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 TorsionQuadraticForm(q):
r"""
Create a torsion quadratic form module from a rational matrix.
The resulting quadratic form takes values in `\QQ / \ZZ`
or `\QQ / 2 \ZZ` (depending on ``q``).
If it takes values modulo `2`, then it is non-degenerate.
In any case the bilinear form is non-degenerate.
INPUT:
- ``q`` -- a symmetric rational matrix
EXAMPLES::
sage: q1 = Matrix(QQ,2,[1,1/2,1/2,1])
sage: TorsionQuadraticForm(q1)
Finite quadratic module over Integer Ring with invariants (2, 2)
Gram matrix of the quadratic form with values in Q/2Z:
[ 1 1/2]
[1/2 1]
In the following example the quadratic form is degenerate.
But the bilinear form is still non-degenerate::
sage: q2 = diagonal_matrix(QQ,[1/4,1/3])
sage: TorsionQuadraticForm(q2)
Finite quadratic module over Integer Ring with invariants (12,)
Gram matrix of the quadratic form with values in Q/Z:
[7/12]
TESTS::
sage: TorsionQuadraticForm(matrix.diagonal([3/4,3/8,3/8]))
Finite quadratic module over Integer Ring with invariants (4, 8, 8)
Gram matrix of the quadratic form with values in Q/2Z:
[3/4 0 0]
[ 0 3/8 0]
[ 0 0 3/8]
"""
q = matrix(QQ, q)
if q.nrows() != q.ncols():
raise ValueError("the input must be a square matrix")
if q != q.transpose():
raise ValueError("the input must be a symmetric matrix")
Q, d = q._clear_denom()
S, U, V = Q.smith_form()
D = U * q * V
Q = FreeQuadraticModule(ZZ, q.ncols(), inner_product_matrix=d**2 * q)
denoms = [D[i, i].denominator() for i in range(D.ncols())]
rels = Q.span(diagonal_matrix(ZZ, denoms) * U)
return TorsionQuadraticModule((1/d)*Q, (1/d)*rels, modulus=1)
def twist(self, s):
r"""
Return the torsion quadratic module with quadratic form scaled by ``s``.
If the old form was defined modulo `n`, then the new form is defined
modulo `n s`.
INPUT:
- ``s`` - a rational number
EXAMPLES::
sage: q = TorsionQuadraticForm(matrix.diagonal([3/9, 1/9]))
sage: q.twist(-1)
Finite quadratic module over Integer Ring with invariants (3, 9)
Gram matrix of the quadratic form with values in Q/Z:
[2/3 0]
[ 0 8/9]
This form is defined modulo `3`::
sage: q.twist(3)
Finite quadratic module over Integer Ring with invariants (3, 9)
Gram matrix of the quadratic form with values in Q/3Z:
[ 1 0]
[ 0 1/3]
The next form is defined modulo `4`::
sage: q.twist(4)
Finite quadratic module over Integer Ring with invariants (3, 9)
Gram matrix of the quadratic form with values in Q/4Z:
[4/3 0]
[ 0 4/9]
"""
s = self.base_ring().fraction_field()(s)
n = self.V().degree()
inner_product_matrix = s * self.V().inner_product_matrix()
ambient = FreeQuadraticModule(self.base_ring(), n,
inner_product_matrix)
V = ambient.span(self.V().basis())
W = ambient.span(self.W().basis())
return TorsionQuadraticModule(V, W)
def TorsionQuadraticForm(q):
r"""
Create a torsion quadratic form module from a rational matrix.
The resulting quadratic form takes values in `\QQ / \ZZ`
or `\QQ / 2 \ZZ` (depending on ``q``).
If it takes values modulo `2`, then it is non-degenerate.
In any case the bilinear form is non-degenerate.
INPUT:
- ``q`` -- a symmetric rational matrix
EXAMPLES::
sage: q1 = Matrix(QQ,2,[1,1/2,1/2,1])
sage: TorsionQuadraticForm(q1)
Finite quadratic module over Integer Ring with invariants (2, 2)
Gram matrix of the quadratic form with values in Q/2Z:
[ 1 1/2]
[1/2 1]
In the following example the quadratic form is degenerate.
But the bilinear form is still non-degenerate::
sage: q2 = diagonal_matrix(QQ,[1/4,1/3])
sage: TorsionQuadraticForm(q2)
Finite quadratic module over Integer Ring with invariants (12,)
Gram matrix of the quadratic form with values in Q/Z:
[7/12]
"""
q = matrix(QQ, q)
if q.nrows() != q.ncols():
raise ValueError("the input must be a square matrix")
if q != q.transpose():
raise ValueError("the input must be a symmetric matrix")
Q, d = q._clear_denom()
S, U, V = Q.smith_form()
D = U * q * V
Q = FreeQuadraticModule(ZZ, q.ncols(), inner_product_matrix=d**2 * q)
denoms = [D[i,i].denominator() for i in range(D.ncols())]
rels = Q.span(diagonal_matrix(ZZ, denoms) * U)
return TorsionQuadraticModule((1/d)*Q, (1/d)*rels)
def twist(self, s):
r"""
Return the torsion quadratic module with quadratic form scaled by ``s``.
If the old form was defined modulo `n`, then the new form is defined
modulo `n s`.
INPUT:
- ``s`` - a rational number
EXAMPLES::
sage: q = TorsionQuadraticForm(matrix.diagonal([3/9, 1/9]))
sage: q.twist(-1)
Finite quadratic module over Integer Ring with invariants (3, 9)
Gram matrix of the quadratic form with values in Q/Z:
[2/3 0]
[ 0 8/9]
This form is defined modulo `3`::
sage: q.twist(3)
Finite quadratic module over Integer Ring with invariants (3, 9)
Gram matrix of the quadratic form with values in Q/3Z:
[ 1 0]
[ 0 1/3]
The next form is defined modulo `4`::
sage: q.twist(4)
Finite quadratic module over Integer Ring with invariants (3, 9)
Gram matrix of the quadratic form with values in Q/4Z:
[4/3 0]
[ 0 4/9]
"""
s = self.base_ring().fraction_field()(s)
n = self.V().degree()
inner_product_matrix = s * self.V().inner_product_matrix()
ambient = FreeQuadraticModule(self.base_ring(), n, inner_product_matrix)
V = ambient.span(self.V().basis())
W = ambient.span(self.W().basis())
return TorsionQuadraticModule(V, W)
def IntegralLattice(inner_product_matrix, basis=None):
r"""
Return the integral lattice spanned by ``basis`` in the ambient space.
A lattice is a finitely generated free abelian group `L \cong \Z^r` equipped
with a non-degenerate, symmetric bilinear form `L \times L \colon \rightarrow \Z`.
Here, lattices have an ambient quadratic space `\Q^n` and a distinguished basis.
INPUT:
- ``inner_product_matrix`` -- a symmetric matrix over the rationals
- ``basis`` -- a list of elements of ambient or a matrix
Output:
A lattice in the ambient space defined by the inner_product_matrix.
Unless specified, the basis of the lattice is the standard basis.
EXAMPLES::
sage: from sage.modules.free_quadratic_module_integer_symmetric import IntegralLattice
sage: IntegralLattice(Matrix(ZZ,2,2,[0,1,1,0]))
Lattice of degree 2 and rank 2 over Integer Ring
Basis matrix:
[1 0]
[0 1]
Inner product matrix:
[0 1]
[1 0]
We can specify a basis as well::
sage: IntegralLattice(Matrix(ZZ,2,2,[0,1,1,0]),basis=[vector([1,1])])
Lattice of degree 2 and rank 1 over Integer Ring
Basis matrix:
[1 1]
Inner product matrix:
[0 1]
[1 0]
"""
if basis is None:
basis = matrix.identity(ZZ, inner_product_matrix.ncols())
if inner_product_matrix != inner_product_matrix.transpose():
raise ValueError("Argument inner_product_matrix must be symmetric\n%s" % inner_product_matrix)
A = FreeQuadraticModule(ZZ, inner_product_matrix.ncols(),
inner_product_matrix=inner_product_matrix)
return FreeQuadraticModule_integer_symmetric(ambient=A, basis=basis,
inner_product_matrix=inner_product_matrix,
already_echelonized=False)
def IntegralLattice(data, basis=None):
r"""
Return the integral lattice spanned by ``basis`` in the ambient space.
A lattice is a finitely generated free abelian group `L \cong \ZZ^r`
equipped with a non-degenerate, symmetric bilinear form
`L \times L \colon \rightarrow \ZZ`. Here, lattices have an
ambient quadratic space `\QQ^n` and a distinguished basis.
INPUT:
The input is a descriptor of the lattice and a (optional) basis.
- ``data`` -- can be one of the following:
* a symmetric matrix over the rationals -- the inner product matrix
* an integer -- the dimension for an Euclidean lattice
* a symmetric Cartan type or anything recognized by
:class:`CartanMatrix` (see also
:mod:`Cartan types <sage.combinat.root_system.cartan_type>`)
-- for a root lattice
* the string ``"U"`` or ``"H"`` -- for hyperbolic lattices
- ``basis`` -- (optional) a matrix whose rows form a basis of the
lattice, or a list of module elements forming a basis
OUTPUT:
A lattice in the ambient space defined by the inner_product_matrix.
Unless specified, the basis of the lattice is the standard basis.
EXAMPLES::
sage: H5 = Matrix(ZZ, 2, [2,1,1,-2])
sage: IntegralLattice(H5)
Lattice of degree 2 and rank 2 over Integer Ring
Basis matrix:
[1 0]
[0 1]
Inner product matrix:
[ 2 1]
[ 1 -2]
A basis can be specified too::
sage: IntegralLattice(H5, Matrix([1,1]))
Lattice of degree 2 and rank 1 over Integer Ring
Basis matrix:
[1 1]
Inner product matrix:
[ 2 1]
[ 1 -2]
We can define an Euclidean lattice just by its dimension::
sage: IntegralLattice(3)
Lattice of degree 3 and rank 3 over Integer Ring
Basis matrix:
[1 0 0]
[0 1 0]
[0 0 1]
Inner product matrix:
[1 0 0]
[0 1 0]
[0 0 1]
Here is an example of the `A_2` root lattice in Euclidean space::
sage: basis = Matrix([[1,-1,0], [0,1,-1]])
sage: A2 = IntegralLattice(3, basis)
sage: A2
Lattice of degree 3 and rank 2 over Integer Ring
Basis matrix:
[ 1 -1 0]
[ 0 1 -1]
Inner product matrix:
[1 0 0]
[0 1 0]
[0 0 1]
sage: A2.gram_matrix()
[ 2 -1]
[-1 2]
We use ``"U"`` or ``"H"`` for defining a hyperbolic lattice::
sage: L1 = IntegralLattice("U")
sage: L1
Lattice of degree 2 and rank 2 over Integer Ring
Basis matrix:
[1 0]
[0 1]
Inner product matrix:
[0 1]
[1 0]
sage: L1 == IntegralLattice("H")
True
We can construct root lattices by specifying their type
(see :mod:`Cartan types <sage.combinat.root_system.cartan_type>`
and :class:`CartanMatrix`)::
sage: IntegralLattice(["E", 7])
Lattice of degree 7 and rank 7 over Integer Ring
Basis matrix:
[1 0 0 0 0 0 0]
[0 1 0 0 0 0 0]
[0 0 1 0 0 0 0]
[0 0 0 1 0 0 0]
[0 0 0 0 1 0 0]
[0 0 0 0 0 1 0]
[0 0 0 0 0 0 1]
Inner product matrix:
[ 2 0 -1 0 0 0 0]
[ 0 2 0 -1 0 0 0]
[-1 0 2 -1 0 0 0]
[ 0 -1 -1 2 -1 0 0]
[ 0 0 0 -1 2 -1 0]
[ 0 0 0 0 -1 2 -1]
[ 0 0 0 0 0 -1 2]
sage: IntegralLattice(["A", 2])
Lattice of degree 2 and rank 2 over Integer Ring
Basis matrix:
[1 0]
[0 1]
Inner product matrix:
[ 2 -1]
[-1 2]
sage: IntegralLattice("D3")
Lattice of degree 3 and rank 3 over Integer Ring
Basis matrix:
[1 0 0]
[0 1 0]
[0 0 1]
Inner product matrix:
[ 2 -1 -1]
[-1 2 0]
[-1 0 2]
sage: IntegralLattice(["D", 4])
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]
We can specify a basis as well::
sage: G = Matrix(ZZ, 2, [0,1,1,0])
sage: B = [vector([1,1])]
sage: IntegralLattice(G, basis=B)
Lattice of degree 2 and rank 1 over Integer Ring
Basis matrix:
[1 1]
Inner product matrix:
[0 1]
[1 0]
sage: IntegralLattice(["A", 3], [[1,1,1]])
Lattice of degree 3 and rank 1 over Integer Ring
Basis matrix:
[1 1 1]
Inner product matrix:
[ 2 -1 0]
[-1 2 -1]
[ 0 -1 2]
sage: IntegralLattice(4, [[1,1,1,1]])
Lattice of degree 4 and rank 1 over Integer Ring
Basis matrix:
[1 1 1 1]
Inner product matrix:
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
sage: IntegralLattice("A2", [[1,1]])
Lattice of degree 2 and rank 1 over Integer Ring
Basis matrix:
[1 1]
Inner product matrix:
[ 2 -1]
[-1 2]
TESTS::
sage: IntegralLattice(["A", 1, 1])
Traceback (most recent call last):
...
ValueError: lattices must be nondegenerate; use FreeQuadraticModule instead
sage: IntegralLattice(["D", 3, 1])
Traceback (most recent call last):
...
ValueError: lattices must be nondegenerate; use FreeQuadraticModule instead
"""
if is_Matrix(data):
inner_product_matrix = data
elif isinstance(data, Integer):
inner_product_matrix = matrix.identity(ZZ, data)
elif data == "U" or data == "H":
inner_product_matrix = matrix([[0, 1], [1, 0]])
else:
inner_product_matrix = CartanMatrix(data)
if basis is None:
basis = matrix.identity(ZZ, inner_product_matrix.ncols())
if inner_product_matrix != inner_product_matrix.transpose():
raise ValueError("the inner product matrix must be symmetric\n%s" %
inner_product_matrix)
A = FreeQuadraticModule(ZZ,
inner_product_matrix.ncols(),
inner_product_matrix=inner_product_matrix)
return FreeQuadraticModule_integer_symmetric(
ambient=A,
basis=basis,
inner_product_matrix=A.inner_product_matrix(),
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 twist(self, s, discard_basis=False):
r"""
Return the lattice with inner product matrix scaled by ``s``.
INPUT:
- ``s`` -- a nonzero integer
- ``discard_basis`` -- a boolean (default: ``False``).
If ``True``, then the lattice returned is equipped
with the standard basis.
EXAMPLES::
sage: L = IntegralLattice("A4")
sage: L.twist(3)
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:
[ 6 -3 0 0]
[-3 6 -3 0]
[ 0 -3 6 -3]
[ 0 0 -3 6]
sage: L = IntegralLattice(3,[[2,1,0],[0,1,1]])
sage: L
Lattice of degree 3 and rank 2 over Integer Ring
Basis matrix:
[2 1 0]
[0 1 1]
Inner product matrix:
[1 0 0]
[0 1 0]
[0 0 1]
sage: L.twist(1)
Lattice of degree 3 and rank 2 over Integer Ring
Basis matrix:
[2 1 0]
[0 1 1]
Inner product matrix:
[1 0 0]
[0 1 0]
[0 0 1]
sage: L.twist(1, True)
Lattice of degree 2 and rank 2 over Integer Ring
Basis matrix:
[1 0]
[0 1]
Inner product matrix:
[5 1]
[1 2]
"""
try:
s = self.base_ring()(s)
except TypeError:
raise ValueError(
"the scaling factor must be an element of the base ring.")
if s == 0:
raise ValueError("the scaling factor must be non zero")
if discard_basis:
return IntegralLattice(s * self.gram_matrix())
else:
n = self.degree()
inner_product_matrix = s * self.inner_product_matrix()
ambient = FreeQuadraticModule(self.base_ring(), n,
inner_product_matrix)
return FreeQuadraticModule_integer_symmetric(
ambient=ambient,
basis=self.basis(),
inner_product_matrix=inner_product_matrix)
def tensor_product(self, other, discard_basis=False):
r"""
Return the tensor product of ``self`` and ``other``.
INPUT:
- ``other`` -- an integral lattice
- ``discard_basis`` -- a boolean (default: ``False``). If ``True``, then the lattice
returned is equipped with the standard basis.
EXAMPLES::
sage: L = IntegralLattice("D3", [[1,-1,0], [0,1,-1]])
sage: L1 = L.tensor_product(L)
sage: L2 = L.tensor_product(L, True)
sage: L1
Lattice of degree 9 and rank 4 over Integer Ring
Basis matrix:
[ 1 -1 0 -1 1 0 0 0 0]
[ 0 1 -1 0 -1 1 0 0 0]
[ 0 0 0 1 -1 0 -1 1 0]
[ 0 0 0 0 1 -1 0 -1 1]
Inner product matrix:
[ 4 -2 -2 -2 1 1 -2 1 1]
[-2 4 0 1 -2 0 1 -2 0]
[-2 0 4 1 0 -2 1 0 -2]
[-2 1 1 4 -2 -2 0 0 0]
[ 1 -2 0 -2 4 0 0 0 0]
[ 1 0 -2 -2 0 4 0 0 0]
[-2 1 1 0 0 0 4 -2 -2]
[ 1 -2 0 0 0 0 -2 4 0]
[ 1 0 -2 0 0 0 -2 0 4]
sage: L1.gram_matrix()
[ 36 -12 -12 4]
[-12 24 4 -8]
[-12 4 24 -8]
[ 4 -8 -8 16]
sage: L2
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:
[ 36 -12 -12 4]
[-12 24 4 -8]
[-12 4 24 -8]
[ 4 -8 -8 16]
"""
if not isinstance(other, FreeQuadraticModule_integer_symmetric):
raise ValueError("other (=%s) must be an integral lattice" % other)
if discard_basis:
gram_matrix = self.gram_matrix().tensor_product(
other.gram_matrix())
return IntegralLattice(gram_matrix)
else:
inner_product_matrix = self.inner_product_matrix().tensor_product(
other.inner_product_matrix())
basis_matrix = self.basis_matrix().tensor_product(
other.basis_matrix())
n = self.degree()
m = other.degree()
ambient = FreeQuadraticModule(self.base_ring(), m * n,
inner_product_matrix)
return FreeQuadraticModule_integer_symmetric(
ambient=ambient,
basis=basis_matrix,
inner_product_matrix=ambient.inner_product_matrix())
def IntegralLattice(data, basis=None):
r"""
Return the integral lattice spanned by ``basis`` in the ambient space.
A lattice is a finitely generated free abelian group `L \cong \ZZ^r`
equipped with a non-degenerate, symmetric bilinear form
`L \times L \colon \rightarrow \ZZ`. Here, lattices have an
ambient quadratic space `\QQ^n` and a distinguished basis.
INPUT:
The input is a descriptor of the lattice and a (optional) basis.
- ``data`` -- can be one of the following:
* a symmetric matrix over the rationals -- the inner product matrix
* an integer -- the dimension for a euclidian lattice
* a symmetric Cartan type or anything recognized by
:class:`CartanMatrix` (see also
:mod:`Cartan types <sage.combinat.root_system.cartan_type>`)
-- for a root lattice
* the string ``"U"`` or ``"H"`` -- for hyperbolic lattices
- ``basis`` -- (optional) a matrix whose rows form a basis of the
lattice, or a list of module elements forming a basis
OUTPUT:
A lattice in the ambient space defined by the inner_product_matrix.
Unless specified, the basis of the lattice is the standard basis.
EXAMPLES::
sage: H5 = Matrix(ZZ, 2, [2,1,1,-2])
sage: IntegralLattice(H5)
Lattice of degree 2 and rank 2 over Integer Ring
Basis matrix:
[1 0]
[0 1]
Inner product matrix:
[ 2 1]
[ 1 -2]
A basis can be specified too::
sage: IntegralLattice(H5, Matrix([1,1]))
Lattice of degree 2 and rank 1 over Integer Ring
Basis matrix:
[1 1]
Inner product matrix:
[ 2 1]
[ 1 -2]
We can define a Euclidian lattice just by its dimension::
sage: IntegralLattice(3)
Lattice of degree 3 and rank 3 over Integer Ring
Basis matrix:
[1 0 0]
[0 1 0]
[0 0 1]
Inner product matrix:
[1 0 0]
[0 1 0]
[0 0 1]
Here is an example of the `A_2` root lattice in Euclidian space::
sage: basis = Matrix([[1,-1,0], [0,1,-1]])
sage: A2 = IntegralLattice(3, basis)
sage: A2
Lattice of degree 3 and rank 2 over Integer Ring
Basis matrix:
[ 1 -1 0]
[ 0 1 -1]
Inner product matrix:
[1 0 0]
[0 1 0]
[0 0 1]
sage: A2.gram_matrix()
[ 2 -1]
[-1 2]
We use ``"U"`` or ``"H"`` for defining a hyperbolic lattice::
sage: L1 = IntegralLattice("U")
sage: L1
Lattice of degree 2 and rank 2 over Integer Ring
Basis matrix:
[1 0]
[0 1]
Inner product matrix:
[0 1]
[1 0]
sage: L1 == IntegralLattice("H")
True
We can construct root lattices by specifying their type
(see :mod:`Cartan types <sage.combinat.root_system.cartan_type>`
and :class:`CartanMatrix`)::
sage: IntegralLattice(["E", 7])
Lattice of degree 7 and rank 7 over Integer Ring
Basis matrix:
[1 0 0 0 0 0 0]
[0 1 0 0 0 0 0]
[0 0 1 0 0 0 0]
[0 0 0 1 0 0 0]
[0 0 0 0 1 0 0]
[0 0 0 0 0 1 0]
[0 0 0 0 0 0 1]
Inner product matrix:
[ 2 0 -1 0 0 0 0]
[ 0 2 0 -1 0 0 0]
[-1 0 2 -1 0 0 0]
[ 0 -1 -1 2 -1 0 0]
[ 0 0 0 -1 2 -1 0]
[ 0 0 0 0 -1 2 -1]
[ 0 0 0 0 0 -1 2]
sage: IntegralLattice(["A", 2])
Lattice of degree 2 and rank 2 over Integer Ring
Basis matrix:
[1 0]
[0 1]
Inner product matrix:
[ 2 -1]
[-1 2]
sage: IntegralLattice("D3")
Lattice of degree 3 and rank 3 over Integer Ring
Basis matrix:
[1 0 0]
[0 1 0]
[0 0 1]
Inner product matrix:
[ 2 -1 -1]
[-1 2 0]
[-1 0 2]
sage: IntegralLattice(["D", 4])
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]
We can specify a basis as well::
sage: G = Matrix(ZZ, 2, [0,1,1,0])
sage: B = [vector([1,1])]
sage: IntegralLattice(G, basis=B)
Lattice of degree 2 and rank 1 over Integer Ring
Basis matrix:
[1 1]
Inner product matrix:
[0 1]
[1 0]
sage: IntegralLattice(["A", 3], [[1,1,1]])
Lattice of degree 3 and rank 1 over Integer Ring
Basis matrix:
[1 1 1]
Inner product matrix:
[ 2 -1 0]
[-1 2 -1]
[ 0 -1 2]
sage: IntegralLattice(4, [[1,1,1,1]])
Lattice of degree 4 and rank 1 over Integer Ring
Basis matrix:
[1 1 1 1]
Inner product matrix:
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
sage: IntegralLattice("A2", [[1,1]])
Lattice of degree 2 and rank 1 over Integer Ring
Basis matrix:
[1 1]
Inner product matrix:
[ 2 -1]
[-1 2]
TESTS::
sage: IntegralLattice(["A", 1, 1])
Traceback (most recent call last):
...
ValueError: lattices must be nondegenerate; use FreeQuadraticModule instead
sage: IntegralLattice(["D", 3, 1])
Traceback (most recent call last):
...
ValueError: lattices must be nondegenerate; use FreeQuadraticModule instead
"""
if is_Matrix(data):
inner_product_matrix = data
elif isinstance(data, Integer):
inner_product_matrix = matrix.identity(ZZ, data)
elif data == "U" or data == "H":
inner_product_matrix = matrix([[0,1],[1,0]])
else:
inner_product_matrix = CartanMatrix(data)
if basis is None:
basis = matrix.identity(ZZ, inner_product_matrix.ncols())
if inner_product_matrix != inner_product_matrix.transpose():
raise ValueError("the inner product matrix must be symmetric\n%s"
% inner_product_matrix)
A = FreeQuadraticModule(ZZ,
inner_product_matrix.ncols(),
inner_product_matrix=inner_product_matrix)
return FreeQuadraticModule_integer_symmetric(ambient=A,
basis=basis,
inner_product_matrix=A.inner_product_matrix(),
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]
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 tensor_product(self, other, discard_basis=False):
r"""
Return the tensor product of ``self`` and ``other``.
INPUT:
- ``other`` -- an integral lattice
- ``discard_basis`` -- a boolean (default: ``False``). If ``True``, then the lattice
returned is equipped with the standard basis.
EXAMPLES::
sage: L = IntegralLattice("D3", [[1,-1,0], [0,1,-1]])
sage: L1 = L.tensor_product(L)
sage: L2 = L.tensor_product(L, True)
sage: L1
Lattice of degree 9 and rank 4 over Integer Ring
Basis matrix:
[ 1 -1 0 -1 1 0 0 0 0]
[ 0 1 -1 0 -1 1 0 0 0]
[ 0 0 0 1 -1 0 -1 1 0]
[ 0 0 0 0 1 -1 0 -1 1]
Inner product matrix:
[ 4 -2 -2 -2 1 1 -2 1 1]
[-2 4 0 1 -2 0 1 -2 0]
[-2 0 4 1 0 -2 1 0 -2]
[-2 1 1 4 -2 -2 0 0 0]
[ 1 -2 0 -2 4 0 0 0 0]
[ 1 0 -2 -2 0 4 0 0 0]
[-2 1 1 0 0 0 4 -2 -2]
[ 1 -2 0 0 0 0 -2 4 0]
[ 1 0 -2 0 0 0 -2 0 4]
sage: L1.gram_matrix()
[ 36 -12 -12 4]
[-12 24 4 -8]
[-12 4 24 -8]
[ 4 -8 -8 16]
sage: L2
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:
[ 36 -12 -12 4]
[-12 24 4 -8]
[-12 4 24 -8]
[ 4 -8 -8 16]
"""
if not isinstance(other, FreeQuadraticModule_integer_symmetric):
raise ValueError("other (=%s) must be an integral lattice" % other)
if discard_basis:
gram_matrix = self.gram_matrix().tensor_product(other.gram_matrix())
return IntegralLattice(gram_matrix)
else:
inner_product_matrix = self.inner_product_matrix().tensor_product(other.inner_product_matrix())
basis_matrix = self.basis_matrix().tensor_product(other.basis_matrix())
n = self.degree()
m = other.degree()
ambient = FreeQuadraticModule(self.base_ring(), m * n, inner_product_matrix)
return FreeQuadraticModule_integer_symmetric(ambient=ambient,
basis=basis_matrix,
inner_product_matrix=ambient.inner_product_matrix())
