lie groups and symmetric spaces. Academic Press, NewYork, 1978.Google Scholar
10. Hooft, G.'t, A planar diagram theory for strong interactions. Nuclear Phys. B 72(1974), 461–473.Google Scholar
11. Itzykson, C. and Zuber, J.-B., Matrix integration and combinatorics of modular groups. Commun. Math. Phys. 134(1990), 197–207.Google Scholar
12. Jack, H., A class of symmetric functions with a parameter. Proc. Roy. Soc. Edinburgh Sect.A 69A(1970), 1–18.Google Scholar
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. Jackson, D.M., The genus series for maps. J. Pure and Appl. Algebra 105(1995), 293–297.Google Scholar
15. James, A.T., Zonal polynomials of the real positive definite matrices. Ann. Math. 74(1961), 475–501.Google Scholar
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. Macdonald, I.G., Symmetric functions and Hall polynomials. Clarendon Press, Oxford, 1979.Google Scholar
18. Macdonald, I.G., Private communication, June 1995.Google Scholar
19. Mehta, M.L., Random matrices. Academic Press, New York, 1967.Google Scholar
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. Okounkov, A., Proof of a conjecture of Goulden and Jackson. Canad. J. Math (this issue).Google Scholar
22. Penner, R.C., Perturbative series and the moduli space of Riemann surfaces. J. Diff.Geometry 27(1988), 35–53.Google Scholar
23. Richmond, L.B. and Wormald, N. C., Almost allmaps are asymmetric. J.Combin. Theory Ser.B 63(1995), 1–7.Google Scholar
24. Stanley, R.P., Some combinatorial properties of Jack symmetric functions. Advances inMath. 77(1989), 76–115.Google Scholar
25. Szegö, G., Orthogonal polynomials. Amer. Math. Soc. Colloq. Publ. 23, Providence, RI, 1939.Google Scholar
26. Tutte, W.T., Graph theory. Encyclopedia of Math. and its Applications 21, Addison-Wesley, London, 1984.Google Scholar
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