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Is there a program (in Macaulay2 or singular or Gap or whatever) which explicitly gives the ring structure of $Spec[V]^G$ where $V$ is an explicitly given affine variety (over $\mathbb{C}$) and $G$ a reductive group acting on $V$ ?

As a background: in supersymmetric quantum field theory, we often come across so-called Seiberg dualities, one of whose consequences is that two quotients $Spec[V]^G$ and $Spec[W]^H$ are the same. Therefore it's important for us to check the equality, but doing this by hand is often extremely tedious. So I'm looking for a way to automate it.

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Perhaps the Singular library called "rinvar_lib" (the documentation for which is in "Appendix D.7 Invariant Theory" in the manual) has the features that you need. –  user2490 Sep 1 '11 at 13:58
This seems way too hard. If G=SL2 and V=Sym^d(C^2) that's the ring of invariants for binary forms. I think it is only known up to d=10. –  Abdelmalek Abdesselam Sep 1 '11 at 19:46
@James-Parson Thanks, I'll have a look. @Abdelmalek Yes, but I was asking if there's an algorithm implemented in a computer algebra system. If that algorithm finishes in a reasonable amount of time is a separate question... –  Yuji Tachikawa Sep 2 '11 at 5:21
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2 Answers

In principle if one has an a priori degree bound on the generators for the ring of invariants then one has a brute force algorithm. Such a bound was found by Popov. See for instance:

H. Derksen, H. Kraft, Constructive Invariant Theory, Algèbre non commutative, groupes quantiques et invariants (Reims, 1995), 221-244, Sémin. Congr. 2, Soc. Math. France, Paris, 1997.

Another general algorithm is in: H. Derksen, Computation of reductive group invariants, Adv. in Math. 141 (1999), 366-384. There is also this book.

If you want an algorithm which runs in a reasonable time, then you probably need an algorithm which is custom-made for the problem at hand, rather than the most general algorithm. For the case of binary forms I mentioned, I believe the most efficient algorithm is the one given by Gordan in 1868, but it has never been run on computer.

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The search term here is "Invariant Theory", plus something like "algorithm" or "computational". The two standard books are

Algorithms in Invariant Theory, by Bernd Sturmfels

Computational Invariant Theory, by Harm Derksen and Gregor Kemper.

I must admit that I haven't read either, although I can generally say that Bernd and Harm are both good writers.

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