2
$\begingroup$

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.

1 Answers 1

2

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?

  • 0
    Yes I think I can and thank you for your hint. Could you please suggest some references about invariant polynomials for more study? – 2017-02-06
  • 1
    What kind of reference do you want? A favorite of mine is "symmetric functions and Hall polynomials" by Macdonald. There is also likely work by Broue (pertaining to complex reflection groups). I'm not that up to date on this literature. – 2017-02-08
  • 0
    Thank you very much. I think this to references that you recommended, would be useful – 2017-02-08