def direct_sum(self, M):
r"""
Return the direct sum of this lattice with ``M``.
INPUT:
- ``M`` -- a module over `\ZZ`
EXAMPLES::
sage: A = IntegralLattice(1)
sage: A.direct_sum(A)
Lattice of degree 2 and rank 2 over Integer Ring
Basis matrix:
[1 0]
[0 1]
Inner product matrix:
[1 0]
[0 1]
"""
IM = matrix.block_diagonal(
[self.inner_product_matrix(),
M.inner_product_matrix()])
ambient = FreeQuadraticModule(ZZ, self.degree() + M.degree(), IM)
smzero = matrix.zero(self.rank(), M.degree())
mszero = matrix.zero(M.rank(), self.degree())
basis = self.basis_matrix().augment(smzero).stack(
mszero.augment(M.basis_matrix()))
ipm = ambient.inner_product_matrix()
return FreeQuadraticModule_integer_symmetric(ambient=ambient,
basis=basis,
inner_product_matrix=ipm,
already_echelonized=False)
def direct_sum(self, M):
r"""
Return the direct sum of this lattice with ``M``.
INPUT:
- ``M`` -- a module over `\ZZ`
EXAMPLES::
sage: A = IntegralLattice(1)
sage: A.direct_sum(A)
Lattice of degree 2 and rank 2 over Integer Ring
Basis matrix:
[1 0]
[0 1]
Inner product matrix:
[1 0]
[0 1]
"""
IM = matrix.block_diagonal([self.inner_product_matrix(),
M.inner_product_matrix()])
ambient = FreeQuadraticModule(ZZ,
self.degree() + M.degree(), IM)
smzero = matrix.zero(self.rank(), M.degree())
mszero = matrix.zero(M.rank(), self.degree())
basis = self.basis_matrix().augment(smzero).stack(
mszero.augment(M.basis_matrix()))
ipm = ambient.inner_product_matrix()
return FreeQuadraticModule_integer_symmetric(ambient=ambient,
basis=basis,
inner_product_matrix=ipm,
already_echelonized=False)
def IntegralLatticeDirectSum(Lattices, return_embeddings=False):
r"""
Return the direct sum of the lattices contained in the list ``Lattices``.
INPUT:
- ``Lattices`` -- a list of lattices ``[L_1,...,L_n]``
- ``return_embeddings`` -- (default: ``False``) a boolean
OUTPUT:
The direct sum of the `L_i` if ``return_embeddings`` is ``False`` or
the tuple ``[L, phi]`` where `L` is the direct sum of `L_i` and ``phi``
is the list of embeddings from `L_i` to `L`.
EXAMPLES::
sage: from sage.modules.free_quadratic_module_integer_symmetric import IntegralLatticeDirectSum
sage: L1 = IntegralLattice("D4")
sage: L2 = IntegralLattice("A3", [[1, 1, 2]])
sage: L3 = IntegralLattice("A4", [[0, 1, 1, 2], [1, 2, 3, 1]])
sage: Lattices = [L1, L2, L3]
sage: IntegralLatticeDirectSum([L1, L2, L3])
Lattice of degree 11 and rank 7 over Integer Ring
Basis matrix:
[1 0 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0]
[0 0 0 0 1 1 2 0 0 0 0]
[0 0 0 0 0 0 0 0 1 1 2]
[0 0 0 0 0 0 0 1 2 3 1]
Inner product matrix:
[ 2 -1 0 0 0 0 0 0 0 0 0]
[-1 2 -1 -1 0 0 0 0 0 0 0]
[ 0 -1 2 0 0 0 0 0 0 0 0]
[ 0 -1 0 2 0 0 0 0 0 0 0]
[ 0 0 0 0 2 -1 0 0 0 0 0]
[ 0 0 0 0 -1 2 -1 0 0 0 0]
[ 0 0 0 0 0 -1 2 0 0 0 0]
[ 0 0 0 0 0 0 0 2 -1 0 0]
[ 0 0 0 0 0 0 0 -1 2 -1 0]
[ 0 0 0 0 0 0 0 0 -1 2 -1]
[ 0 0 0 0 0 0 0 0 0 -1 2]
sage: [L, phi] = IntegralLatticeDirectSum([L1, L2, L3], True)
sage: LL3 = L.sublattice(phi[2].image().basis_matrix())
sage: L3.discriminant() == LL3.discriminant()
True
sage: x = L3([1, 2, 3, 1])
sage: phi[2](x).inner_product(phi[2](x)) == x.inner_product(x)
True
TESTS::
sage: IntegralLatticeDirectSum([IntegralLattice("D4")])
Lattice of degree 4 and rank 4 over Integer Ring
Basis matrix:
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
Inner product matrix:
[ 2 -1 0 0]
[-1 2 -1 -1]
[ 0 -1 2 0]
[ 0 -1 0 2]
sage: L1 = IntegralLattice(2 * matrix.identity(2), [[1/2, 1/2]])
sage: L2 = IntegralLattice("A3", [[1, 1, 2]])
sage: [L, phi] = IntegralLatticeDirectSum([L1, L2], True)
sage: L
Lattice of degree 5 and rank 2 over Integer Ring
Basis matrix:
[1/2 1/2 0 0 0]
[ 0 0 1 1 2]
Inner product matrix:
[ 2 0 0 0 0]
[ 0 2 0 0 0]
[ 0 0 2 -1 0]
[ 0 0 -1 2 -1]
[ 0 0 0 -1 2]
"""
for L in Lattices:
if not isinstance(L, FreeQuadraticModule_integer_symmetric):
raise ValueError("Lattices must be a list of lattices")
N = len(Lattices)
dims = [L_i.dimension() for L_i in Lattices]
degrees = [L_i.degree() for L_i in Lattices]
degree_tot = sum(degrees)
sum_degree = [sum(degrees[:i]) for i in range(N + 1)]
inner_product_list = [copy(L_i.inner_product_matrix()) for L_i in Lattices]
IM = matrix.block_diagonal(inner_product_list)
ambient = FreeQuadraticModule(ZZ, degree_tot, inner_product_matrix=IM)
basis = [
matrix.block(1, 3, [
matrix.zero(dims[i], sum_degree[i]), Lattices[i].basis_matrix(),
matrix.zero(dims[i], sum_degree[-1] - sum_degree[i + 1])
]) for i in range(N)
]
basis_matrix = matrix.block(N, 1, basis)
ipm = ambient.inner_product_matrix()
direct_sum = FreeQuadraticModule_integer_symmetric(
ambient=ambient,
basis=basis_matrix,
inner_product_matrix=ipm,
already_echelonized=False)
if not return_embeddings:
return direct_sum
sum_dims = [sum(dims[:i]) for i in range(N + 1)]
phi = [
Lattices[i].hom(direct_sum.basis()[sum_dims[i]:sum_dims[i + 1]])
for i in range(N)
]
return [direct_sum, phi]
def orthogonal_group(self, gens=None, is_finite=None):
"""
Return the orthogonal group of this lattice as a matrix group.
The elements are isometries of the ambient vector space
which preserve this lattice. They are represented by
matrices with respect to the standard basis.
INPUT:
- ``gens`` -- a list of matrices (default:``None``)
- ``is_finite`` -- bool (default: ``None``) If set to ``True``,
then the group is placed in the category of finite groups. Sage does not check this.
OUTPUT:
The matrix group generated by ``gens``.
If ``gens`` is not specified, then generators of the full
orthogonal group of this lattice are computed. They are
continued as the identity on the orthogonal complement of
the lattice in its ambient space. Currently, we can only
compute the orthogonal group for positive definite lattices.
EXAMPLES::
sage: A4 = IntegralLattice("A4")
sage: Aut = A4.orthogonal_group()
sage: Aut
Group of isometries with 5 generators (
[-1 0 0 0] [0 0 0 1] [-1 -1 -1 0] [ 1 0 0 0] [ 1 0 0 0]
[ 0 -1 0 0] [0 0 1 0] [ 0 0 0 -1] [-1 -1 -1 -1] [ 0 1 0 0]
[ 0 0 -1 0] [0 1 0 0] [ 0 0 1 1] [ 0 0 0 1] [ 0 0 1 1]
[ 0 0 0 -1], [1 0 0 0], [ 0 1 0 0], [ 0 0 1 0], [ 0 0 0 -1]
)
The group acts from the right on the lattice and its discriminant group::
sage: x = A4.an_element()
sage: g = Aut.an_element()
sage: g
[ 1 1 1 0]
[ 0 0 -1 0]
[ 0 0 1 1]
[ 0 -1 -1 -1]
sage: x*g
(1, 1, 1, 0)
sage: (x*g).parent()==A4
True
sage: (g*x).parent()
Vector space of dimension 4 over Rational Field
sage: y = A4.discriminant_group().an_element()
sage: y*g
(1)
If the group is finite we can compute the usual things::
sage: Aut.order()
240
sage: conj = Aut.conjugacy_classes_representatives()
sage: len(conj)
14
sage: Aut.structure_description()
'C2 x S5'
The lattice can live in a larger ambient space::
sage: A2 = IntegralLattice(matrix.identity(3),Matrix(ZZ,2,3,[1,-1,0,0,1,-1]))
sage: A2.orthogonal_group()
Group of isometries with 3 generators (
[-1/3 2/3 2/3] [ 2/3 2/3 -1/3] [1 0 0]
[ 2/3 -1/3 2/3] [ 2/3 -1/3 2/3] [0 0 1]
[ 2/3 2/3 -1/3], [-1/3 2/3 2/3], [0 1 0]
)
It can be negative definite as well::
sage: A2m = IntegralLattice(-Matrix(ZZ,2,[2,1,1,2]))
sage: G = A2m.orthogonal_group()
sage: G.order()
12
If the lattice is indefinite, sage does not know how to compute generators.
Can you teach it?::
sage: U = IntegralLattice(Matrix(ZZ,2,[0,1,1,0]))
sage: U.orthogonal_group()
Traceback (most recent call last):
...
NotImplementedError: currently, we can only compute generators for orthogonal groups over definite lattices.
But we can define subgroups::
sage: S = IntegralLattice(Matrix(ZZ,2,[2, 3, 3, 2]))
sage: f = Matrix(ZZ,2,[0,1,-1,3])
sage: S.orthogonal_group([f])
Group of isometries with 1 generator (
[ 0 1]
[-1 3]
)
TESTS:
We can handle the trivial group::
sage: S = IntegralLattice(Matrix(ZZ,2,[2, 3, 3, 2]))
sage: S.orthogonal_group([])
Group of isometries with 1 generator (
[1 0]
[0 1]
)
"""
from sage.categories.groups import Groups
from sage.groups.matrix_gps.isometries import GroupOfIsometries
sig = self.signature_pair()
if gens is None:
gens = []
if sig[1] == 0 or sig[0] == 0: # definite
from sage.quadratic_forms.quadratic_form import QuadraticForm
is_finite = True
# Compute transformation matrix to the ambient module.
L = self.overlattice(self.ambient_module().gens())
Orthogonal = L.orthogonal_complement(self)
B = self.basis_matrix().stack(Orthogonal.basis_matrix())
if sig[0] == 0: # negative definite
q = QuadraticForm(ZZ, -2 * self.gram_matrix())
else: # positive definite
q = QuadraticForm(ZZ, 2 * self.gram_matrix())
identity = matrix.identity(Orthogonal.rank())
for g in q.automorphism_group().gens():
g = g.matrix().T
# We continue g as identity on the orthogonal complement.
g = matrix.block_diagonal([g, identity])
g = B.inverse() * g * B
gens.append(g)
else: # indefinite
raise NotImplementedError(
"currently, we can only compute generators "
"for orthogonal groups over definite lattices.")
deg = self.degree()
base = self.ambient_vector_space().base_ring()
inv_bil = self.inner_product_matrix()
if is_finite:
cat = Groups().Finite()
else:
cat = Groups()
D = self.discriminant_group()
G = GroupOfIsometries(deg,
base,
gens,
inv_bil,
category=cat,
invariant_submodule=self,
invariant_quotient_module=D)
return G
def IntegralLatticeDirectSum(Lattices, return_embeddings=False):
r"""
Return the direct sum of the lattices contained in the list ``Lattices``.
INPUT:
- ``Lattices`` -- a list of lattices ``[L_1,...,L_n]``
- ``return_embeddings`` -- (default: ``False``) a boolean
OUTPUT:
The direct sum of the `L_i` if ``return_embeddings`` is ``False`` or
the tuple ``[L, phi]`` where `L` is the direct sum of `L_i` and ``phi``
is the list of embeddings from `L_i` to `L`.
EXAMPLES::
sage: from sage.modules.free_quadratic_module_integer_symmetric import IntegralLatticeDirectSum
sage: L1 = IntegralLattice("D4")
sage: L2 = IntegralLattice("A3", [[1, 1, 2]])
sage: L3 = IntegralLattice("A4", [[0, 1, 1, 2], [1, 2, 3, 1]])
sage: Lattices = [L1, L2, L3]
sage: IntegralLatticeDirectSum([L1, L2, L3])
Lattice of degree 11 and rank 7 over Integer Ring
Basis matrix:
[1 0 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0]
[0 0 0 0 1 1 2 0 0 0 0]
[0 0 0 0 0 0 0 0 1 1 2]
[0 0 0 0 0 0 0 1 2 3 1]
Inner product matrix:
[ 2 -1 0 0 0 0 0 0 0 0 0]
[-1 2 -1 -1 0 0 0 0 0 0 0]
[ 0 -1 2 0 0 0 0 0 0 0 0]
[ 0 -1 0 2 0 0 0 0 0 0 0]
[ 0 0 0 0 2 -1 0 0 0 0 0]
[ 0 0 0 0 -1 2 -1 0 0 0 0]
[ 0 0 0 0 0 -1 2 0 0 0 0]
[ 0 0 0 0 0 0 0 2 -1 0 0]
[ 0 0 0 0 0 0 0 -1 2 -1 0]
[ 0 0 0 0 0 0 0 0 -1 2 -1]
[ 0 0 0 0 0 0 0 0 0 -1 2]
sage: [L, phi] = IntegralLatticeDirectSum([L1, L2, L3], True)
sage: LL3 = L.sublattice(phi[2].image().basis_matrix())
sage: L3.discriminant() == LL3.discriminant()
True
sage: x = L3([1, 2, 3, 1])
sage: phi[2](x).inner_product(phi[2](x)) == x.inner_product(x)
True
TESTS::
sage: IntegralLatticeDirectSum([IntegralLattice("D4")])
Lattice of degree 4 and rank 4 over Integer Ring
Basis matrix:
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
Inner product matrix:
[ 2 -1 0 0]
[-1 2 -1 -1]
[ 0 -1 2 0]
[ 0 -1 0 2]
sage: L1 = IntegralLattice(2 * matrix.identity(2), [[1/2, 1/2]])
sage: L2 = IntegralLattice("A3", [[1, 1, 2]])
sage: [L, phi] = IntegralLatticeDirectSum([L1, L2], True)
sage: L
Lattice of degree 5 and rank 2 over Integer Ring
Basis matrix:
[1/2 1/2 0 0 0]
[ 0 0 1 1 2]
Inner product matrix:
[ 2 0 0 0 0]
[ 0 2 0 0 0]
[ 0 0 2 -1 0]
[ 0 0 -1 2 -1]
[ 0 0 0 -1 2]
"""
for L in Lattices:
if not isinstance(L, FreeQuadraticModule_integer_symmetric):
raise ValueError("Lattices must be a list of lattices")
N = len(Lattices)
dims = [L_i.dimension() for L_i in Lattices]
degrees = [L_i.degree() for L_i in Lattices]
dim_tot = sum(dims)
degree_tot = sum(degrees)
sum_degree = [sum(degrees[:i]) for i in range(N+1)]
inner_product_list = [copy(L_i.inner_product_matrix()) for L_i in Lattices]
IM = matrix.block_diagonal(inner_product_list)
ambient = FreeQuadraticModule(ZZ,
degree_tot,
inner_product_matrix=IM)
basis = [matrix.block(1, 3, [matrix.zero(dims[i], sum_degree[i]),
Lattices[i].basis_matrix(),
matrix.zero(dims[i], sum_degree[-1] - sum_degree[i+1])
]) for i in range(N)]
basis_matrix = matrix.block(N, 1, basis)
ipm = ambient.inner_product_matrix()
direct_sum = FreeQuadraticModule_integer_symmetric(ambient=ambient,
basis=basis_matrix,
inner_product_matrix=ipm,
already_echelonized=False)
if not return_embeddings:
return direct_sum
sum_dims = [sum(dims[:i]) for i in range(N+1)]
phi = [Lattices[i].hom(direct_sum.basis()[sum_dims[i]:sum_dims[i+1]])
for i in range(N)]
return [direct_sum, phi]
def orthogonal_group(self, gens=None, is_finite=None):
"""
Return the orthogonal group of this lattice as a matrix group.
The elements are isometries of the ambient vector space
which preserve this lattice. They are represented by
matrices with respect to the standard basis.
INPUT:
- ``gens`` -- a list of matrices (default:``None``)
- ``is_finite`` -- bool (default: ``None``) If set to ``True``,
then the group is placed in the category of finite groups. Sage does not check this.
OUTPUT:
The matrix group generated by ``gens``.
If ``gens`` is not specified, then generators of the full
orthogonal group of this lattice are computed. They are
continued as the identity on the orthogonal complement of
the lattice in its ambient space. Currently, we can only
compute the orthogonal group for positive definite lattices.
EXAMPLES::
sage: A4 = IntegralLattice("A4")
sage: Aut = A4.orthogonal_group()
sage: Aut
Group of isometries with 5 generators (
[-1 0 0 0] [0 0 0 1] [-1 -1 -1 0] [ 1 0 0 0] [ 1 0 0 0]
[ 0 -1 0 0] [0 0 1 0] [ 0 0 0 -1] [-1 -1 -1 -1] [ 0 1 0 0]
[ 0 0 -1 0] [0 1 0 0] [ 0 0 1 1] [ 0 0 0 1] [ 0 0 1 1]
[ 0 0 0 -1], [1 0 0 0], [ 0 1 0 0], [ 0 0 1 0], [ 0 0 0 -1]
)
The group acts from the right on the lattice and its discriminant group::
sage: x = A4.an_element()
sage: g = Aut.an_element()
sage: g
[ 1 1 1 0]
[ 0 0 -1 0]
[ 0 0 1 1]
[ 0 -1 -1 -1]
sage: x*g
(1, 1, 1, 0)
sage: (x*g).parent()==A4
True
sage: (g*x).parent()
Vector space of dimension 4 over Rational Field
sage: y = A4.discriminant_group().an_element()
sage: y*g
(1)
If the group is finite we can compute the usual things::
sage: Aut.order()
240
sage: conj = Aut.conjugacy_classes_representatives()
sage: len(conj)
14
sage: Aut.structure_description() # optional - database_gap
'C2 x S5'
The lattice can live in a larger ambient space::
sage: A2 = IntegralLattice(matrix.identity(3),Matrix(ZZ,2,3,[1,-1,0,0,1,-1]))
sage: A2.orthogonal_group()
Group of isometries with 3 generators (
[-1/3 2/3 2/3] [ 2/3 2/3 -1/3] [1 0 0]
[ 2/3 -1/3 2/3] [ 2/3 -1/3 2/3] [0 0 1]
[ 2/3 2/3 -1/3], [-1/3 2/3 2/3], [0 1 0]
)
It can be negative definite as well::
sage: A2m = IntegralLattice(-Matrix(ZZ,2,[2,1,1,2]))
sage: G = A2m.orthogonal_group()
sage: G.order()
12
If the lattice is indefinite, sage does not know how to compute generators.
Can you teach it?::
sage: U = IntegralLattice(Matrix(ZZ,2,[0,1,1,0]))
sage: U.orthogonal_group()
Traceback (most recent call last):
...
NotImplementedError: currently, we can only compute generators for orthogonal groups over definite lattices.
But we can define subgroups::
sage: S = IntegralLattice(Matrix(ZZ,2,[2, 3, 3, 2]))
sage: f = Matrix(ZZ,2,[0,1,-1,3])
sage: S.orthogonal_group([f])
Group of isometries with 1 generator (
[ 0 1]
[-1 3]
)
TESTS:
We can handle the trivial group::
sage: S = IntegralLattice(Matrix(ZZ,2,[2, 3, 3, 2]))
sage: S.orthogonal_group([])
Group of isometries with 1 generator (
[1 0]
[0 1]
)
"""
from sage.categories.groups import Groups
from sage.groups.matrix_gps.isometries import GroupOfIsometries
sig = self.signature_pair()
if gens is None:
gens = []
if sig[1]==0 or sig[0]==0: #definite
from sage.quadratic_forms.quadratic_form import QuadraticForm
is_finite = True
# Compute transformation matrix to the ambient module.
L = self.overlattice(self.ambient_module().gens())
Orthogonal = L.orthogonal_complement(self)
B = self.basis_matrix().stack(Orthogonal.basis_matrix())
if sig[0] == 0: #negative definite
q = QuadraticForm(ZZ, -2*self.gram_matrix())
else: # positve definite
q = QuadraticForm(ZZ, 2*self.gram_matrix())
identity = matrix.identity(Orthogonal.rank())
for g in q.automorphism_group().gens():
g = g.matrix().T
# We continue g as identity on the orthogonal complement.
g = matrix.block_diagonal([g, identity])
g = B.inverse()*g*B
gens.append(g)
else: #indefinite
raise NotImplementedError(
"currently, we can only compute generators "
"for orthogonal groups over definite lattices.")
deg = self.degree()
base = self.ambient_vector_space().base_ring()
inv_bil = self.inner_product_matrix()
if is_finite:
cat = Groups().Finite()
else:
cat = Groups()
D = self.discriminant_group()
G = GroupOfIsometries(deg,
base,
gens,
inv_bil,
category=cat,
invariant_submodule=self,
invariant_quotient_module=D)
return G
