FFT like theorems for tensor product

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 $\mathbb C[Sym^mV]^{S_n}$, $m \geq 2$ ?

• +1 for the audacity to ask something as hopeless as this. Mar 24 '12 at 3:53
• I thought this was going to be about fast Fourier transforms. Mar 24 '12 at 4:07
• One could break up into irreducible representations of $S_n$ and then try to answer for those. Is there a reason this might fail to give the best possible answer? Mar 24 '12 at 4:21