> 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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