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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Alexandrov, A., Mironov, A., Morozov, A., Natanzon, S.: On KP-integrable Hurwitz functions. JHEP. doi:10.1007/JHEP11(2014)080
Ambjørn J., Chekhov L.: The matrix model for dessins d’enfants. Ann. Inst. Henri Poincaré, Comb. Phys. Interact. 1, 337–361 (2014)
Ambjørn J., Chekhov L.: A matrix model for hypergeometric Hurwitz numbers. Theor. Math. Phys. 181, 1486–1498 (2014)
Borot G., Eynard B., Mulase M., Safnuk B.: A matrix model for Hurwitz numbers and topological recursion. J. Geom. Phys. 61, 522–540 (2011)
Bertola M., Prats-Ferrer A.: Topological expansion for the Cauchy two-matrix model. J. Phys. A Math. Theor. 42, 335201 (2009)
Bertola M., Gekhtman M., Szmigielski J.: The Cauchy two-matrix model. Commun. Math. Phys. 287, 983–1014 (2009)
Corteel S., Goupil A., Schaeffer G.: Content evaluation and class symmetric functions. Adv. Math. 188(2), 315–336 (2004)
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)
Diaconis, P., Greene, C.: Applications of Murphy’s elements. Stanford Technical Report, vol. 335 (1989)
Eynard B., Orantin N.: Topological recursion in enumerative geometry and random matrices. J. Phys. A 42, 293001 (2009)
Farahat H.K., Higman G.: The centers of symmetric group rings. Proc. Roy. Soc. London Ser. A 250, 212–221 (1959)
Guay-Paquet, M., Harnad, J.: Generating functions for weighted Hurwitz numbers. arXiv:1408.6766
Goulden I.P.: A differential operator for symmetric functions and the combinatorics of multiplying transpositions. Trans. Am. Math. Soc. 344(1), 421–440 (1994)
Goulden I.P., Guay-Paquet M., Novak J.: Monotone Hurwitz numbers and the HCIZ Integral. Ann. Math. Blaise Pascal 21, 71–99 (2014)
Goulden, I.P., Guay-Paquet, M., Novak, J.: Toda equations and piecewise polynomiality for mixed double Hurwitz numbers. arXiv:1307.2137
Harish-Chandra.: Differential operators on a semisimple Lie algebra. Am. J. Math. 79, 87–120 (1957)
Harnad, J.: Multispecies weighted Hurwitz numbers. arXiv:1504.07512
Harnad, J.: Quantum Hurwitz numbers and Macdonald polynomials. arXiv:1504.03311
Harnad, J.: Weighted Hurwitz numbers and hypergeometric τ-functions: an overview. arXiv:1504.03408
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)
Harnad J., Orlov A.Yu.: Scalar products of symmetric functions and matrix integrals. Theor. Math. Phys. 137, 1676–90 (2003)
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)
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)
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
Itzykson C., Zuber J.-B.: The planar approximation. II. J. Math. Phys. 21, 411–21 (1980)
Jucys A.A.: Symmetric polynomials and the center of the symmetric group ring. Rep. Math. Phys. 5(1), 107–112 (1974)
Kazarian M.: KP hierarchy for Hodge integrals. Adv. Math. 221, 1–21 (2009)
Kazarian, M., Zograf, P.: Virasoro constraints and topological recursion for Grothendieck’s dessin counting. arXiv:1406.5976
Kontsevich M.: Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys. 147, 1–23 (1992)
Macdonald, I.G.: Symmetric functions and hall polynomials. Clarendon Press, Oxford. (1995)
Murphy G.E.: A new construction of Young’s seminormal representation of the symmetric groups. J. Algebra 69, 287–297 (1981)
Morozov A., Shakirov Sh.: On equivalence of two Hurwitz matrix models. Mod. Phys. Lett. A 24, 2659–2666 (2009)
Natanzon, S.M., Orlov, A.Yu.: Hurwitz numbers and BKP hierarchy. arXiv:1407.8323
Natanzon, S.M., Orlov, A.Yu.: BKP and projective Hurwitz numbers. arXiv:1501.01283
Okounkov A.: Toda equations for Hurwitz numbers. Math. Res. Lett. 7, 447–453 (2000)
Orlov A.Yu., Scherbin D.M.: Hypergeometric solutions of soliton equations. Theor. Math. Phys. 128, 906–926 (2001)
Pandharipande R.: The Toda equations and the Gromov–Witten theory of the Riemann sphere. Lett. Math. Phys. 53, 59–74 (2000)
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)
Takebe T.: Representation theoretical meaning of the initial value problem for the Toda lattice hierarchy I. Lett. Math. Phys. 21, 77–84 (1991)
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)
Zograf, P.: Enumeration of Grothendieck’s dessins and KP hierarchy. Int. Math. Res. Notices (2015). doi:10.1093/imrn/rnv077. arXiv:1312.2538
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)
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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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DOI: https://doi.org/10.1007/s11005-015-0756-z