Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 097, 19 pages      arXiv:1504.07512      https://doi.org/10.3842/SIGMA.2015.097
Contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet

Multispecies Weighted Hurwitz Numbers

J. Harnad ab
a) Centre de recherches mathématiques, Université de Montréal, C.P. 6128, succ. Centre-ville, Montréal (QC) H3C 3J7, Canada
b) Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke W., Montréal (QC) H4B 1R6, Canada

Received March 31, 2015, in final form November 16, 2015; Published online December 02, 2015; Misprints corrected December 10, 2015

Abstract
The construction of hypergeometric $2D$ Toda $\tau$-functions as generating functions for weighted Hurwitz numbers is extended to multispecies families. Both the enumerative geometrical significance of multispecies weighted Hurwitz numbers, as weighted enumerations of branched coverings of the Riemann sphere, and their combinatorial significance in terms of weighted paths in the Cayley graph of $S_n$ are derived. The particular case of multispecies quantum weighted Hurwitz numbers is studied in detail.

Key words: weighted Hurwitz number; $\tau$-function; multispecies.

pdf (465 kb)   tex (25 kb)       [previous version:  pdf (465 kb)   tex (25 kb)]

References

  1. Bouchard V., Mariño M., Hurwitz numbers, matrix models and enumerative geometry, in From Hodge Theory to Integrability and TQFT: ${\rm tt}^*$-Geometry, Proc. Sympos. Pure Math., Vol. 78, Amer. Math. Soc., Providence, RI, 2008, 263-283, arXiv:0709.1458.
  2. Diaconis P., Greene C., Applications of Murphy's elements, Stanford Technical Report, no. 335, 1989.
  3. Eynard B., Invariants of spectral curves and intersection theory of moduli spaces of complex curves, Commun. Number Theory Phys. 8 (2014), 541-588, arXiv:1110.2949.
  4. Eynard B., Orantin N., Invariants of algebraic curves and topological expansion, Commun. Number Theory Phys. 1 (2007), 347-452, math-ph/0702045.
  5. Eynard B., Orantin N., Topological recursion in enumerative geometry and random matrices, J. Phys. A: Math. Theor. 42 (2009), 293001, 117 pages, arXiv:0811.3531.
  6. Faddeev L.D., Kashaev R.M., Quantum dilogarithm, Modern Phys. Lett. A 9 (1994), 427-434, hep-th/9310070.
  7. Frobenius G., Über die Charaktere der symmetrischen Gruppe, Sitzungsber. Königl. Preuss. Akad. Wiss. (1900), 516-534.
  8. Frobenius G., Über die charakterische Einheiten der symmetrischen Gruppe, Sitzungsber. Königl. Preuss. Akad. Wiss. (1903), 328-358.
  9. Goulden I.P., Jackson D.M., The KP hierarchy, branched covers, and triangulations, Adv. Math. 219 (2008), 932-951, arXiv:0803.3980.
  10. Gross K.I., Richards D.S.P., Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions, Trans. Amer. Math. Soc. 301 (1987), 781-811.
  11. Guay-Paquet M., Harnad J., 2D Toda $\tau$-functions as combinatorial generating functions, Lett. Math. Phys. 105 (2015), 827-852.
  12. Guay-Paquet M., Harnad J., Generating functions for weighted Hurwitz numbers, arXiv:1408.6766.
  13. Harnad J., Quantum Hurwitz numbers and Macdonald polynomials, arXiv:1504.03311.
  14. Harnad J., Weighted Hurwitz numbers and hypergeometric $\tau$-functions: an overview, arXiv:1504.03408.
  15. Harnad J., Orlov A.Yu., Hypergeometric $\tau$-functions, Hurwitz numbers and enumeration of paths, Comm. Math. Phys. 338 (2015), 267-284, arXiv:1407.7800.
  16. Hurwitz A., Ueber Riemann'sche Fläsche mit gegebnise Verzweigungspunkten, Math. Ann. 39 (1891), 1-60.
  17. Hurwitz A., Ueber die Anzahl der Riemann'sche Fläsche mit gegebnise Verzweigungspunkten, Math. Ann. 55 (1902), 53-66.
  18. Jucys A.-A.A., Symmetric polynomials and the center of the symmetric group ring, Rep. Math. Phys. 5 (1974), 107-112.
  19. Lando S.K., Zvonkin A.K., Graphs on surfaces and their applications, Encyclopaedia of Mathematical Sciences, Vol. 141, Springer-Verlag, Berlin, 2004.
  20. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.
  21. Murphy G.E., A new construction of Young's seminormal representation of the symmetric groups, J. Algebra 69 (1981), 287-297.
  22. Natanzon S.M., Orlov A.Yu., Hurwitz numbers and BKP hierarchy, arXiv:1407.8323.
  23. Natanzon S.M., Orlov A.Yu., BKP and projective Hurwitz numbers, arXiv:1501.01283.
  24. Okounkov A., Toda equations for Hurwitz numbers, Math. Res. Lett. 7 (2000), 447-453, math.AG/0004128.
  25. Orlov A.Yu., Shcherbin D.M., Hypergeometric solutions of soliton equations, Theoret. and Math. Phys. 128 (2001), 906-926.
  26. Pandharipande R., The Toda equations and the Gromov-Witten theory of the Riemann sphere, Lett. Math. Phys. 53 (2000), 59-74, math.AG/9912166.
  27. Schur I., Neue Begründung der Theorie der Gruppencharaktere, Sitzungsber. Königl. Preuss. Akad. Wiss. (1904), 406-432.
  28. Takasaki K., Initial value problem for the Toda lattice hierarchy, in Group Representations and Systems of Differential Equations (Tokyo, 1982), Adv. Stud. Pure Math., Vol. 4, North-Holland, Amsterdam, 1984, 139-163.
  29. Takebe T., Representation theoretical meaning of the initial value problem for the Toda lattice hierarchy. I, Lett. Math. Phys. 21 (1991), 77-84.
  30. Ueno K., Takasaki K., Toda lattice hierarchy, in Group Representations and Systems of Differential Equations (Tokyo, 1982), Adv. Stud. Pure Math., Vol. 4, North-Holland, Amsterdam, 1984, 1-95.

Previous article  Next article   Contents of Volume 11 (2015)