Definition: Let $G$ be a group acting on $R_n:=K[x_1,\dots,x_n]$ with $$\begin{aligned} G\times R_n &\rightarrow R_n\\ (g,f) &\mapsto f^g \end{aligned} $$ where $(f^g)$ acts on a point $p$ in the follow way: $$(f^g)(p)=f(g\cdot p)$$ Let $I\subset R_n$ an ideal. $I$ is a G-invariant ideal if and only if for all $g \in G$ and for all $f \in I$, then $f^g \in I$.
Goals: My aim is to look for an example of such an ideal, for $R_n$ and $G$ fixed.
The most immediate thing that came to my mind is to take $G:=S_n$. But then which ideal of $R_n$ can I take to be $G$-invariant?
Consider che polynomials of the form
$h_k(x_1, \dots, x_n)= \sum_{i_1\dots i_k \in [n]} x_{i_1}\dots x_{i_k}$
It is easy to observe that these polynomials are $S_n$ invariants (the action on $h_k$ comes from the action of $S_n$ on the sets of the form $\{i_1\dots i_k\} $)
The ideal generated by the $h_k$ for $k \in [n]$ is $S_n$ invariant.