A question about finding a system of invariants for a subgroup $H$ of the symmetric group $S_n$

Question

If $H = S_n$ then then the fundamental symmetric polynomials 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?

Answer 1

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. 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.