def __init__(self, n):
"""
Initialize ``self``.
EXAMPLES::
sage: G = groups.matrix.BinaryDihedral(4)
sage: TestSuite(G).run()
"""
self._n = n
if n % 2 == 0:
R = CyclotomicField(2 * n)
zeta = R.gen()
i = R.gen()**(n // 2)
else:
R = CyclotomicField(4 * n)
zeta = R.gen()**2
i = R.gen()**n
MS = MatrixSpace(R, 2)
zero = R.zero()
gens = [MS([zeta, zero, zero, ~zeta]), MS([zero, i, i, zero])]
from sage.libs.gap.libgap import libgap
gap_gens = [libgap(matrix_gen) for matrix_gen in gens]
gap_group = libgap.Group(gap_gens)
FinitelyGeneratedMatrixGroup_gap.__init__(self,
ZZ(2),
R,
gap_group,
category=Groups().Finite())
def __init__(self, domain, prefix):
"""
EXAMPLES::
sage: G = WeylGroup(['B',3])
sage: TestSuite(G).run()
"""
self._domain = domain
if self.cartan_type().is_affine():
category = AffineWeylGroups()
elif self.cartan_type().is_finite():
category = FiniteWeylGroups()
else:
category = WeylGroups()
self.n = domain.dimension() # Really needed?
self._prefix = prefix
# FinitelyGeneratedMatrixGroup_gap takes plain matrices as input
gens_matrix = [self.morphism_matrix(self.domain().simple_reflection(i))
for i in self.index_set()]
from sage.libs.all import libgap
libgap_group = libgap.Group(gens_matrix)
degree = ZZ(self.domain().dimension())
ring = self.domain().base_ring()
FinitelyGeneratedMatrixGroup_gap.__init__(
self, degree, ring, libgap_group, category=category)
def __init__(self, domain, prefix):
"""
EXAMPLES::
sage: G = WeylGroup(['B',3])
sage: TestSuite(G).run()
sage: cm = CartanMatrix([[2,-5,0],[-2,2,-1],[0,-1,2]])
sage: W = WeylGroup(cm)
sage: TestSuite(W).run() # long time
"""
self._domain = domain
if self.cartan_type().is_affine():
category = AffineWeylGroups()
elif self.cartan_type().is_finite():
category = FiniteWeylGroups()
else:
category = WeylGroups()
if self.cartan_type().is_irreducible():
category = category.Irreducible()
self.n = domain.dimension() # Really needed?
self._prefix = prefix
# FinitelyGeneratedMatrixGroup_gap takes plain matrices as input
gens_matrix = [self.morphism_matrix(self.domain().simple_reflection(i))
for i in self.index_set()]
from sage.libs.all import libgap
libgap_group = libgap.Group(gens_matrix)
degree = ZZ(self.domain().dimension())
ring = self.domain().base_ring()
FinitelyGeneratedMatrixGroup_gap.__init__(
self, degree, ring, libgap_group, category=category)
def __init__(self, n=1, R=0):
"""
Initialize ``self``.
EXAMPLES::
sage: H = groups.matrix.Heisenberg(n=2, R=5)
sage: TestSuite(H).run() # long time
sage: H = groups.matrix.Heisenberg(n=2, R=4)
sage: TestSuite(H).run() # long time
sage: H = groups.matrix.Heisenberg(n=3)
sage: TestSuite(H).run(max_runs=30, skip="_test_elements") # long time
sage: H = groups.matrix.Heisenberg(n=2, R=GF(4))
sage: TestSuite(H).run() # long time
"""
def elementary_matrix(i, j, val, MS):
elm = copy(MS.one())
elm[i, j] = val
elm.set_immutable()
return elm
self._n = n
self._ring = R
# We need the generators of the ring as a commutative additive group
if self._ring is ZZ:
ring_gens = [self._ring.one()]
else:
if self._ring.cardinality() == self._ring.characteristic():
ring_gens = [self._ring.one()]
else:
# This is overkill, but is the only way to ensure
# we get all of the elements
ring_gens = list(self._ring)
dim = ZZ(n + 2)
MS = MatrixSpace(self._ring, dim)
gens_x = [
elementary_matrix(0, j, gen, MS) for j in range(1, dim - 1)
for gen in ring_gens
]
gens_y = [
elementary_matrix(i, dim - 1, gen, MS) for i in range(1, dim - 1)
for gen in ring_gens
]
gen_z = [elementary_matrix(0, dim - 1, gen, MS) for gen in ring_gens]
gens = gens_x + gens_y + gen_z
from sage.libs.gap.libgap import libgap
gap_gens = [libgap(single_gen) for single_gen in gens]
gap_group = libgap.Group(gap_gens)
cat = Groups().FinitelyGenerated()
if self._ring in Rings().Finite():
cat = cat.Finite()
FinitelyGeneratedMatrixGroup_gap.__init__(self,
ZZ(dim),
self._ring,
gap_group,
category=cat)
def __init__(self, n):
"""
Initialize ``self``.
EXAMPLES::
sage: G = groups.matrix.BinaryDihedral(4)
sage: TestSuite(G).run()
"""
self._n = n
if n % 2 == 0:
R = CyclotomicField(2*n)
zeta = R.gen()
i = R.gen()**(n//2)
else:
R = CyclotomicField(4*n)
zeta = R.gen()**2
i = R.gen()**n
MS = MatrixSpace(R, 2)
zero = R.zero()
gens = [ MS([zeta, zero, zero, ~zeta]), MS([zero, i, i, zero]) ]
from sage.libs.gap.libgap import libgap
gap_gens = [libgap(matrix_gen) for matrix_gen in gens]
gap_group = libgap.Group(gap_gens)
FinitelyGeneratedMatrixGroup_gap.__init__(self, ZZ(2), R, gap_group, category=Groups().Finite())
def __init__(self, n=1, R=0):
"""
Initialize ``self``.
EXAMPLES::
sage: H = groups.matrix.Heisenberg(n=2, R=5)
sage: TestSuite(H).run() # long time
sage: H = groups.matrix.Heisenberg(n=2, R=4)
sage: TestSuite(H).run() # long time
sage: H = groups.matrix.Heisenberg(n=3)
sage: TestSuite(H).run(max_runs=30, skip="_test_elements") # long time
sage: H = groups.matrix.Heisenberg(n=2, R=GF(4))
sage: TestSuite(H).run() # long time
"""
def elementary_matrix(i, j, val, MS):
elm = copy(MS.one())
elm[i,j] = val
elm.set_immutable()
return elm
self._n = n
self._ring = R
# We need the generators of the ring as a commutative additive group
if self._ring is ZZ:
ring_gens = [self._ring.one()]
else:
if self._ring.cardinality() == self._ring.characteristic():
ring_gens = [self._ring.one()]
else:
# This is overkill, but is the only way to ensure
# we get all of the elements
ring_gens = list(self._ring)
dim = ZZ(n + 2)
MS = MatrixSpace(self._ring, dim)
gens_x = [elementary_matrix(0, j, gen, MS)
for j in range(1, dim-1) for gen in ring_gens]
gens_y = [elementary_matrix(i, dim-1, gen, MS)
for i in range(1, dim-1) for gen in ring_gens]
gen_z = [elementary_matrix(0, dim-1, gen, MS) for gen in ring_gens]
gens = gens_x + gens_y + gen_z
from sage.libs.gap.libgap import libgap
gap_gens = [libgap(single_gen) for single_gen in gens]
gap_group = libgap.Group(gap_gens)
cat = Groups().FinitelyGenerated()
if self._ring in Rings().Finite():
cat = cat.Finite()
FinitelyGeneratedMatrixGroup_gap.__init__(self, ZZ(dim), self._ring,
gap_group, category=cat)
def _subgroup_constructor(self, libgap_subgroup):
"""
Return a finitely generated subgroup.
See
:meth:`sage.groups.libgap_wrapper.ParentLibGAP._subgroup_constructor`
for details.
TESTS::
sage: SL2Z = SL(2,ZZ)
sage: S, T = SL2Z.gens()
sage: G = SL2Z.subgroup([T^2]); G # indirect doctest
Matrix group over Integer Ring with 1 generators (
[1 2]
[0 1]
)
sage: G.ambient() is SL2Z
True
"""
from sage.groups.matrix_gps.finitely_generated import FinitelyGeneratedMatrixGroup_gap
return FinitelyGeneratedMatrixGroup_gap(self.degree(),
self.base_ring(),
libgap_subgroup,
ambient=self)
def __init__(self,
degree,
base_ring,
gens,
invariant_bilinear_form,
category=None,
check=True,
invariant_submodule=None,
invariant_quotient_module=None):
r"""
Create this orthogonal group from the input.
TESTS::
sage: from sage.groups.matrix_gps.isometries import GroupOfIsometries
sage: bil = Matrix(ZZ,2,[3,2,2,3])
sage: gens = [-Matrix(ZZ,2,[0,1,1,0])]
sage: cat = Groups().Finite()
sage: O = GroupOfIsometries(2, ZZ, gens, bil, category=cat)
sage: TestSuite(O).run()
"""
from copy import copy
G = copy(invariant_bilinear_form)
G.set_immutable()
self._invariant_bilinear_form = G
self._invariant_submodule = invariant_submodule
self._invariant_quotient_module = invariant_quotient_module
if check:
I = invariant_submodule
Q = invariant_quotient_module
for f in gens:
self._check_matrix(f)
if (not I is None) and I * f != I:
raise ValueError("the submodule is not preserved")
if not Q is None and (Q.W() != Q.W() * f
or Q.V() * f != Q.V()):
raise ValueError("the quotient module is not preserved")
if len(gens) == 0: # handle the trivial group
gens = [G.parent().identity_matrix()]
from sage.libs.gap.libgap import libgap
gap_gens = [libgap(matrix_gen) for matrix_gen in gens]
gap_group = libgap.Group(gap_gens)
FinitelyGeneratedMatrixGroup_gap.__init__(self,
degree,
base_ring,
gap_group,
category=category)
def __init__(self, degree, base_ring,
gens, invariant_bilinear_form,
category=None, check=True,
invariant_submodule=None,
invariant_quotient_module=None):
r"""
Create this orthogonal group from the input.
TESTS::
sage: from sage.groups.matrix_gps.isometries import GroupOfIsometries
sage: bil = Matrix(ZZ,2,[3,2,2,3])
sage: gens = [-Matrix(ZZ,2,[0,1,1,0])]
sage: cat = Groups().Finite()
sage: O = GroupOfIsometries(2, ZZ, gens, bil, category=cat)
sage: TestSuite(O).run()
"""
from copy import copy
G = copy(invariant_bilinear_form)
G.set_immutable()
self._invariant_bilinear_form = G
self._invariant_submodule = invariant_submodule
self._invariant_quotient_module = invariant_quotient_module
if check:
I = invariant_submodule
Q = invariant_quotient_module
for f in gens:
self._check_matrix(f)
if (not I is None) and I*f != I:
raise ValueError("the submodule is not preserved")
if not Q is None and (Q.W() != Q.W()*f or Q.V()*f != Q.V()):
raise ValueError("the quotient module is not preserved")
if len(gens) == 0: # handle the trivial group
gens = [G.parent().identity_matrix()]
from sage.libs.gap.libgap import libgap
gap_gens = [libgap(matrix_gen) for matrix_gen in gens]
gap_group = libgap.Group(gap_gens)
FinitelyGeneratedMatrixGroup_gap.__init__(self,
degree,
base_ring,
gap_group,
category=category)
