Lüroth's theorem for fields of rational functions in infinitely many permuted variables

Lüroth’s theorem describes the dominant maps from rational curves over a field. In this note we study those dominant rational maps from cartesian powers $X^{\mathrm{\Psi }}$ of geometrically irreducible varieties $X$ over a field $k$ for infinite sets $\mathrm{\Psi }$ that are equivariant with respect to all permutations of the factors $X$. At least some of such maps arise as compositions $h:X^{\mathrm{\Psi }}\stackrel{f^{\mathrm{\Psi }}}{\to }{Y}^{\mathrm{\Psi }}\to H\mathrm{\backslash }{Y}^{\mathrm{\Psi }}$, where $X\stackrel{f}{\to }Y$ is a dominant $k$-map and $H$ is a group of birational automorphisms of $Y\mathrm{\mid }k$, acting diagonally on $Y^{\mathrm{\Psi }}$. In characteristic 0, we show that this construction, when properly modified, gives all dominant equivariant maps from $X^{\mathrm{\Psi }}$, if $\dim X=1$. For arbitrary $X$, the results are only partial. Also, a somewhat similar problem of describing the equivariant integral schemes over $X^{\mathrm{\Psi }}$ of finite type is touched very briefly.