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$S_k$ action on $A/I$
Let $S_2$ be a finite group of order $2$ and let $S_2$ act on $k[x,y]$ by interchanging $x$ and $y$, where $k=\overline{k}$. Then since
$$
R = \left( \dfrac{k[x,y]}{(x+y)}
\right)^{S_2}
= ...
