# Determination of the quotient of an affine variety by a reductive group by computer

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

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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