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Isotropy group for the subset of Grassmannian

Consider a complex Grassmannian $Gr_{k}(C^{n})$, which is a symmetric space with symmetry group $U(n)$ (i.e. unitary group). Consider a subspace $S_{0}$ of the Grassmannian determined by the canonical ...
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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} = ...