need to recall the way these generators are constructed. For clarity, we stick to the notations of [@BMR] ; in this paper, the authors introduce 4 different spaces, $\mathcal{M}(r+1) = \{ (z_0,\dots,z_r) \in \C^{r+1} \ | \ z_i \neq z_j \}$, $\mathcal{M}^{\#}(m,r) = \{ (z_1,\dots,z_r) \in \C^r \ | \ z_i \neq 0,
z_i /z_j \not\in \mu_m \}$, $\mathcal{M}(e,r) = \{ (z_1,\dots,z_r) \in \C^r \
| \ z_i \not\in \mu_e z_j \}$, and $\mathcal{M}^{\#}(r) = \{ (z_1,\dots,z_r) \in \C^r \ | \ z_i \neq 0 \}$, where $\mu_n$ denotes the set of $n$-th roots of 1 in $\C$. We have a Galois covering $r : \mathcal{M}^{\#}(m,r)
\to \mathcal{M}^{\#}(r) = \mathcal{M}^{\#}(m,r) / (\mu_m)^r$, a locally trivial fibration $p : \mathcal{M}(r+1) \to \mathcal{M}^{\#}(r)$ with fiber $\C$ given by $(z_0,\dots,z_r) \mapsto (z_0 - z_1,\dots,
z_0-z_r)$, and a natural action of $\mathfrak{S}_r$ on $\mathcal{M}(r+1)$ that leaves the $(r+1)$-st coordinate fixed. We choose a fixed point $x \in \mathcal{M}(r+1)/\mathfrak{S}_r$, and a lift $\widetilde{p(x)}$ of $p(x) \in \mathcal{M}^{\#}(r)/\mathfrak{S}_r$ in $\mathcal{M}^{\#}(d,r)/G(d,1,r) = (\mathcal{M}(d,r)/(\mu_m)^r)/\mathfrak{S}_r$. We get an isomorphism $\psi : \pi_1(\mathcal{M}^{\#}(d,r)/G(d,1,r),
\widetilde{p(x)}) \to \pi_1(\mathcal{M}(r+1)/\mathfrak{S}_r,x)$ by composing the isomorphisms induced by $r$ and $p$. $$\xymatrix{
 \pi_1(\mathcal{M}^{\#}(d,r)/G(d,1,r), \widetilde{p(x)}) \ar[dr]_{r}^{\simeq} \ar[rr]^{\psi} & &
\ar[dl]^{p}_{\simeq} \pi_1(\mathcal{M}(r+1)/\mathfrak{S}_r, x) \\
 & \pi_1(\mathcal{M}^{\#}(r)/\mathfrak{S}_r, p(x)) & 