def automorphism_group(self):
"""
Returns the group of permutations whose action on this structure
leave it fixed.
EXAMPLES::
sage: p = PermutationGroupElement((2,3,4))
sage: P = species.PermutationSpecies()
sage: a = P.structures(["a", "b", "c", "d"]).random_element(); a
['a', 'c', 'b', 'd']
sage: a.automorphism_group()
Permutation Group with generators [(2,3), (1,4)]
::
sage: [a.transport(perm) for perm in a.automorphism_group()]
[['a', 'c', 'b', 'd'],
['a', 'c', 'b', 'd'],
['a', 'c', 'b', 'd'],
['a', 'c', 'b', 'd']]
"""
from sage.groups.all import SymmetricGroup, PermutationGroup
S = SymmetricGroup(len(self._labels))
p = self.permutation_group_element()
return PermutationGroup(S.centralizer(p).gens())
def automorphism_group(self):
"""
EXAMPLES::
sage: p = PermutationGroupElement((2,3))
sage: S = species.SetSpecies()
sage: F = S * S
sage: a = F.structures([1,2,3,4]).random_element(); a
{1}*{2, 3, 4}
sage: a.automorphism_group()
Permutation Group with generators [(2,3), (2,3,4)]
::
sage: [a.transport(g) for g in a.automorphism_group()]
[{1}*{2, 3, 4},
{1}*{2, 3, 4},
{1}*{2, 3, 4},
{1}*{2, 3, 4},
{1}*{2, 3, 4},
{1}*{2, 3, 4}]
::
sage: a = F.structures([1,2,3,4]).random_element(); a
{2, 3}*{1, 4}
sage: [a.transport(g) for g in a.automorphism_group()]
[{2, 3}*{1, 4}, {2, 3}*{1, 4}, {2, 3}*{1, 4}, {2, 3}*{1, 4}]
"""
from sage.groups.all import PermutationGroupElement, PermutationGroup, SymmetricGroup
from sage.misc.misc import uniq
from sage.combinat.species.misc import change_support
left, right = self._list
n = len(self._labels)
#Get the supports for each of the sides
l_support = self._subset._list
r_support = self._subset.complement()._list
#Get the automorphism group for the left object and
#make it have the correct support. Do the same to the
#right side.
l_aut = change_support(left.automorphism_group(), l_support)
r_aut = change_support(right.automorphism_group(), r_support)
identity = PermutationGroupElement([])
gens = l_aut.gens() + r_aut.gens()
gens = [g for g in gens if g != identity]
gens = uniq(gens) if len(gens) > 0 else [[]]
return PermutationGroup(gens)
def change_support(perm, support, change_perm=None):
"""
Changes the support of a permutation defined on [1, ..., n] to
support.
EXAMPLES::
sage: from sage.combinat.species.misc import change_support
sage: p = PermutationGroupElement((1,2,3)); p
(1,2,3)
sage: change_support(p, [3,4,5])
(3,4,5)
"""
if change_perm is None:
change_perm = prod([PermutationGroupElement((i+1,support[i])) for i in range(len(support)) if i+1 != support[i]], PermutationGroupElement([], SymmetricGroup(support)))
if isinstance(perm, PermutationGroup_generic):
return PermutationGroup([change_support(g, support, change_perm) for g in perm.gens()])
return change_perm*perm*~change_perm
def automorphism_group(self):
"""
Returns the group of permutations whose action on this subset leave
it fixed.
EXAMPLES::
sage: F = species.SubsetSpecies()
sage: a = F.structures([1,2,3,4])[6]; a
{1, 3}
sage: a.automorphism_group()
Permutation Group with generators [(2,4), (1,3)]
::
sage: [a.transport(g) for g in a.automorphism_group()]
[{1, 3}, {1, 3}, {1, 3}, {1, 3}]
"""
from sage.groups.all import SymmetricGroup, PermutationGroup
a = SymmetricGroup(self._list)
b = SymmetricGroup(self.complement()._list)
return PermutationGroup(a.gens() + b.gens())
def automorphism_group(self):
"""
Returns the group of permutations whose action on this structure
leave it fixed.
EXAMPLES::
sage: P = species.CycleSpecies()
sage: a = P.structures([1, 2, 3, 4]).random_element(); a
(1, 2, 3, 4)
sage: a.automorphism_group()
Permutation Group with generators [(1,2,3,4)]
::
sage: [a.transport(perm) for perm in a.automorphism_group()]
[(1, 2, 3, 4), (1, 2, 3, 4), (1, 2, 3, 4), (1, 2, 3, 4)]
"""
from sage.groups.all import SymmetricGroup, PermutationGroup
S = SymmetricGroup(len(self._labels))
p = self.permutation_group_element()
return PermutationGroup(S.centralizer(p).gens())
