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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