I understand the basics of invariant theory in commutative algebra. Assume we are working over a polynomial ring in n variables over a field of characteristic 0. What I’m looking for are examples or results of an inverse process. I was thinking along the lines of given a homogeneous polynomial f find the largest (maximal?) subgroup G of $\mathrm{SL}(n, k)$ for which f is invariant under the action of G on the polynomial ring. Some examples of types of questions I have are:
If such a G could be found given f then would $k[x_{1}, ..., x_{n}]^{G}$ be the smallest subring of $k[x_{1}, ..., x_{n}]$ containing f that is an invariant subring for some subgroup of $\mathrm{SL}(n, k)$?
EDIT: Here’s an example of what I’m thinking.
Consider $\mathbb{C}[x,y]$ and let $f$ and $h$ be homogeneous polynomials. Let $G_{f}$ be the largest subgroup of $\mathrm{SL}(2,\mathbb{C})$ that leaves $f$ invariant and let $G_h$ be similarly defined. Let S be a subring of $\mathbb{C}[x,y]$ that contains $f$ and $g$ that is a ring of invariants for some subgroup $G$ of $\mathrm{SL}(n, k)$. I want to conclude that $$\mathbb{C}[x,y]^{G_{f}\cap G_{h}} \subset S $$ Would this be the same thing as saying that $G_{f}\cap G_{h}$ contains $G$ as a subgroup?
I’m working with $\mathrm{SL}(n,k)$ because I want to say there is a unique largest subgroup with for with a given homogeneous polynomial is invariant. For instance the polynomial $x^2+y^2$ is invariant under the Klein 4 subgroup of $\mathrm{GL}(2, \mathbb{C})$ and under the cyclic group of order 4. It seems to me that restricting to $\mathrm{SL}(2,k)$ may allow one to say there is a largest subgroup but I don’t know this to be true.