Let's continue this discussion at PR #40186 <https://github.com/sagemath/sage/pull/40186> (or someone can make a new issue).
On Tuesday, June 17, 2025 at 11:02:28 AM UTC-4 disne...@gmail.com wrote: > I don't know if you figured this out, but I think you can do everything > you want using the GAP functionality, here is your n=2, N=4 example: > > gap> N := 4; > 4 > gap> n := 2; > 2 > gap> SN := SymmetricGroup(N); > > Sym( [ 1 .. 4 ] ) > gap> Sn := SymmetricGroup(n); > Sym( [ 1 .. 2 ] ) > gap> G := GroupHomomorphismByImages(Sn, SN, [(1,2)], [(1,2)(3,4)]); > [ (1,2) ] -> [ (1,2)(3,4) ] > gap> F := OnPoints; > function( pnt, elm ) ... end > gap> H := function(pnt, g) > > return F(pnt, Image(G, g)); > > end; > function( pnt, g ) ... end > gap> F((1,2,3,4), (1,2)); > (1,3,4,2) > gap> H((1,2,3,4), (1,2)); > (1,4,3,2) > > Here I have used GroupHomomorphismByImages to define the action of Sn on N > elements, but you could equally write this as a GAP action and then > construct the homomorphism using the action. For the GAP links to explain > these functions see > https://docs.gap-system.org/doc/ref/chap41.html > https://docs.gap-system.org/doc/ref/chap40.html > > Hope that helps, > > Linden > On Monday, June 2, 2025 at 9:38:08 PM UTC+1 axio...@yahoo.de wrote: > >> > Indeed, you have S_n as a subgroup of S_N, because it acts on Y, with >> |Y|=N. >> > (and this is something you could have said explicitly). >> >> That's interesting, I never looked at functorial composition of species >> as restriction of group actions, but that's very promising. >> I'm sorry I wasn't clear, most likely this is because I don't understand >> things well enough myself. >> >> I think it matters that we want the image of S_n under the the action of >> G as a subgroup of S_N. >> >> Unfortunately, I have some difficulties understanding how group actions >> work in gap, I just posted a question on the gap forum mailing list. >> >> I hope that `FactorCosetAction` does exactly what I did naively in >> `PolynomialSpeciesElement.action`, but apparently, I do not understand >> `OrbitStabilizer` well enough. Here is what I tried to get started: >> >> gap> G := SymmetricGroup(4); >> Sym( [ 1 .. 4 ] ) >> >> gap> H := Group([(1,2,3,4)]); >> Group([ (1,2,3,4) ]) >> >> gap> a := FactorCosetAction(G, H); >> <action epimorphism> >> >> gap> Image(a); >> Group([ (2,5,4,3), (1,4)(2,6)(3,5) ]) >> >> gap> OrbitStabilizer(G, 1, GeneratorsOfGroup(G), >> GeneratorsOfGroup(Image(a))); >> rec( orbit := [ 1, 4, 3, 2, 5, 6 ], stabilizer := Group([ (1,2,3,4) ]) ) >> >> gap> OrbitStabilizer(G, 1, a); >> rec( orbit := [ 1, 4, 2, 3 ], stabilizer := Group([ (2,4), (3,4) ]) ) >> >> I would have expected that the last two calls give the same result. >> Perhaps 1 should be replaced with a coset in the second call, but I >> couldn't figure it out. >> >> Martin >> >> On Monday, 2 June 2025 at 17:46:50 UTC+2 dim...@gmail.com wrote: >> >> On Mon, Jun 2, 2025 at 3:10 AM 'Martin R' via sage-devel >> <sage-...@googlegroups.com> wrote: >> > >> > Here is one idea: perhaps I can rewrite >> `PolynomialSpeciesElement.action` so that it actually returns a gap action. >> (it makes my brain spin every time I have to translate between species and >> group actions) >> >> Indeed, you have S_n as a subgroup of S_N, because it acts on Y, with >> |Y|=N. >> (and this is something you could have said explicitly). >> Then, you have an action of S_N (on X). So all you do is you restrict >> the latter action to its subgroup S_n (it could be any subgroup of >> S_N, not necessarily isomorphic to S_n). >> In GAP it's usual to encode actions as >> https://docs.gap-system.org/doc/ref/chap41.html >> >> HTH >> Dima >> >> >> > >> > On Sunday, 1 June 2025 at 11:21:46 UTC+2 Martin R wrote: >> >> >> >> Ideally, I would like to define this in terms of generators of >> associated permutation groups. It is easy to see that it is sufficient to >> implement it, when F is a transitive action. Note, however, that H is not >> necessarily transitive, even if F is. >> >> >> >> To see (part of) the problem, replace the set Y in the example with a >> set of 2k pairs (i.e., in the example above, we have k=2), and F by the >> action of S_{2k} that relabels cycles. We then obtain an action on a set >> with (2k-1)! elements. (It's not hard to see that it has 2^{k-1} (k-1)! >> fixed points.) >> >> >> >> A priori, the actions F and G are given as formal sums of permutation >> groups (i.e., the stabilizer subgroups), which is quite efficient. In the >> example, F corresponds to the cyclic permutation group with generator >> (1,...,2k). To implement the above, I turn then into actual actions. In >> particular, I actually compute the sets X and Y, and they are simply too >> large. >> >> >> >> However, even optimizing the helper function _stabilizer_subgroups >> would be very interesting. >> >> >> >> Best wishes, >> >> >> >> Martin >> >> On Sunday, 1 June 2025 at 10:53:00 UTC+2 Martin R wrote: >> >>> >> >>> > I think the point is that Y is assumed to be {1,2,...,N}. So the >> action G is a homomorphism from S_n to S_N. The action F is a homomorphism >> from S_N to the symmetric group on X. We can compose these two >> homomorphisms to get an action of S_n on X. >> >>> >> >>> Yes, that's correct. >> >>> >> >>> Here is an example that can be done by hand. Examples are easier to >> understand when X and Y are sets of combinatorial objects. I replaced them >> with numbers above for simplicity of the definition. >> >>> >> >>> Let N=4 and let n=2. Let G be the action of S_2 on >> Y={(12),(21),(ab),(ba)} which switches the positions in the tuples, defined >> by >> >>> (1,2) *_G (1,2) = (2,1) and >> >>> (1,2) *_G (a,b) = (b,a). >> >>> >> >>> Let F be the action of S_4 on cycles (or cyclic permutations, if you >> prefer) of length 4, that relabels the elements of a cycle (or, if you >> prefer, acting by conjugation). That is., X={(1,2,3,4), (1,2,4,3), >> (1,3,2,4), (1,3,4,2), (1,4,2,3), (1,4,3,2)}, and, for example, >> >>> >> >>> (2,1) *_F (1,2,3,4) = (2,1,3,4) >> >>> >> >>> Then the functorial composition is an action of S_2 on the set X. The >> most intuitive way to see it is to think of X as the set of cyclic >> permutations of Y, and S_2 acts by swapping the positions in the tuples. >> For example: >> >>> >> >>> (1,2) *_H ((12), (ab), (21), (ba)) = ((21), (ba), (12), (ab), so >> that's a fixed point, whereas >> >>> (1,2) *_H ((12), (21), (ab), (ba)) = ((21), (12), (ba), (ab) is not. >> >>> >> >>> Does this help? >> >>> On Sunday, 1 June 2025 at 01:39:18 UTC+2 dmo...@deductivepress.ca >> wrote: >> >>>> >> >>>> I was confused, too, but I think the point is that Y is assumed to >> be {1,2,...,N}. So the action G is a homomorphism from S_n to S_N. The >> action F is a homomorphism from S_N to the symmetric group on X. We can >> compose these two homomorphisms to get an action of S_n on X. >> >>>> >> >>>> On May 31, 2025, at 6:06 PM, Dima Pasechnik <dim...@gmail.com> >> wrote: >> >>>> >> >>>> On Sat, May 31, 2025 at 2:57 PM 'Martin R' via sage-devel >> >>>> <sage-...@googlegroups.com> wrote: >> >>>> >> >>>> >> >>>> Dear permutation group / gap experts! >> >>>> >> >>>> I would enjoy some expert help to implement the so called functorial >> composition of species. The operation is easy to define even without >> mentioning combinatorial species, as follows: >> >>>> >> >>>> Let Y = {1,2,...,N} >> >>>> >> >>>> Let G: S_n x Y -> Y be a (left) action of the symmetric group S_n on >> Y. >> >>>> Let F: S_N x X -> X be a (left) action of S_N on X. >> >>>> >> >>>> Then we can define an action H: S_n x X -> X as follows: >> >>>> >> >>>> For pi in S_n, let G_pi be the permutation of Y induced by the >> action G. Then, >> >>>> >> >>>> pi *_H x := G_pi *_F x. >> >>>> >> >>>> >> >>>> Should the last x be y? Anyhow, I am completely lost here - are X >> and >> >>>> Y arbitrary? Or is Y a subset of X? >> >>>> An example might help. >> >>>> >> >>>> Dima >> >>>> >> >>>> >> >>>> Currently, https://github.com/sagemath/sage/pull/40186 implements >> this in a very naive way. It involves three functions: >> >>>> >> >>>> >> sage.rings.lazy_species.FunctorialCompositionSpeciesElement.__init__.coefficient >> >> >> >>>> (implementing the above) >> >>>> sage.rings.species.PolynomialSpeciesElement.action >> >>>> (used to turn a species into the corresponding action) >> >>>> sage.rings.species._stabilizer_subgroups >> >>>> (used to turn an action into a combinatorial species - i.e., a >> formal sum of stabilizer subgroups) >> >>>> >> >>>> I would not be surprised if all three of them could be improved by >> applying some permutation-group-knowledge which I am lacking. In fact, I >> tried to code this quickly, so it is quite likely that I even missed the >> most obvious things and did it completely backwards. >> >>>> >> >>>> Best wishes, >> >>>> >> >>>> Martin >> >>>> >> >>>> -- >> >>>> You received this message because you are subscribed to the Google >> Groups "sage-devel" group. >> >>>> To unsubscribe from this group and stop receiving emails from it, >> send an email to sage-devel+...@googlegroups.com. >> >>>> To view this discussion visit >> https://groups.google.com/d/msgid/sage-devel/31413384-37a7-4666-8099-7e526ad03dd9n%40googlegroups.com. >> >> >> >>>> >> >>>> >> >>>> -- >> >>>> You received this message because you are subscribed to the Google >> Groups "sage-devel" group. >> >>>> To unsubscribe from this group and stop receiving emails from it, >> send an email to sage-devel+...@googlegroups.com. >> >>>> >> >>>> To view this discussion visit >> https://groups.google.com/d/msgid/sage-devel/CAAWYfq0B1%3DDsnU1dEFAf%3DZbsVTDViDVjZAkYYoWL19NzFagUFA%40mail.gmail.com. >> >> >> >>>> >> >>>> >> > -- >> > You received this message because you are subscribed to the Google >> Groups "sage-devel" group. >> > To unsubscribe from this group and stop receiving emails from it, send >> an email to sage-devel+...@googlegroups.com. >> > To view this discussion visit >> https://groups.google.com/d/msgid/sage-devel/09e510ca-e281-413a-9ab0-fd97a4735af4n%40googlegroups.com. >> >> >> >> -- You received this message because you are subscribed to the Google Groups "sage-devel" group. 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