# Symmetric functions for a subgroup of permutations

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 $g=|G|$.

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