The celebrated Chevalley–Shephard–Todd theorem says that $\mathbb C[V]^{S\_n}$ is a polynomial algebra and gives the generators of this algebra, where $V$ is the standard (or natural) representation of the symmetric group $S\_n$. I am just curious to know for what other representations of $S\_n$ the generators of this algebra is known ? When is this algebra a polynomial algebra ? 



Invariants of Symmetric group
Let $n \ge 7$. If $V$ is an irreducible representation of $S\_n$ such that $\mathbb{C}[V]^{S\_n}$ is a polynomial algebra then either $V$ is the trivial representation, the sign representation or the $(n-1)$-dimensional standard representation. 


*Outline Proof*: Let $\rho : S\_n \rightarrow \mathrm{GL}(V)$ be an irreducible representation affording the irreducible character $\chi^\lambda$ where $\lambda$ is a partition of $n$. Suppose that $V$ is $d$-dimensional. By the Chevalley-Shephard-Todd theorem, $\mathbb{C}[V]^{S\_n}$ is a polynomial algebra if and only if $\rho(S\_n)$ is generated by pseudo-reflections. If $\rho(g)$ is a pseudo-reflection then $\rho(g)$ is similar to a diagonal matrix $\mathrm{diag}(1,1,\ldots,1,\zeta)$ where $\zeta$ is a root of unity. Hence $\chi(g) = d-1 + \zeta$. However, the irreducible characters of symmetric groups are real valued, so if $g \not= 1\_{S\_n}$ then $g$ is an involution and $\chi^\lambda(g) = d-2$. 


It therefore suffices to show that if $n \ge 7$ and $g \in S\_n$ is an involution such that $\chi^\lambda(g) = \chi^\lambda(1)-2$ then $g$ is a transposition and either $\lambda = (1^n)$ or $\lambda = (n-1,1)$. This follows by induction using the Murnaghan-Nakayama rule. (The details are fiddly but routine.) 


**Edit:** Geoff Robinson shows in his answer (posted at the same time as mine) that $g$ is a product of at most $3$ transpositions. This leads to a quick inductive proof: suppose that $\lambda$ has two removable boxes, whose removal gives partitions $\mu$ and $\nu$ of $n-1$. If $\mu \not= (n-1)$ and $\nu \not= (n-1)$ then, by induction, we have $\chi^\lambda(g) \le \chi^\lambda(1)-4$. Hence $\lambda = (n-1,1)$. In the remaining case $\lambda$ is a rectangular partition, and provided $n\ge 8$, we can repeat this argument after removing two boxes from $\lambda$ in two different ways.


For smaller $n$ there are two exceptional cases, corresponding to the partitions $(2,2)$ of $4$ and $(2,2,2)$ of $6$. The former representation is $2$-dimensional and is obtained from the standard representation of $S\_3$ via the quotient map $S\_4 \rightarrow S\_3$. The latter is $5$-dimensional, and can be obtained by applying the outer automorphism of $S\_6$ to the standard $5$-dimensional representation of $S\_6$; the relevant character value is $\chi^{(2,2,2)}(g) = 3$ where $g = (12)(34)(56) \in S\_6$.


