论文标题
2d Toda $τ$功能，加权Hurwitz编号和Cayley图：行列式表示和递归公式
2D Toda $τ$ Functions, Weighted Hurwitz Numbers and the Cayley Graph: Determinant Representation and Recursion Formula
论文作者
论文摘要
我们将KP $τ$函数的决定性表示形式概括为2D TODA $τ$函数的情况。加权Hurwitz数字的生成功能是2D Toda $τ$函数的参数族；为此，我们给出了加权Hurwitz数字的决定性代表。然后，我们可以获得加权Hurwitz数字$ H^d_ {g}（σ，ω）$的有限维方程系统，具有相同的尺寸$ | = | = |ω| = n $。使用此方程式系统，我们计算了具有尺寸$ 0，\，1，\，2 $的加权Hurwitz数字的值，并给出一个递归公式来计算更高维度加权的Hurwitz数字。对于任何给定的加权生成函数$ g（z）$，当$ d = 0 $时，加权的Hurwitz编号将退化为Hurwitz编号。我们获得了Hurwitz数字的矩阵表示。对称组的Cayley图中加权路径的生成函数是2D TODA $τ$函数的参数族。为此，我们获得了Cayley图中加权路径的决定性表示。
We generalize the determinant representation of the KP $τ$ 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 $H^d_{G}(σ,ω)$ with the same dimension $|σ|=|ω|=n$. Using this equation system, we calculated the value of the weighted Hurwitz numbers with dimension $0,\,1,\,2$ 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 $τ$ functions; for which we obtain a determinant representation of weighted paths in the Cayley graph.
