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

TL;DR

This paper extends the determinant representation of KP $ au$ functions to 2D Toda $ au$ functions, providing new formulas and recursion methods for weighted Hurwitz numbers and paths in Cayley graphs, with applications to finite-dimensional systems.

Contribution

It introduces a determinant representation and recursion formulas for weighted Hurwitz numbers within the 2D Toda $ au$ functions framework, generalizing previous results and linking to Cayley graph paths.

Findings

Derived a finite-dimensional equation system for weighted Hurwitz numbers.

Established a determinant representation for weighted Hurwitz numbers.

Provided a recursion formula for higher-dimensional weighted Hurwitz numbers.

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 $H^d_{G}(\sigma,\omega)$ 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.