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 (4)

• On KP-integrable Hurwitz functions
• The matrix model for hypergeometric Hurwitz numbers
• Enumerative geometry, tau-functions and Heisenberg-Virasoro algebra
• Explicit closed algebraic formulas for Orlov-Scherbin $n$-point functions