Integrable structures of specialized hypergeometric tau functions. (English) Zbl 1508.37100
Summary: Okounkov’s generating function of the double Hurwitz numbers of the Riemann sphere is a hypergeometric tau function of the 2D Toda hierarchy in the sense of A. Yu. Orlov and D. M. Scherbin [Theor. Math. Phys. 128, No. 1, 906–926 (2001; Zbl 0992.37063); translation from Teor. Mat. Fiz. 128, No. 1, 84–108 (2001); Physica D 152–153, 51–65 (2001; Zbl 0988.37091)]. This tau function turns into a tau function of the lattice KP hierarchy by specializing one of the two sets of time variables to constants. When these constants are particular values, the specialized tau functions become solutions of various reductions of the lattice KP hierarchy, such as the lattice Gelfand-Dickey hierarchy, the Bogoyavlensky-Itoh-Narita lattice and the Ablowitz-Ladik hierarchy. These reductions contain previously unknown integrable hierarchies as well.
MSC:
| 37K60 | Lattice dynamics; integrable lattice equations |
| 39A36 | Integrable difference and lattice equations; integrability tests |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 37K20 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions |
| 33C80 | Connections of hypergeometric functions with groups and algebras, and related topics |
| 05E05 | Symmetric functions and generalizations |
