On Tue, May 3, 2022 at 7:12 AM 'Martin R' via sage-devel
<sage-devel@googlegroups.com> wrote:
>
> I implemented (well, the implementation is trivial) the following, and I'd
> like feedback. I am not completely sure whether the interface for the second
> variant, where the generators of the acting group are required, is ideal, but
> I think it looks useable.
>
> A more serious problem is that orbits are currently computed twice: once when
> creating the generators of the permutation group, and another time, when
> asking for them.
>
> Martin
>
Hi Martin:
Thanks for programming this. I'd like to test it's functionality but
don't know your function's syntax. Something like
def PermutationGroup2(gens=None, gap_group=None, domain=None,
canonicalize=True, category=None, action=None):
maybe?
- David
> """
> ...
> We can create a permutation group from a group action::
>
> sage: A = lambda x: (2*x) % 6
> sage: X = [0,1,2,3,4,5]
> sage: G = PermutationGroup(action=A, domain=X)
> sage: G.orbits()
> [[0], [1, 2, 4], [3], [5]]
>
> sage: A = lambda g, x: vector(g*x, immutable=True)
> sage: X = [vector(x, immutable=True) for x in GF(3)^2]
> sage: G = SL(2,3); G.gens()
> (
> [1 1] [0 1]
> [0 1], [2 0]
> )
> sage: H = PermutationGroup(G.gens(), action=A, domain=X)
> sage: H.orbits()
> [[(0, 0)], [(1, 0), (0, 2), (2, 2), (2, 0), (1, 2), (2, 1), (0, 1),
> (1, 1)]]
> sage: H.gens()
> [((0,1),(1,1),(2,1))((0,2),(2,2),(1,2)),
> ((1,0),(0,2),(2,0),(0,1))((1,1),(1,2),(2,2),(2,1))]
> ...
> """
> if not is_ExpectElement(gens) and hasattr(gens, '_permgroup_'):
> return gens._permgroup_()
> if gens is not None and not isinstance(gens, (tuple, list, GapElement)):
> raise TypeError("gens must be a tuple, list, or GapElement")
> gap_group = kwds.get("gap_group", None)
> domain = kwds.get("domain", None)
> canonicalize = kwds.get("canonicalize", True)
> category = kwds.get("category", None)
> action = kwds.get("action", None)
> if action is not None:
> if domain is None:
> raise ValueError("you must specify the domain for an action")
> from sage.combinat.cyclic_sieving_phenomenon import
> orbit_decomposition
> if gap_group is not None:
> raise ValueError("gap_group is not supported with action")
> if gens is None and gap_group is None:
> gens = [tuple(o) for o in orbit_decomposition(domain, action)]
> else:
> gens = [[tuple(o) for o in orbit_decomposition(domain, lambda x:
> action(g, x))]
> for g in gens]
> if args:
> from sage.misc.superseded import deprecation
> deprecation(31510, "gap_group, domain, canonicalize, category will
> become keyword only")
> if len(args) > 4:
> raise ValueError("invalid input")
> args = list(args)
> gap_group = args.pop(0)
> if args:
> domain = args.pop(0)
> if args:
> canonicalize = args.pop(0)
> if args:
> category = args.pop(0)
> return PermutationGroup_generic(gens=gens, gap_group=gap_group,
> domain=domain,
> canonicalize=canonicalize,
> category=category)
>
>
> On Monday, 2 May 2022 at 23:01:42 UTC+2 David Joyner wrote:
>>
>> On Mon, May 2, 2022 at 11:24 AM 'Martin R' via sage-devel
>> <sage-...@googlegroups.com> wrote:
>> >
>> > I am actually not sure anymore, which methods or functionality this class
>> > should provide.
>> >
>> > Would it possibly be better to enhance PermutationGroup with an additional
>> > optional "from_action" and "from_cyclic_action" argument? Eg.:
>> >
>> > PermutationGroup(domain = X, cyclic_action = lambda x: f(x))
>> >
>> > PermutationGroup(domain = X, group_action = (G, lambda g, x: f(g, x)))
>> >
>>
>> I like this idea!
>>
>> > Martin
>> > On Monday, 2 May 2022 at 14:20:57 UTC+2 kcrisman wrote:
>> >>
>> >> On Sunday, May 1, 2022 at 7:32:01 PM UTC-4 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.
>> >>>
>> >>
>> >> Hmm, maybe a tutorial is needed for this. I would imagine that a lot of
>> >> people who aren't familiar with how to create a new wrapper class would
>> >> be the ones who need this. Of course, the FiniteGroupAction suggestion
>> >> sounds quite welcome, too, though really having both options would be
>> >> best.
>> >
>> > --
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>
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