A square root of Hurwitz numbers

  • Junho LeeEmail author


We exhibit a generating function of spin Hurwitz numbers analogous to (disconnected) double Hurwitz numbers that is a tau function of the two-component BKP (2-BKP) hierarchy and is a square root of a tau function of the two-component KP (2-KP) hierarchy defined by related Hurwitz numbers.

Mathematics Subject Classification

14H70 53D45 57M12 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The author sincerely thank the referee for comments and suggestions that helped to improve the presentation of this paper.


  1. 1.
    Alexandrov, A., Zabrodin, A.: Free fermions and tau-functions. J. Geom. Phys. 67, 37–80 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Transformation groups for soliton equations, IV. A new hierarchy of soliton equations of KP-type. Physica D 4, 343–365 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Gunningham, S.: Spin Hurwitz numbers and topological quantum field theory. Geom. Topol. 20(4), 1859–1907 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ivanov, V.N.: Gaussian limit for projective characters of large symmetric groups. J. Math. Sci. 121(3), 2330–2344 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ivanov, V.N.: Plancherel measure on shifted Young diagrams, pp. 73–86. Representation theory, dynamical systems, and asymptotic combinatorics, Providence (2006)zbMATHGoogle Scholar
  6. 6.
    Ivanov, V.N., Olshanski, G.: Kerov’s central limit theorem for the Plancherel measure on Young diagrams, Symmetric functions 2001: surveys of developments and perspectives, pp. 93–151. Kluwer Academic Publishers, Dordrecht (2002). arXiv:math/0304010
  7. 7.
    Józefiak, T.: A class of projective representations of hyperoctahedral groups and Schur Q -functions, Topics in Algebra, pp. 317–326. Banach Center Publications (PWN-Polish Scientific Publishers), Warsaw (1990)zbMATHGoogle Scholar
  8. 8.
    Kac, V.: Infinite dimensional Lie algebras. Cambridge University Press, Cambridge (1990)CrossRefzbMATHGoogle Scholar
  9. 9.
    Kerov, S., Olshanski, G.: Polynomial functions on the set of Young diagrams. C. R. Acad. Sci. Paris Ser. I Math 319(2), 121–126 (1994)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kiem, Y.-H., Li, J.: Low degree GW invariants of spin surfaces. Pure Appl. Math. Q. 7(4), 1449–1476 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kiem, Y.-H., Li, J.: Low degree GW invariants of spin surfaces II. Sci. China Math. 54(8), 1679–1706 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kool, M., Thomas, R.P.: Stable pairs with descendents on local surfaces I: the vertical component, arXiv:1605.02576
  13. 13.
    Lee, J.: Sum formulas for local Gromov-Witten invariants of spin curves. Trans. Amer. Math. Soc. 365(1), 459–490 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lee, J.: A note on Gunningham’s formula, arXiv:1407.0055. To appear in Bull. Aust. Math. Soc
  15. 15.
    Lee, J., Parker, T.H.: A structure theorem for the Gromov-Witten invariants of Kähler surfaces. J. Differ. Geom. 77(3), 483–513 (2007)CrossRefzbMATHGoogle Scholar
  16. 16.
    Lee, J., Parker, T.H.: Spin Hurwitz numbers and the Gromov-Witten invariants of Kahler surfaces. Commun. Anal. Geom. 21(5), 1015–1060 (2013)CrossRefzbMATHGoogle Scholar
  17. 17.
    Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford University Press, New York (1995)zbMATHGoogle Scholar
  18. 18.
    Maulik, D., Pandharipande, R.: New calculations in Gromov-Witten theory. Pure Appl. Math. Q. 4(2), 469–500 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Okounkov, A.: Infinite wedge and random partitions. Selecta Math. (N.S.) 7(1), 57–81 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Okounkov, A.: Toda equations for Hurwitz numbers. Math. Res. Lett. 7(4), 447–453 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Okounkov, A., Pandharipande, R.: Gromov-Witten theory, Hurwitz theory, and completed cycles. Ann. of Math. 163(2), 517–560 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Okounkov, A., Pandharipande, R.: The equivariant Gromov-Witten theory of \({\mathbb{P}}^1\). Ann. of Math. 163(2), 561–605 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Orlov, A. Yu., Shiota, T., Takasaki, K.: Pfaffian structures and certain solutions to BKP hierarchies I. Sum over partitions, arXiv:1201.4518
  24. 24.
    Sergeev, A.N.: Tensor algebra of the identity representation as a module over Lie superalgebras \(Gl(n, m)\) and \(Q(n)\). Mat. Sb. 123(3), 422–430 (1984)MathSciNetGoogle Scholar
  25. 25.
    Takasaki, K.: Dispersionless Hirota equations of two-component BKP hierarchy. SIGMA Symmetry Integrability Geometry. Methods Appl 2, 057 (2006)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Ueno, K., Takasaki, K.: Toda lattice hierarchy, group representations and systems of differential equations (Tokyo, 1982), Adv. Stud. Pure Math. 4, 1–95 (1984)Google Scholar
  27. 27.
    Wassermann, A.J.: Automorphic actions of compact groups on operator algebras, Thesis (Ph.D.) University of Pennsylvania, (1981)Google Scholar
  28. 28.
    You, Yuching: Polynomial solutions of the BKP hierarchy and projective representations of symmetric groups, Infinite-dimensional Lie algebras and groups. Adv. Ser. Math. Phys. 7, 449–464 (1989)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Central FloridaOrlandoUSA

Personalised recommendations