Let $X$ be a rational variety and let $G$ be a finite group acting on $X$. Let us consider the diagonal action of $G$ over the product $X^{h} = X\times...\times X$,

$$G\times(X\times...\times X)\rightarrow X\times...\times X,\quad (g,x,...,x)\mapsto (gx,...,gx).$$

Let $Y= X^{h}/G$ be the quotient. **Are there hypothesis under which $Y$ is rational ?**