def right_coset_representatives(self):
r"""
Return the right coset representatives of ``self``.
EXAMPLES::
sage: W = ReflectionGroup(['A',2]) # optional - gap3
sage: for w in W: # optional - gap3
....: rcr = w.right_coset_representatives() # optional - gap3
....: print("%s %s"%(w.reduced_word(), # optional - gap3
....: [v.reduced_word() for v in rcr])) # optional - gap3
[] [[], [2], [1], [2, 1], [1, 2], [1, 2, 1]]
[2] [[], [2], [1]]
[1] [[], [1], [1, 2]]
[1, 2] [[]]
[2, 1] [[]]
[1, 2, 1] [[], [2], [2, 1]]
"""
from sage.combinat.root_system.reflection_group_complex import _gap_return
W = self.parent()
T = W.reflections()
T_fix = [i + 1 for i in T.keys() if self.fix_space().is_subspace(T[i].fix_space())]
S = str(
gap3(
"ReducedRightCosetRepresentatives(%s,ReflectionSubgroup(%s,%s))"
% (W._gap_group._name, W._gap_group._name, T_fix)
)
)
return sage_eval(_gap_return(S, coerce_obj="W"), locals={"self": self, "W": W})
def __init__(self, cartan_type):
"""
Construct this Coxeter group as a Sage permutation group, by
fetching the permutation representation of the generators from
Chevie's database.
TESTS::
sage: from sage.combinat.root_system.coxeter_group import CoxeterGroupAsPermutationGroup
sage: W = CoxeterGroupAsPermutationGroup(CartanType(["H",3])) # optional - chevie
sage: TestSuite(W).run() # optional - chevie
"""
assert cartan_type.is_finite()
assert cartan_type.is_irreducible()
self._semi_simple_rank = cartan_type.n
from sage.interfaces.gap3 import gap3
gap3._start()
gap3.load_package("chevie")
self._gap_group = gap3('CoxeterGroup("%s",%s)'%(cartan_type.letter,cartan_type.n))
# Following #9032, x.N is an alias for x.numerical_approx in every Sage object ...
N = self._gap_group.__getattr__("N").sage()
generators = [str(x) for x in self._gap_group.generators]
self._is_positive_root = [None] + [ True ] * N + [False]*N
PermutationGroup_generic.__init__(self, gens = generators,
category = Category.join([FinitePermutationGroups(), FiniteCoxeterGroups()]))
def right_coset_representatives(self):
r"""
Return the right coset representatives of ``self``.
EXAMPLES::
sage: W = ReflectionGroup(['A',2]) # optional - gap3
sage: for w in W: # optional - gap3
....: rcr = w.right_coset_representatives() # optional - gap3
....: print("%s %s"%(w.reduced_word(), # optional - gap3
....: [v.reduced_word() for v in rcr])) # optional - gap3
[] [[], [2], [1], [2, 1], [1, 2], [1, 2, 1]]
[2] [[], [2], [1]]
[1] [[], [1], [1, 2]]
[1, 2] [[]]
[2, 1] [[]]
[1, 2, 1] [[], [2], [2, 1]]
"""
from sage.combinat.root_system.reflection_group_complex import _gap_return
W = self.parent()
T = W.reflections()
T_fix = [i + 1 for i in T.keys()
if self.fix_space().is_subspace(T[i].fix_space())]
S = str(gap3('ReducedRightCosetRepresentatives(%s,ReflectionSubgroup(%s,%s))' % (W._gap_group._name, W._gap_group._name, T_fix)))
return sage_eval(_gap_return(S, coerce_obj='W'),
locals={'self': self, 'W': W})
def __init__(self, cartan_type):
"""
Construct this Coxeter group as a Sage permutation group, by
fetching the permutation representation of the generators from
Chevie's database.
TESTS::
sage: from sage.combinat.root_system.coxeter_group import CoxeterGroupAsPermutationGroup
sage: W = CoxeterGroupAsPermutationGroup(CartanType(["H",3])) # optional - chevie
sage: TestSuite(W).run() # optional - chevie
"""
assert cartan_type.is_finite()
assert cartan_type.is_irreducible()
self._semi_simple_rank = cartan_type.n
from sage.interfaces.gap3 import gap3
gap3._start()
gap3.load_package("chevie")
self._gap_group = gap3('CoxeterGroup("%s",%s)' %
(cartan_type.letter, cartan_type.n))
# Following #9032, x.N is an alias for x.numerical_approx in every Sage object ...
N = self._gap_group.__getattr__("N").sage()
generators = [str(x) for x in self._gap_group.generators]
self._is_positive_root = [None] + [True] * N + [False] * N
PermutationGroup_generic.__init__(self,
gens=generators,
category=Category.join([
FinitePermutationGroups(),
FiniteCoxeterGroups()
]))
def right_coset_representatives(self, J):
r"""
Return the right coset representatives of ``self`` for the
parabolic subgroup generated by the simple reflections in ``J``.
EXAMPLES::
sage: W = ReflectionGroup(["A",3]) # optional - gap3
sage: for J in Subsets([1,2,3]): W.right_coset_representatives(J) # optional - gap3
[(), (2,5)(3,9)(4,6)(8,11)(10,12), (1,4)(2,8)(3,5)(7,10)(9,11),
(1,7)(2,4)(5,6)(8,10)(11,12), (1,2,10)(3,6,5)(4,7,8)(9,12,11),
(1,4,6)(2,3,11)(5,8,9)(7,10,12), (1,6,4)(2,11,3)(5,9,8)(7,12,10),
(1,7)(2,6)(3,9)(4,5)(8,12)(10,11),
(1,10,2)(3,5,6)(4,8,7)(9,11,12), (1,2,3,12)(4,5,10,11)(6,7,8,9),
(1,5,9,10)(2,12,8,6)(3,4,7,11), (1,6)(2,9)(3,8)(5,11)(7,12),
(1,8)(2,7)(3,6)(4,10)(9,12), (1,10,9,5)(2,6,8,12)(3,11,7,4),
(1,12,3,2)(4,11,10,5)(6,9,8,7), (1,3)(2,12)(4,10)(5,11)(6,8)(7,9),
(1,5,12)(2,9,4)(3,10,8)(6,7,11), (1,8,11)(2,5,7)(3,12,4)(6,10,9),
(1,11,8)(2,7,5)(3,4,12)(6,9,10), (1,12,5)(2,4,9)(3,8,10)(6,11,7),
(1,3,7,9)(2,11,6,10)(4,8,5,12), (1,9,7,3)(2,10,6,11)(4,12,5,8),
(1,11)(3,10)(4,9)(5,7)(6,12), (1,9)(2,8)(3,7)(4,11)(5,10)(6,12)]
[(), (2,5)(3,9)(4,6)(8,11)(10,12), (1,4)(2,8)(3,5)(7,10)(9,11),
(1,2,10)(3,6,5)(4,7,8)(9,12,11), (1,4,6)(2,3,11)(5,8,9)(7,10,12),
(1,6,4)(2,11,3)(5,9,8)(7,12,10), (1,2,3,12)(4,5,10,11)(6,7,8,9),
(1,5,9,10)(2,12,8,6)(3,4,7,11), (1,6)(2,9)(3,8)(5,11)(7,12),
(1,3)(2,12)(4,10)(5,11)(6,8)(7,9),
(1,5,12)(2,9,4)(3,10,8)(6,7,11), (1,3,7,9)(2,11,6,10)(4,8,5,12)]
[(), (2,5)(3,9)(4,6)(8,11)(10,12), (1,7)(2,4)(5,6)(8,10)(11,12),
(1,4,6)(2,3,11)(5,8,9)(7,10,12),
(1,7)(2,6)(3,9)(4,5)(8,12)(10,11),
(1,10,2)(3,5,6)(4,8,7)(9,11,12), (1,2,3,12)(4,5,10,11)(6,7,8,9),
(1,10,9,5)(2,6,8,12)(3,11,7,4), (1,12,3,2)(4,11,10,5)(6,9,8,7),
(1,8,11)(2,5,7)(3,12,4)(6,10,9), (1,12,5)(2,4,9)(3,8,10)(6,11,7),
(1,11)(3,10)(4,9)(5,7)(6,12)]
[(), (1,4)(2,8)(3,5)(7,10)(9,11), (1,7)(2,4)(5,6)(8,10)(11,12),
(1,2,10)(3,6,5)(4,7,8)(9,12,11), (1,6,4)(2,11,3)(5,9,8)(7,12,10),
(1,10,2)(3,5,6)(4,8,7)(9,11,12), (1,5,9,10)(2,12,8,6)(3,4,7,11),
(1,8)(2,7)(3,6)(4,10)(9,12), (1,12,3,2)(4,11,10,5)(6,9,8,7),
(1,3)(2,12)(4,10)(5,11)(6,8)(7,9),
(1,11,8)(2,7,5)(3,4,12)(6,9,10), (1,9,7,3)(2,10,6,11)(4,12,5,8)]
[(), (2,5)(3,9)(4,6)(8,11)(10,12), (1,4,6)(2,3,11)(5,8,9)(7,10,12),
(1,2,3,12)(4,5,10,11)(6,7,8,9)]
[(), (1,4)(2,8)(3,5)(7,10)(9,11), (1,2,10)(3,6,5)(4,7,8)(9,12,11),
(1,6,4)(2,11,3)(5,9,8)(7,12,10), (1,5,9,10)(2,12,8,6)(3,4,7,11),
(1,3)(2,12)(4,10)(5,11)(6,8)(7,9)]
[(), (1,7)(2,4)(5,6)(8,10)(11,12), (1,10,2)(3,5,6)(4,8,7)(9,11,12),
(1,12,3,2)(4,11,10,5)(6,9,8,7)]
[()]
"""
from sage.combinat.root_system.reflection_group_complex import _gap_return
J_inv = [self._index_set_inverse[j] + 1 for j in J]
S = str(
gap3(
"ReducedRightCosetRepresentatives(%s,ReflectionSubgroup(%s,%s))"
% (self._gap_group._name, self._gap_group._name, J_inv)
)
)
return sage_eval(_gap_return(S), locals={"self": self})
def right_coset_representatives(self, J):
r"""
Return the right coset representatives of ``self`` for the
parabolic subgroup generated by the simple reflections in ``J``.
EXAMPLES::
sage: W = ReflectionGroup(["A",3]) # optional - gap3
sage: for J in Subsets([1,2,3]): W.right_coset_representatives(J) # optional - gap3
[(), (2,5)(3,9)(4,6)(8,11)(10,12), (1,4)(2,8)(3,5)(7,10)(9,11),
(1,7)(2,4)(5,6)(8,10)(11,12), (1,2,10)(3,6,5)(4,7,8)(9,12,11),
(1,4,6)(2,3,11)(5,8,9)(7,10,12), (1,6,4)(2,11,3)(5,9,8)(7,12,10),
(1,7)(2,6)(3,9)(4,5)(8,12)(10,11),
(1,10,2)(3,5,6)(4,8,7)(9,11,12), (1,2,3,12)(4,5,10,11)(6,7,8,9),
(1,5,9,10)(2,12,8,6)(3,4,7,11), (1,6)(2,9)(3,8)(5,11)(7,12),
(1,8)(2,7)(3,6)(4,10)(9,12), (1,10,9,5)(2,6,8,12)(3,11,7,4),
(1,12,3,2)(4,11,10,5)(6,9,8,7), (1,3)(2,12)(4,10)(5,11)(6,8)(7,9),
(1,5,12)(2,9,4)(3,10,8)(6,7,11), (1,8,11)(2,5,7)(3,12,4)(6,10,9),
(1,11,8)(2,7,5)(3,4,12)(6,9,10), (1,12,5)(2,4,9)(3,8,10)(6,11,7),
(1,3,7,9)(2,11,6,10)(4,8,5,12), (1,9,7,3)(2,10,6,11)(4,12,5,8),
(1,11)(3,10)(4,9)(5,7)(6,12), (1,9)(2,8)(3,7)(4,11)(5,10)(6,12)]
[(), (2,5)(3,9)(4,6)(8,11)(10,12), (1,4)(2,8)(3,5)(7,10)(9,11),
(1,2,10)(3,6,5)(4,7,8)(9,12,11), (1,4,6)(2,3,11)(5,8,9)(7,10,12),
(1,6,4)(2,11,3)(5,9,8)(7,12,10), (1,2,3,12)(4,5,10,11)(6,7,8,9),
(1,5,9,10)(2,12,8,6)(3,4,7,11), (1,6)(2,9)(3,8)(5,11)(7,12),
(1,3)(2,12)(4,10)(5,11)(6,8)(7,9),
(1,5,12)(2,9,4)(3,10,8)(6,7,11), (1,3,7,9)(2,11,6,10)(4,8,5,12)]
[(), (2,5)(3,9)(4,6)(8,11)(10,12), (1,7)(2,4)(5,6)(8,10)(11,12),
(1,4,6)(2,3,11)(5,8,9)(7,10,12),
(1,7)(2,6)(3,9)(4,5)(8,12)(10,11),
(1,10,2)(3,5,6)(4,8,7)(9,11,12), (1,2,3,12)(4,5,10,11)(6,7,8,9),
(1,10,9,5)(2,6,8,12)(3,11,7,4), (1,12,3,2)(4,11,10,5)(6,9,8,7),
(1,8,11)(2,5,7)(3,12,4)(6,10,9), (1,12,5)(2,4,9)(3,8,10)(6,11,7),
(1,11)(3,10)(4,9)(5,7)(6,12)]
[(), (1,4)(2,8)(3,5)(7,10)(9,11), (1,7)(2,4)(5,6)(8,10)(11,12),
(1,2,10)(3,6,5)(4,7,8)(9,12,11), (1,6,4)(2,11,3)(5,9,8)(7,12,10),
(1,10,2)(3,5,6)(4,8,7)(9,11,12), (1,5,9,10)(2,12,8,6)(3,4,7,11),
(1,8)(2,7)(3,6)(4,10)(9,12), (1,12,3,2)(4,11,10,5)(6,9,8,7),
(1,3)(2,12)(4,10)(5,11)(6,8)(7,9),
(1,11,8)(2,7,5)(3,4,12)(6,9,10), (1,9,7,3)(2,10,6,11)(4,12,5,8)]
[(), (2,5)(3,9)(4,6)(8,11)(10,12), (1,4,6)(2,3,11)(5,8,9)(7,10,12),
(1,2,3,12)(4,5,10,11)(6,7,8,9)]
[(), (1,4)(2,8)(3,5)(7,10)(9,11), (1,2,10)(3,6,5)(4,7,8)(9,12,11),
(1,6,4)(2,11,3)(5,9,8)(7,12,10), (1,5,9,10)(2,12,8,6)(3,4,7,11),
(1,3)(2,12)(4,10)(5,11)(6,8)(7,9)]
[(), (1,7)(2,4)(5,6)(8,10)(11,12), (1,10,2)(3,5,6)(4,8,7)(9,11,12),
(1,12,3,2)(4,11,10,5)(6,9,8,7)]
[()]
"""
from sage.combinat.root_system.reflection_group_element import _gap_return
J_inv = [self._index_set_inverse[j] + 1 for j in J]
S = str(
gap3(
'ReducedRightCosetRepresentatives(%s,ReflectionSubgroup(%s,%s))'
% (self._gap_group._name, self._gap_group._name, J_inv)))
return sage_eval(_gap_return(S), locals={'self': self})