$E_W$ given by $$E_W=\sum_{k=1}^N q_k v_k \frac{{{\partial}}}{\d v_k}.$$ The conformal dimension is $\delta=1-q_l$.

The coordinates $v_1,\dots,v_N$ are not flat whenever $N \ge 3$.

There exist unique [*global*]{} flat coordinates $t^i(v_1,\ldots,v_N)$ on ${\mathbb C}^N$ such that $$t^i(v_1,\ldots,v_N)=v_i+O(v_*^2).$$ They satisfy the quasi-homogeneity condition $$\begin{gathered}
\label{eq:homogeneity of flat coordinates}
E_W(t^i(v_1,\ldots,v_N))=q_i t^i(v_1,\ldots,v_N),\end{gathered}$$ and, hence, the Euler vector field in the flat coordinates $t^i$ is given by $$E_W=\sum_{i=1}^N q_i t^i\frac{\d}{\d t^i}.$$ The Frobenius manifold structure in the flat coordinates $t^i$ is described by a polynomial potential $F_W(t^1,\ldots,t^N)\in{\mathbb C}[t^1,\ldots,t^N]$, which we fix by requiring that it doesn’t contain monomials of degree less than $3$. Then the polynomial $F_W$ satisfies the condition $$\begin{gathered}
\label{eq:homogeneity of potential for Coxeter}
E_W(F_W)=(3-\delta)F_W.\end{gathered}$$

Explicit formulas for the flat coordinates of the Saito Frobenius manifolds of simple singularities are given in [@NY98]. For the $A_N$-case the formula is $$t^\gamma(v_1,\ldots,v_N) = \hspace{-0.2cm}\sum_{\substack{\alpha_1,\ldots,\alpha_N\ge 0\\ \sum (N+2-i)\alpha_i = N+2-\gamma}}\hspace{-0.2cm}\frac{1}{N+1-\gamma}\prod_{k=0}^{\sum\alpha_i-1}\left(N+1-\gamma-k(N+1)\right)\frac{\prod v_i^{\alpha_i}}{\prod \alpha_i!}, \quad 1\le\gamma \le N.$$ For the $D_N$-case the formula is $$\begin{aligned}
&t^\gamma(v_1,\ldots,v_N) = \hspace{-0.2cm}\sum_{\substack{\alpha_1,\ldots,\alpha_{N-1}\ge 0\\ \sum (N-i)\alpha_i = N-\gamma}}\hspace{-0.1cm} \left(-\frac{1}{2}\right)^{\sum\alpha_i-1} \prod_{k=0}^{\sum\alpha_i-2} \left(2\gamma-1 + 2k(N-1) \right) \frac{\prod v_i^{\alpha_i}}{\prod \alpha_i!}, \quad 1\le\gamma \le N-1,\label{eq:Dn flat coordinates}\\
&t^N(v_1,\ldots,v_N)=v_N.\notag\end{aligned}$$ Note that in this case the coordinates $t^1,\ldots,t^{N-1}$ depend only on $v_1,\ldots,v_{N-1}$.

Extended $r$-spin theory and the open WDVV 