2 problems

Let $K[X^1,\dots,X^k]$ be the polynomial ring in $k$ sets of countably many variables, with $\operatorname{Sym}(\infty)$ permuting the indices. Let $I\subset K[X^1,\dots,X^k]$ be a…

Let $K[X]$ be the polynomial ring in countably many variables, with the symmetric group acting by permuting variables. A homogeneous ideal $I\subset K[X]$ is symmetric if it is inv…