Ring of Invariants of symmetric group 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 the vector space is indexed by the elements of $S_n$. Then what are the generators for the ring of invariants ? I guess the elementary symmetric polynomial in $n!$ variables generate the ring but I am not sure. 

The elementary symmetric polynomial do not generate the ring if $n>2$, let $g_1,g_2$ be two distinct elements  of $S_n$, consider $G.(g_1,g_2)=\{gg_1,gg_2),g\in G\}$ the polynomial $\sum X_{gg_1}X_{gg_2}$ is invariant by $G$ but not invariant by $S_{n!}$.
