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 ?
1 Answer
Knowledge of these generators would allow one to have a better understanding of the graph isomorphism problem; this is explained e.g. in the book by Derksen and Kemper "Computational invariant theory".
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).
As well, while computing explicit generators is hard, computing the Molien series for the ring of invariants is much easier (you just need the character of the representation), and they give you a good hint on what to expect from degrees (nothing too good, that is...).

$\begingroup$ Could you please give me a reference where the generators are written explicitly for $n=5$ ? $\endgroup$– MathewJun 27, 2016 at 8:38