Reading this work https://arxiv.org/pdf/2012.06452.pdf, I'm wondering if it's true that any fundamental invariant polynomial $\{f_i\}_{i=1}^{N_{inv}}$ can be written as
$$f_i(x) = \sum_{g \in G}\psi(g \cdot x)$$
where $\psi(x) = \prod_{i=1}^n x_i^{b_i}$ for fixed exponents $b_i$. Here it is considered $G \le S_n$ and for $x \in \mathbb{R}^n$,
$$g \cdot x := (x_{\sigma_g(1)}, \dots x_{\sigma_g(n)})$$
is just permuting the coordinates.
So in this case I guess we are considering symmetric polynomials: for example, given $x \in \mathbb{R}^3$ and $S_3$ then I should be able to express the elementary symmetric polynomials (Wikipedia) using this form. In particular we have
$$e_0(x_1,x_2,x_3)=1 , $$
$$e_1(x_1,x_2,x_3) = x_1+x_2+x_3 ,$$
$$e_2(x_1,x_2,x_3) = x_1x_2 + x_1x_3 ,$$
$$e_3(x_1,x_2,x_3) = x_1x_2x_3. $$
Is it actually possible to describe all these fundamentals by assuming any $G \le S_3$? Can you eventually help me to see that?