# How do I find the orbits given the ring of invariants?

Suppose I have a group $G$ acting linearly on a vector space $V$ and I know the ring of invariants $K[V]^{G}$ (i.e., I know the subring of $K[V]$ which is fixed pointwise under the induced action). My questions are:

1. Under what conditions does the ring of invariants determine the orbits (I think I read something about how you can do this for reductive groups but I'm not sure if that is true and if it is how you actually go about doing it)?

2. How do you calculate representatives for the orbits in cases where it is determined?

(and in both cases where can I find out some more information)

Personally I am interested in the case where you are considering the action of the whole of $GL_n(V)$ where $V$ is over some finite field. Here the ring of invariants is the so-called Dickson algebra which is the algebra generated by the elementary symmetric polynomials in the dual vectors.

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This question may be too advanced for this site. I would recommend posting this question on Math Overflow (mathoverflow.net). – Jim Belk Jun 21 '11 at 1:39