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2D Toda τ-Functions as Combinatorial Generating Functions

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Abstract

Two methods of constructing 2D Toda τ-functions that are generating functions for certain geometrical invariants of a combinatorial nature are related. The first involves generation of paths in the Cayley graph of the symmetric group S n by multiplication of the conjugacy class sums \({C_\lambda \in \mathbf{C}[S_n]}\) in the group algebra by elements of an abelian group of central elements. Extending the characteristic map to the tensor product \({\mathbf{C}[S_n] \otimes \mathbf{C}[S_n]}\) leads to double expansions in terms of power sum symmetric functions, in which the coefficients count the number of such paths. Applying the same map to sums over the orthogonal idempotents leads to diagonal double Schur function expansions that are identified as τ-functions of hypergeometric type. The second method is the standard construction of τ-functions as vacuum-state matrix elements of products of vertex operators in a fermionic Fock space with elements of the abelian group of convolution symmetries. A homomorphism between these two group actions is derived and shown to be intertwined by the characteristic map composed with fermionization. Applications include Okounkov’s generating function for double Hurwitz numbers, which count branched coverings of the Riemann sphere with specified ramification profiles at two branch points, and only simple branching at all the others, and various analogous combinatorial counting functions.

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References

  1. Alexandrov, A., Mironov, A., Morozov, A., Natanzon, S.: On KP-integrable Hurwitz functions. JHEP. doi:10.1007/JHEP11(2014)080

  2. Ambjørn J., Chekhov L.: The matrix model for dessins d’enfants. Ann. Inst. Henri Poincaré, Comb. Phys. Interact. 1, 337–361 (2014)

    Article  Google Scholar 

  3. Ambjørn J., Chekhov L.: A matrix model for hypergeometric Hurwitz numbers. Theor. Math. Phys. 181, 1486–1498 (2014)

    Article  Google Scholar 

  4. Borot G., Eynard B., Mulase M., Safnuk B.: A matrix model for Hurwitz numbers and topological recursion. J. Geom. Phys. 61, 522–540 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. Bertola M., Prats-Ferrer A.: Topological expansion for the Cauchy two-matrix model. J. Phys. A Math. Theor. 42, 335201 (2009)

    Article  ADS  Google Scholar 

  6. Bertola M., Gekhtman M., Szmigielski J.: The Cauchy two-matrix model. Commun. Math. Phys. 287, 983–1014 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Corteel S., Goupil A., Schaeffer G.: Content evaluation and class symmetric functions. Adv. Math. 188(2), 315–336 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Transformation groups for soliton equations. In: Sato, M. (ed.) Non-linear integrable systems classical and quantum theory, Proceedings of RIMS Symposium (1981), World Scientific (1983)

  9. Diaconis, P., Greene, C.: Applications of Murphy’s elements. Stanford Technical Report, vol. 335 (1989)

  10. Eynard B., Orantin N.: Topological recursion in enumerative geometry and random matrices. J. Phys. A 42, 293001 (2009)

    Article  MathSciNet  Google Scholar 

  11. Farahat H.K., Higman G.: The centers of symmetric group rings. Proc. Roy. Soc. London Ser. A 250, 212–221 (1959)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  12. Guay-Paquet, M., Harnad, J.: Generating functions for weighted Hurwitz numbers. arXiv:1408.6766

  13. Goulden I.P.: A differential operator for symmetric functions and the combinatorics of multiplying transpositions. Trans. Am. Math. Soc. 344(1), 421–440 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  14. Goulden I.P., Guay-Paquet M., Novak J.: Monotone Hurwitz numbers and the HCIZ Integral. Ann. Math. Blaise Pascal 21, 71–99 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  15. Goulden, I.P., Guay-Paquet, M., Novak, J.: Toda equations and piecewise polynomiality for mixed double Hurwitz numbers. arXiv:1307.2137

  16. Harish-Chandra.: Differential operators on a semisimple Lie algebra. Am. J. Math. 79, 87–120 (1957)

  17. Harnad, J.: Multispecies weighted Hurwitz numbers. arXiv:1504.07512

  18. Harnad, J.: Quantum Hurwitz numbers and Macdonald polynomials. arXiv:1504.03311

  19. Harnad, J.: Weighted Hurwitz numbers and hypergeometric τ-functions: an overview. arXiv:1504.03408

  20. Harnad, J., Orlov, A.Yu.: Matrix integrals as Borel sums of Schur function expansions. In: Abenda, S., Gaeta, G., Walcher, S. (eds.) Symmetries sand perturbation theory, SPT2002, World Scientific, Singapore (2003)

  21. Harnad J., Orlov A.Yu.: Scalar products of symmetric functions and matrix integrals. Theor. Math. Phys. 137, 1676–90 (2003)

  22. Harnad J., Orlov A.Yu.: Fermionic construction of partition functions for two-matrix models and perturbative Schur function expansions. J. Phys. A 39, 8783–8809 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  23. Harnad, J., Orlov, A.Yu.: Convolution symmetries of integrable hierarchies, matrix models and tau functions. In: Deift, P., Forrester, P. (eds.) Random Matrix Theory, Interacting Particle Systems, and Integrable Systems, MSRI publications, vol. 65, pp. 247–276. Cambridge University Press (2014)

  24. Harnad, J., Orlov, A.Yu.: Hypergeometric τ-functions, Hurwitz numbers and enumeration of paths. Commun. Math. Phys. (2015). doi:10.1007/s00220-015-2329-5. arXiv:1407.7800

  25. Itzykson C., Zuber J.-B.: The planar approximation. II. J. Math. Phys. 21, 411–21 (1980)

  26. Jucys A.A.: Symmetric polynomials and the center of the symmetric group ring. Rep. Math. Phys. 5(1), 107–112 (1974)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  27. Kazarian M.: KP hierarchy for Hodge integrals. Adv. Math. 221, 1–21 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  28. Kazarian, M., Zograf, P.: Virasoro constraints and topological recursion for Grothendieck’s dessin counting. arXiv:1406.5976

  29. Kontsevich M.: Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys. 147, 1–23 (1992)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  30. Macdonald, I.G.: Symmetric functions and hall polynomials. Clarendon Press, Oxford. (1995)

  31. Murphy G.E.: A new construction of Young’s seminormal representation of the symmetric groups. J. Algebra 69, 287–297 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  32. Morozov A., Shakirov Sh.: On equivalence of two Hurwitz matrix models. Mod. Phys. Lett. A 24, 2659–2666 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  33. Natanzon, S.M., Orlov, A.Yu.: Hurwitz numbers and BKP hierarchy. arXiv:1407.8323

  34. Natanzon, S.M., Orlov, A.Yu.: BKP and projective Hurwitz numbers. arXiv:1501.01283

  35. Okounkov A.: Toda equations for Hurwitz numbers. Math. Res. Lett. 7, 447–453 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  36. Orlov A.Yu., Scherbin D.M.: Hypergeometric solutions of soliton equations. Theor. Math. Phys. 128, 906–926 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  37. Pandharipande R.: The Toda equations and the Gromov–Witten theory of the Riemann sphere. Lett. Math. Phys. 53, 59–74 (2000)

    Article  MathSciNet  Google Scholar 

  38. Takasaki, K.: Initial value problem for the Toda lattice hierarchy. In: Group representation and systems of differential equations, Adv. Stud. in Pure Math. vol. 4, pp. 139–163 (1984)

  39. Takebe T.: Representation theoretical meaning of the initial value problem for the Toda lattice hierarchy I. Lett. Math. Phys. 21, 77–84 (1991)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  40. Ueno, K., Takasaki, K.: Toda Lattice hierarchy. In: Group representation and systems of differential equations, Adv. Stud. in Pure Math. vol. 4, pp. 1–95 (1984)

  41. Zograf, P.: Enumeration of Grothendieck’s dessins and KP hierarchy. Int. Math. Res. Notices (2015). doi:10.1093/imrn/rnv077. arXiv:1312.2538

  42. Zinn-Justin P., Zuber J.-B.: On some integrals over the U(N) unitary group and their large N limit. J. Phys. A 36, 3173–3194 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Université du Québecà Montréal, 201 Av du Président-Kennedy, Montréal, QC, H2X 3Y7, Canada

    Mathieu Guay-Paquet

  2. Centre de recherches mathématiques, Université de Montréal, C. P. 6128, succ. centre ville, Montréal, QC, H3C 3J7, Canada

    J. Harnad

  3. Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke W., Montréal, QC, H4B 1R6, Canada

    J. Harnad

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  1. Mathieu Guay-Paquet
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Correspondence to J. Harnad.

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Work supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds de recherche du Québec – Nature et technologies (FRQNT).

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Guay-Paquet, M., Harnad, J. 2D Toda τ-Functions as Combinatorial Generating Functions. Lett Math Phys 105, 827–852 (2015). https://doi.org/10.1007/s11005-015-0756-z

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