I am interested in the following question: let $n\geq 2$ be an integer and consider a product of $n$ copies of the projective line:
$X=\mathbb{P}^1 \times \cdots \times \mathbb{P}^1$
(say everything is defined over an algebraically closed field of characteristic zero). Consider the automorphism $\sigma: X \to X$ sending $(t_1, \ldots, t_n)$ to $(t_n, t_1, \ldots, t_{n-1})$. Is it possible to construct equivariant morphism from $X$ to $\mathbb{P}^1$ for the trivial action on the later. If so, which are the easier ones? I guess the answer has to do with symmetric functions but I have not managed to get things written down.
Thanks for your help!