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I am working on an exercise and not sure where to start.

Let $K=(\mathbb{C}^2)^{n+3}$. The special linear group $SL_2$ acts naturally on each $\mathbb{C}^2$ and hence on $K$. Let $R$ be the field of rational functions on $K$. The claim is that the subfield $F$ of $K$ which is invariant under the action of $SL_2$ is generated by the Plucker coordinates $\det(v_i,v_j)$, where each $v_i\in\mathbb{C}^2$.

It is clear to me that each Plucker coordinate is in fact $SL_2$ invariant, but I am unclear on how to prove that $F$ is actually generated by these coordinates.

Thanks!

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