2D Toda τ functions, weighted Hurwitz numbers and the Cayley graph: Determinant representation and recursion formula

Abstract

We generalize the determinant representation of the Kadomtsev–Petviashvili τ functions to the case of the 2D Toda τ functions. The generating functions for the weighted Hurwitz numbers are a parametric family of 2D Toda τ 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 HGd(σ,ω) with the same dimension |σ| = |ω| = n. Using this equation system, we calculated the value of the weighted Hurwitz numbers with dimension 0, 1, 2, 3 and give a recursion formula for calculating the higher dimensional weighted Hurwitz numbers. Finally, we get a matrix representation for the Hurwitz numbers and obtain a determinant representation of weighted paths in the Cayley graph.