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

Journal of Mathematical Physics（2023）

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.

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Key words

hurwitz numbers,determinant representation,cayley graph

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