Suppose I have an action of a linear reductive group ($GL(2,\mathbb{C})^2$ in this case) on a complex vector space (of dimension $16$) and I want to compute explicitly the ring of invariants of this representation. I already have the candidates for the generators (certain polynomials of $16$ variables) but I need to find the relations between them.

My question is the following - is it possible to force Magma, Sage or other program to find these polynomial relations (hopefully the whole ideal of relations...) for me? I don't expect them to be complicated, but they are way too big to manage by hand. I know the topic is discussed in several books (Derksen & Kemper, Sturmfels etc.) but are there precise, already implemented algorithms doing this or should one do it by himself?

PS. I saw Algorithms in Invariant Theory topic, but that's not exactly what I'm asking about.

  • 1
    $\begingroup$ It should be possible, but I am not handy with the computers. Many algebraic geometers also use Macaulay (rather Macaulay 2). It should be a simple code: you are mapping one polynomial ring into another and are looking for generators of the kernel ideal. Given the number of variables, I am not sure if the calculation is going to finish in reasonable amount of time, but it is worth a shot. $\endgroup$ – Lev Borisov Dec 6 '13 at 0:28
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    $\begingroup$ If you have a sequence $S$ of polynomials, the Magma command $RelationIdeal(S)$ should give the answer w/o precising any invariant theory. Whether or not it will work in finite time and memory is a different question. $\endgroup$ – ThisNameForSale Dec 6 '13 at 0:32

I once used the following MAGMA-code (just for finite groups, I don't know if it works for general reductive groups).

intrinsic Presentation(R::RngInvar) -> SeqEnum
   {A presentation of the invariant ring R.}

   fund := FundamentalInvariants(R);
   prim := PrimaryInvariants(R);
   sec := IrreducibleSecondaryInvariants(R);
   invar := prim cat sec;
   P := PolynomialRing(BaseRing(R), #fund);
   A := Algebra(R);

   invarpres := [];
   for f in invar do
       b,g := HomogeneousModuleTest(fund,[R!1],f);
       Append(~invarpres, g[1]);
   end for;

   rel := RelationIdeal(R);


   return ideal<P|[phi(r) : r in Basis(rel)]>;

end intrinsic;

Put this into a file, say "pres.m", and then you can do:

> Attach("pres.m");
> G:=MatrixGroup<2,Rationals() | [-1,0,0,-1] >;
> G;
MatrixGroup(2, Rational Field)
[-1  0]
[ 0 -1]
> R:=InvariantRing(G);
> Presentation(R);
Ideal of Polynomial ring of rank 3 over Rational Field
Order: Lexicographical
Variables: $.1, $.2, $.3
    $.1*$.3 - $.2^2

This is an example of the $A_2$-singularity

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As mentioned in Sturmfel's book, it can be done in any computational environment by finding the Gröbner basis of the invariants with respect to lexicographic ordering.
For instance, if you have invariants $$f_1=x_1^2+x_2^2,$$ $$f_2=x_1^2x_2^2,$$ $$f_3=x_1^3x_2+x_2^3x_1,$$ finding a Gröbner basis of the ideal $$I=\langle f_1-x_1^2+x_2^2, \; f_2-x_1^2x_2^2,\; f_3-x_1^3x_2+x_2^3x_1 \rangle$$ with lexicographic ordering $x_1<x_2<x_3<f_1<f_2<f_3$ yields the first generator $f_1^2f_2-4f_2^2-f_3^2$ which is the syzygy you are looking for.
Now, if you have other invariant polynomials that you want to rewrite in the fundamental invariants $f_1,f_2$ and $f_3$, it can be done by determining their normal form reduction modulo the Gröbner basis. For example, $x_1^7x_2-x_2^7x_1 \rightarrow f_1^2f_3-f_2f_3$.

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