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It is a well-known theorem that the ring of polynomials $P \in \mathbb{C}[x_1 , \dots , x_n]$ that are invariant under all the permutations $\sigma \in \mathfrak{S}_n$ is in fact $\mathbb{C}[e_1 , \dots ,e_n]$ where the $e_i$ are the elementary symmetric polynomials, that can be seen as being generated by $$(x-x_1)(x-x_2)\dots (x-x_n). $$ We write this as $$ \mathbb{C}[x_1 , \dots , x_n]^{\mathfrak{S}_n} = \mathbb{C}[e_1 , \dots e_n]. $$

Consider a subgroup $G \subset \mathfrak{S}_n$. Is there a similar description of the ring of polynomials that are invariant under $G$ ?

By similar, I mean the following: is it possible to find explicitly a minimal finite set of polynomials $f_1 , \dots , f_r \in \mathbb{C}[x_1 , \dots , x_n]$ such that $$ \mathbb{C}[x_1 , \dots , x_n]^G = \mathbb{C}[f_1 , \dots , f_r] ? $$ It is known (see for instance here, theorem 1.2) that $r$ can be chosen smaller than $\frac{(g+n)!}{g!n!}$ where $g=|G|$.

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I don't know what you mean by similar description. There is always a finite set of invariant polynomials such that all invariant polynomials are consequences of these being invariant. That is they are generated as ring by these.

However the invariants cannot be expected to be algebraically independent. A necessary and sufficient condition for this to happen is the subgroup be generated by reflections: that is in the linear span of $x_i$'s the representation of this subgroup should be generated by (pseudo) reflections. The sufficiency was proved by Chevalley, and necessity by Shepherd-Todd. See book by Humphreys, or Kane, or Kemper.

EDIT: for computational aspects see the books by Kemper & Derksen, or the book by Sturmfels (Algorithmic Invariant Theory)

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  • $\begingroup$ I have edited the question to be more precise. I'm looking for a way to compute the $f_i$. $\endgroup$ – Antoine Oct 27 '16 at 16:23

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