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I've just started reading about classical invariant theory and I'm not seeing how the general pattern should work, maybe it's obvious I don't know...

Let $k$ be a field with $\mathrm{char}\ k = 0$, and let $V$ be the (usual) three dimensional permutation representation $V$ of $\Sigma_3$, the symmetric group on three objects.

Denote by $k[V]^{\Sigma_3}$ the ring of invariants of $\Sigma_3$ over $V$. Then $k[V]^{\Sigma_3}$ is the polynomial ring $k[s_1,s_2,s_3]$ where $s_1,s_2,s_3$ are the so-called elementary symmetric polynomials in three variables, by the fundamental theorem of symmetric polynomials. Analogously, the case $k[V \oplus V]^{\Sigma_3}$ is given by the polarisations of the elementary symmetric polynomials, as prooved by Weyl (in the general case).

Question: How does the form of the generators change (apart from number) if instead we take the standard two dimensional irreducible representation $U$ (this one) of $\Sigma_3$ i.e. the case $k[U]^{\Sigma_3}$ and similarly $k[U \oplus U]^{\Sigma_3}$?

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