Would you be happy with something like the following, as a unifying and
easily accessible framework? Of course, I am thinking of providing also
methods that convert it (back) into a representation, a combinatorial
species, a permutation group.
class FiniteGroupAction(SageObject):
def __init__(A, X=None, G=None):
r"""
A finite group action.
INPUT:
- ``A`` -- can be one of the following:
* a permutation group, if ``X`` and ``G`` are ``None``
* a bijection on a finite set ``X``, if ``G`` is ``None``
* a map `G\times X\to X`, otherwise.
- ``X`` (optional, default ``None``) -- a finite set
- ``G`` (optional, default ``None``) -- a finite group
EXAMPLES::
sage: G = PermutationGroup([['b','c','a']],
domain=['a','b','c','d'])
sage: a = GroupAction(G)
sage: a.orbits()
[['a', 'b', 'c'], ['d']]
sage: A = lambda x: (2*x) % 6
sage: X = [0,1,2,3,4,5]
sage: a = GroupAction(A, X)
sage: a.orbits()
[[0], [1, 2, 4], [3], [5]]
sage: G = SymmetricGroup(3)
sage: A = lambda g, x: g*x
sage: a = GroupAction(A, G, G)
sage: a.orbits()
[['a', 'b', 'c', 'd']]
On Monday, 2 May 2022 at 01:32:01 UTC+2 Travis Scrimshaw wrote:
> Sorry, I don't know an easy way. I've always just defined them by hand
>>> whenever needed.
>>> However, I agree with you that a better way is needed.
>>>
>>
>> I would love for there to be some standard way to define a group action
>> on a set - preferably maintaining other algebraic properties of the set,
>> such as addition! But I don't know if there is even close to a standard
>> way to do this either.
>>
>
> - There is a standard way to do this, but not a generic method/class for
> it IIRC. You can do this by implementing an _act_on_() method on a wrapper
> class. Yet, this requires some manual input.
> - For cyclic actions, there is the DiscreteDynamicalSystem class
> introduced in https://trac.sagemath.org/ticket/24128.
> - There is also the Representation class in
> modules/with_basis/representation.py if you want to want to extend the
> action on the set to the module with a basis given by that set.
>
> Likely we will want to implement a class SetWithAction that automates a
> collects the orbit_decomposition functions and similar together as methods
> as a single global entry point.
>
> Best,
> Travis
>
>
>
--
You received this message because you are subscribed to the Google Groups
"sage-devel" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to sage-devel+unsubscr...@googlegroups.com.
To view this discussion on the web visit
https://groups.google.com/d/msgid/sage-devel/20c36129-bd92-4252-9a85-a750aea7b038n%40googlegroups.com.
