# Tensor and symmetric invariants of Symmetric group

For the action of $S_n$ on $\mathbb C^n$ the elementary symmetric polynomials generate the ring of polynomial invariants. What are the generators for the action of $S_n$ on $\mathbb C^n \otimes \mathbb C^n$, $Sym^2(\mathbb C^n)$ and $\Lambda^2 \mathbb C^n$ ? Since they are permutation representations the orbit sums generate the ring of invariants but are they known explicitly ? At least for small n's are the degrees of a minimal generating set known ?

IIRC, minimal generators for $Sym^2(\mathbb{C}^n)$ are known for $n\leq 5$. One can probably go a bit further nowadays (these results are more than 10 years old, and computers got bigger).
• Could you please give me a reference where the generators are written explicitly for $n=5$ ? – Mathew Jun 27 '16 at 8:38