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 <dimp...@gmail.com> wrote:
>
> On Sat, May 31, 2025 at 2:57 PM 'Martin R' via sage-devel
> <sage-devel@googlegroups.com <mailto:sage-devel@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
>>
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