Specify the action! By making a group action framework, we would also be
providing the possibility of changing the action to something contrary to
the assumptions of the original developers.... Yes, in fact I think this is
one of the natural reasons for doing an explicit group action framework.
Even for the action of S_n on the set {1,2,3,...,n} one can twist the
'usual' action with an automorphism \phi of S_n, so that \sigma acts on i
by \phi(\sigma)(i).
The 'usual' actions then become special predefined objects, like the
special graphs, maybe summoned up automatically using the
permutation/whatever's __call__ function if it's an idiomatic action like
\sigma(3).
As a category, I would imagine we would have a GroupAction category and/or
a GroupWithAction category, which would put some requirements on the group
and its elements.
I've attached a bit of sample code, which could be used as a base to start
a group action category. (Currently just a class, as I need to go and read
the category tutorials, though...) The examples are at the bottom;
includes products of actions, twisting by a group endomorphism, computing
characters, orbits, checking the action definition, checking transitivity,
and generating the Cayley graph of the action for a given generating
set.....
On Monday, March 25, 2013 2:30:57 PM UTC+3, Volker Braun wrote:
>
> The group action category stuff would be nice, but you would run into
> exactly the same question that Dima asked: What are you going to do if
> there is more than one possible action. You'll have to either use some
> heuristics (take the simpler / less nested action) or raise some exception
> telling the user to explicitly disambiguate between them.
>
>
>
> On Monday, March 25, 2013 8:33:57 AM UTC+1, tom d wrote:
>>
>> Hm, wouldn't this just be a direct product of the individual group
>> actions? It seems to me that we're expecting the permutations to act
>> according to an 'obvious' group action. Should we also expect 'obvious'
>> actions of things like a dihedral group when given a 2-dimensional vector?
>> Probably the answer is to generalize and build up a proper group actions
>> category (with obvious methods passing to representations!).
>
>
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