Dear Mike, Ralf, Robert, and anybody interested,
I think that I finally realised how to proceed concerning isotypes, in general
but also particularly of functorial composition. I guess that all what I'm
writing below is well known to many of us, but so far at least I did not
realise how to put the pieces together. I hope you don't mind me stating the
obvious.
Let's consider functorial composition. Two structures S1 and S2 are
isomorphic, iff there is a bijection s such that S1 = F[G[s]](S1). For
example, we obtain simple graphs with vertices labelled from the set U as
Subsets[Pairs[U]].
So, to generate the isotypes we could
* first determine the action induced by the symmetric group on G[U] (maybe this
can be done by McKay's algorithm?)
* and then determine the isotypes of F using this action. (In the case where
F=Subsets, this should be trivial)
I do not know yet whether this is practical, but at least I think that the
second step points us into the right direction: we (including, in fact, also
Nicolas) really want to have isotypes corresponding to more general actions
than the those corresponding to the symmetric group.
As far as I know we are in pretty good shape: symmetrica does most of the
things we need already: sum and product is doable, I believe also plethysm
("partitional" or "ordinary" substitution).
Some questions and comments:
* how can we represent group actions? We want to have the group action as a
further argument to isomorphismTypes, but usually we will want to say: use
the symmetric group acting on the labels...
* what's the role of the cycle index series? is this a symmetric group only
thing? probably not, I guess that to every species we can associate a group
action induced by relabelling - which is exactly what we need for functorial
composition.
* I'd guess that linear species (instead of finite sets and bijections we
consider finite ordered sets with order preserving bijections) fit in nicely
into this framework: in this case we only have the trivial action. Note that
there are operations on linear species that are not available for ordinary
species, at least integration is an example.
Hoping for comments and ideas,
Martin
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