On the topological type of a set of plane valuations with symmetries

Let $\{C_i:i=1,\dots ,r\}$ be a set of irreducible plane curve singularities. For an action of a finite group $G$, let $\Delta^{L}({t_{a i}})$betheAlexanderpolynomialin$r\vert G\vert$ variables of the algebraic link $(\bigcup\limits_{i=1}^{r}\bigcup\limits_{a\in G}a C_i )\cap S^3_{\varepsilon}$andlet$\zeta(t_1,\ldots, t_r) = \Delta^{L}(t_1,\ldots,t_1,t_2,\ldots,t_2, \ldots,t_r,\ldots,t_r)$with$\vert G\vert$identicalvariablesineachgroup\mathrm{.}(If$r=1$,$\zeta(t)$ is the monodromy zeta function of the function germ $\prod _{a\in G}{a}^{\ast }f$, where $f=0$ is an equation defining the curve $C_1$.) We prove that $\zeta (t_1,\dots ,t_r)$ determines the topological type of the link $L$. We prove an analogous statement for plane divisorial valuations formulated in terms of the Poincar'e series of a set of valuations.