Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T17:05:19.714Z Has data issue: false hasContentIssue false
  • English
  • Français

Maps in Locally Orientable Surfaces, the Double Coset Algebra, and Zonal Polynomials

Published online by Cambridge University Press:  20 November 2018

I. P. Goulden  and
D. M. Jackson
I. P. Goulden
Affiliation:
Department of Combinatorics and Optimization University of Waterloo Waterloo, Ontario N2L 3G1
D. M. Jackson
Affiliation:
Department of Combinatorics and Optimization University of Waterloo Waterloo, Ontario N2L 3G1
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The genus series is the generating series for the number of maps (inequivalent two-cell embeddings of graphs), in locally orientable surfaces, closed and without boundary, with respect to vertex- and face-degrees, number of edges and genus. A hypermap is a face two-colourable map. An expression for the genus series for (rooted) hypermaps is derived in terms of zonal polynomials by using a double coset algebra in conjunction with an encoding of a map as a triple of matchings. The expression is analogous to the one obtained for orientable surfaces in terms of Schur functions.

Keywords

05E0505A1557M15
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 3 , 01 June 1996 , pp. 569 - 584
Copyright
Copyright © Canadian Mathematical Society 1996

References

1. Bergeron, N. and Garsia, A.M., Zonal polynomials and domino tableaux, Discrete Math. 99(1992), 3–15.Google Scholar
2. Goulden, I.P. and Jackson, D.M., Combinatorial constructions for integrals over normally distributed random matrices, Proc. Amer. Math. Soc, 123 (1995), 995–1003.Google Scholar
3. Goulden, I.P., Connection coefficients, matchings, and combinatorial conjectures for Jack symmetric functions, Trans. Amer. Math. Soc. 348 (1996), 873–892.Google Scholar
4. Goulden, I.P.,Combinatorial Enumeration, Wiley Interscience, New York, 1983.Google Scholar
5. Gross, J.L. and Tucker, T.W.,Topological Graph Theory, Wiley Interscience, New York, 1987.Google Scholar
6. Hanlon, P.J., Stanley, R.P., and Stembridge, J.R., Some combinatorial aspects of the spectra of normally distributed random matrices, Contemporary Math. 138 (1992), 151–174.Google Scholar
7. Itzykson, C. and Drouffe, J-M., Statistical field theory, vol. 2, Cambridge Univ. Press, 1990.Google Scholar
8. Jack, H., A class of symmetric polynomials with a parameter, Proc. Roy. Soc. Edinburgh, 69A(1970), 1–18.Google Scholar
9. Jackson, D.M., Perry, M.J. and Visentin, T.I., Factorisations for partition functions for random Hermitian matrix models, Comm. Math. Phys., to appear.Google Scholar
10. Jackson, D.M. and Visentin, T.I., A character theoretic approach to embeddings of rooted maps in an orientable surface of given genus, Trans. Amer. Math. Soc. 322 (1990), 343–363.Google Scholar
11. Jackson, D.M., Character theory and rooted maps in an orientable surface of given genus: face-coloured maps, Trans. Amer. Math. Soc. 322 (1990), 365–376.Google Scholar
12. James, A.T., Zonal polynomials of the real positive definite matrices, Ann. of Math. 74 (1961), 475–501.Google Scholar
13. Jones, G.A. and Thornton, J.S., Operations on maps, and outer automorphisms, J. Combin. Theory B 35 (1983), 93–103.Google Scholar
14. Macdonald, I.G.,Symmetric functions and Hall polynomials, Clarendon Press, Oxford, 1979.Google Scholar
15. Stembridge, J.R., A Maple package for symmetric functions—Version 2, July, 1993. Department of Mathematics, University of Michigan, Ann Arbor, Michigan, 48109–1003.Google Scholar
16. Takemura, A.,Zonal polynomials, Lecture Notes 4, Institute of Mathematical Statistics, Hayward, California, 1984.Google Scholar
17. Tutte, W.T.,Graph Theory, Encyclopedia of Math, and its Applications 21, Addison-Wesley, London, 1984.Google Scholar
18. Vince, A., Combinatorial maps, J. Combin. Theory B 34 (1983), 1–21.Google Scholar
19. Vince, A., Regular combinatorial maps, J. Combin. Theory B 35 (1983), 256–277.Google Scholar