BKP and projective Hurwitz numbers. (English) Zbl 1368.05009
Summary: We consider \(d\)-fold branched coverings of the projective plane \(\mathbb {RP}^2\) and show that the hypergeometric tau function of the BKP hierarchy of Kac and van de Leur is the generating function for weighted sums of the related Hurwitz numbers. In particular, we get the \(\mathbb {RP}^2\) analogues of the \(\mathbb {CP}^1\) generating functions proposed by A. Okounkov [Math. Res. Lett. 7, No. 4, 447–453 (2000; Zbl 0969.37033)] and by I. P. Goulden and D. M. Jackson [Adv. Math. 219, No. 3, 932–951 (2008; Zbl 1158.37026); Proc. Am. Math. Soc. 125, No. 1, 51–60 (1997; Zbl 0861.05006)]. Other examples are Hurwitz numbers weighted by the Hall-Littlewood and by the Macdonald polynomials. We also consider integrals of tau functions which generate Hurwitz numbers related to base surfaces with arbitrary Euler characteristics \(e\), in particular projective Hurwitz numbers \(e=1\).
MSC:
| 05A15 | Exact enumeration problems, generating functions |
| 14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |
| 17B80 | Applications of Lie algebras and superalgebras to integrable systems |
| 35Q51 | Soliton equations |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 37K20 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions |
| 37K30 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures |
Keywords:
Hurwitz numbers; tau functions; BKP; projective plane; Schur polynomials; Hall-Littlewood polynomials; hypergeometric functions; random partitions; random matricesReferences:
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