for any group $G$, a $G$-invariant polynomial over the reals is a multivariate polynomial $f\in \mathbb R[x]$ such that, for every $g\in G$ we have $f(x)=f(\gamma(g)x)$, where $\gamma$ is the general linear group representation of $G$. I want to show that:
A polynomial is $G$-invariant if and only if all its homogeneous components are invariant.
But I cant prove that. Can you give me a hint please?
All the monomials those have same degree is called a homogeneous component. I know that product and sum of two $G$-invariant polynomials is $G$-invariant.
I assume you mean the following: Identify $G$ with a subgroup of $S_n$ (via Cayley's theorem, say). Then, for $\sigma\in G\leq S_n$, we have $$\sigma.x_i=x_{\sigma(i)}.$$ Then note that $$\sigma.(x_{i_1}^{k_1}\cdots x_{i_r}^{k_r})=x_{\sigma(i_1)}^{k_1}\cdots x_{\sigma(i_r)}^{k_r}$$ It follows that $\deg(x_{i_1}^{k_1}\cdots x_{i_r}^{k_r})=k_1+\cdots+k_r=\deg(\sigma.(x_{i_1}^{k_1}\cdots x_{i_r}^{k_r}))$. That is, elements of $G$ act by degree preserving automorphisms of $\mathbb{R}[x_1,\ldots,x_n]$.
Can you prove it from here?