If $H = S\_n$ then then the [*fundamental symmetric polynomials*](https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial) allow to write any $S\_n$-invariant polynomial $f$ as a polynomial expression of these elementary symmetric functions. In other words, $\mathbb{C}[x\_1, \dots ,x\_n]^{S\_n} = \mathbb{C}[e\_1, \dots ,e\_n]$.


Now, given $H \le S\_n$ a subgroup of the symmetric group, is there a general way to compute a system of invariants for $\mathbb{C}[x\_1, \dots , x\_n]^H$ ?


EDIT:


A brute-force approach to find invariants for $\mathbb{C}[x\_1, \dots,x\_n]^H$ might be the Reynolds Operator?


It is defined as:


$$R : \mathbb{R}[x\_1, \dots, x\_n] \rightarrow \mathbb{R}[x\_1, \dots, x\_n]^H$$


$R(f):= \frac{1}{|H|}\sum\_{g \in H}f(g \cdot \textbf{x})$.


So just for simplicity consider $\mathbb{Z}\_3 \le S\_3$ and we define a group action


\begin{align}
\mathbb{Z}\_3 \times V &\rightarrow V\\
(g, \textbf{x}) &\mapsto g \cdot \textbf{x} := (x\_1+g, \dots, x\_n +g)
\end{align}


where $V$ is a real vector space of dimension $d$.


So for example if $d=2$ and we have $f(x,y)=x+y$ then the $\mathbb{Z}\_3$ invariant version of $f$ might be


$$f\_{inv}(x,y)= f(x,y)+f(x+1,y+1)+f(x+2,y+2) = 3x+3y$$


Is this correct?



A question about finding a system of invariants for a subgroup $H$ of the symmetric group $S_n$
It depends what you mean by "compute. The ring $R$ of invariants is spanned as a vector space by symmetrized monomials $\sum\_{h\in H} h\cdot m$, where $m$ is a monomial.
$R$ is generated as a $\mathbb{C}$-algebra by those symmetrized monomials of degree at most $h$. However, there is no nice description of a minimal generating set or the relations (syzygies) among these generators, etc. Various algebra packages can make these computations for reasonably small groups. Some additional information is [here](https://math.mit.edu/~rstan/pubs/pubfiles/38.pdf). See Theorem 1.2 in particular. This paper deals with more general group actions than $H\leq S\_n$, but this extra condition does not affect the difficulty in finding a minimal set of generators.


