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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?

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  • $\begingroup$ What does $x_1+g$ mean? $x_1$ is a real number and $g\in\mathbb{Z}_3$, so it makes no sense to add them. The action you want is $(g,\boldsymbol{x})\mapsto (x_{g(1)},x_{g(2)},x_{g(3)})$, where we are regarding $g$ as a permutation of $1,2,3$. $\endgroup$ Nov 30, 2021 at 21:05
  • $\begingroup$ Is the dimension of the vector space dependent on the number of the elements we have in the group? So it seems that here you have chosen a vector of dimension $3$ to define the action of $\mathbb{Z}_3$. $\endgroup$ Dec 1, 2021 at 15:23

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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.

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  • $\begingroup$ Thanks for the paper reference, it seems pretty interesting. I have also edited my question with additional details and considerations. $\endgroup$ Nov 30, 2021 at 16:01

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