2D Toda $\tau$ Functions, Weighted Hurwitz Numbers and the Cayley Graph: Determinant Representation and Recursion Formula

Abstract

We generalize the determinant representation of the KP $\tau$ functions to the case of the 2D Toda $\tau$ functions. The generating functions for the weighted Hurwitz numbers are a parametric family of 2D Toda $\tau$ functions; for which we give a determinant representation of weighted Hurwitz numbers. Then we can get a finite-dimensional equation system for the weighted Hurwitz numbers $H_G^d(\sigma ,\omega )$ with the same dimension $|\sigma |=|\omega |=n$. Using this equation system, we calculated the value of the weighted Hurwitz numbers with dimension $0,1,2$ and give a recursion formula to calculating the higher dimensional weighted Hurwitz numbers. For any given weighted generating function $G(z)$, the weighted Hurwitz number degenerates into the Hurwitz numbers when $d=0$. We get a matrix representation for the Hurwitz numbers. The generating functions of weighted paths in the Cayley graph of the symmetric group are a parametric family of 2D Toda $\tau$ functions; for which we obtain a determinant representation of weighted paths in the Cayley graph.

References in corpus (18)

• Generalized Kazakov-Migdal-Kontsevich Model: group theory aspects
• On KP-integrable Hurwitz functions
• Free fermions and tau-functions
• Integrability of Hurwitz Partition Functions. I. Summary
• Virasoro constraints and topological recursion for Grothendieck's dessin counting
• Hypergeometric τ-functions, Hurwitz numbers and enumeration of paths
• Multivariate hypergeometric functions as tau functions of Toda lattice and Kadomtsev-Petviashvili equation
• Combinatorial solutions to integrable hierarchies
• On Hurwitz numbers and Hodge integrals
• The matrix model for hypergeometric Hurwitz numbers
• Generating functions for weighted Hurwitz numbers
• Weighted Hurwitz numbers and topological recursion
• Enumerative geometry, tau-functions and Heisenberg-Virasoro algebra
• Explicit closed algebraic formulas for Orlov-Scherbin $n$-point functions
• Fermionic approach to weighted Hurwitz numbers and topological recursion
• Toda Equations and Piecewise Polynomiality for Mixed Double Hurwitz Numbers
• From $r$-Spin Intersection Numbers to Hodge Integrals
• Constellations and $\tau$-functions for rationally weighted Hurwitz numbers