Big +1 to framework for explicitly instantiating group actions.
--
Benjamin Jones
benjaminfjo...@gmail.com
On Mon, Mar 25, 2013 at 11:01 PM, tom d <sdent...@gmail.com> wrote:
> 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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