# Software for computing generators of the invariant rings of the symmetric groups

I wish to compute the generators of the ring of invariants for a symmetric group acting on a polynomial ring using a computer software. For concreteness,consider $S_5$. Then, there are 7 irreducible representations of $S_5$. For each of these, I wish to do the above computations. The method being followed by me is as follows on SageMath.

For each representation, I write down the matrices corresponding to the action of (1,2) and (1,2,3,4,5) on the polynomial ring. As these 2 permutations generate $S_5$, the matrices will generate the required matrix group. Then, using the invariant_generators() function of SageMath, I perform the required computation.

This method worked successfully for the linear representations, standard representations and the 5-dimensional representations. But, for the 6-dimensional representation, the software is taking too much time for computation. I left my computer on overnight, and still, my desired computations weren't complete by morning.

So, my problem is, what is the fastest method in SageMath for computing the generators of the ring of invariants of the action of a finite group on a polynomial ring? Also, is there any other software, available for free usage (unlike MAGMA), which will work better than SageMath for the given computation?

• In case you are in US. MAGMA is free for all U.S. non-profit, non-governmental scientific research or educational institutions since 2013. Someone has arranged a global site license for the whole country! – achille hui May 16 '16 at 10:08
• This SageMath documentation for finitely generated matrix groups seems to point to an underlying GAP implementation (cleverly nicknamed Meataxe). It seems likely that direct access to the underlying routines would facilitate some lower-level optimizations. GAP is also free software (else SageMath could not bundle it) provided under GNU GPL terms. – hardmath May 16 '16 at 12:15
• @hardmath Yeah, I went through that too, but I couldn't find any details for the implementation. – MathManiac May 17 '16 at 9:49
• Simon King's paper (see also the 2013 J. of Sym. Comp. published version) details the invariant_algebra algorithm and suggests the construction of the Reynolds operator as a likely bottleneck. See Sec. D.7.1.5 of the Singular Reference Manual for the implementation details in finvar.lib. – hardmath May 17 '16 at 15:48
• @MathManiac There are some other software you can try. However your question of better is relative and is a matter of perception. You can try Singular or homalg under GAP. In fact these s/w can be used within SAGE as well. IMHO, Singular's invariant theory library (invar.lib) comes the best possible after MAGMA implementation. – Mike V.D.C. Mar 15 at 17:00