> 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
>
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