Question

The fundamental theorem of symmetric functions states that $\mathbb C[V]^{S_n}$ is generated by the elementary symmetric polynomials, where $V$ is the natural representation of $S_n$. Then a theorem of Herman Weyl says that $\mathbb C[\oplus^mV]^{S_n}$ is generated by the polarizations of elementary symmetric polynomials, which is known as the first fundamental theorem (in short FFT) of Invariant theory. Is there any relevant theorem for tensor products or the symmetric algebra ? I mean what are the generators and relations for $\mathbb C[\otimes^mV]^{S_n}$ and $\mathbb C[Sym^mV]^{S_n}$, $m \geq 2$ ?