# Ring of invariants for the regular representation

The symmetric group $S_n$ acts on $\mathbb C^n$ by permuting the coordinates. In this case the ring of invariants is generated by elementary symmetric polynomials in n-variables. Now consider the regular representation of $S_n$, the basis of this vector space is indexed by the elements of $S_n$. Then what are the generators for the ring of invariants ? Is there a reference where the generators are given explicitly ? What are the degrees of the generators ?

To the best of my knowledge this is an open problem. In fact, there is strong evidence that the problem is very hard indeed: Consider the action of $S_n$ on the "two-sets," i.e., on the subsets of $\{1,\ldots,n\}$ of two elements. It is easy to see that this is a subrepresentation of the regular representation. So if generating invariants for the regular representation are known, then these map to generating invariants for the action on the two-sets, so they are also known. But such invariants have only been canclulated for $n \le 5$.