Integrable structures of specialized hypergeometric tau functions

Abstract

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 Orlov and Scherbin. 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.

arXiv commentlatex2e, 21pages, no figure, submitted to proceedings of RIMS workshop "Mathematical structures of integrable systems, its deepening and expansion" (September 9-11, 2019); (v2) many typos are corrected and a few formulae are added after Proposion 8.1

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Cited by in corpus (2)

• Higher genera Catalan numbers and Hirota equations for extended nonlinear Schroedinger hierarchy
• Grothendieck's Dessins d'Enfants in a Web of Dualities. III