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Suppose we have a ring $R$ and a finite group $G$ acting on it, Is there a way to compute the invariant ring $R^G$ explicitly? Infact I am more interested in the case of affine ring and the symmetric group acting on it and I want to have the relations in the invariant ring explicitly.

More presicely, I have an algebra having finitely many generators and finitely many relation and the symmetric group acts on it. I want to have the relations in the invariant ring explicitly. I could not find a way to do so in the books I have seen so far.

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    $\begingroup$ To second Liviu's comment, you should also be more specific about the action of the symmetric group you are interested in. The case of multi-symmetric polynomials where S_n rigidly permutes groups of variables is for instance much more complicated than the obvious action of S_n on C^n. $\endgroup$
    – Abdelmalek Abdesselam
    Commented Oct 24, 2012 at 13:46
  • $\begingroup$ @ Abdelmalek and Liviu I mean the ring of functions on an arbitrary affine variety and the obvious action of the symmetric group. I am going to have a look on the article and book mentioned. Thank you for your responses. $\endgroup$
    – George
    Commented Oct 25, 2012 at 5:20

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Can you be more specific on the meaning of affine ring? More precisely do you mean the ring of function on an arbitrary affine variety or just the ring of functions on $\mathbb{C}^n$? In any case, this nice article of R. Stanley may be a good place to start.

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See the book "Invariant Theory of Finite Groups" by Neusel and Smith, as well as this other book by Derksen and Kemper.

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