def _split_hyperbolic(L) :
cur_cor = 2
Lcor = L + cur_cor * identity_matrix(L.nrows())
while not is_positive_definite(Lcor) :
cur_cor += 2
Lcor = L + cur_cor * identity_matrix(L.nrows())
a = FreeModule(ZZ, L.nrows()).gen(0)
if a * L * a >= 0 :
Lcor_length_inc = max(3, a * L * a)
cur_Lcor_length = Lcor_length_inc
while True :
short_vectors = flatten(QuadraticForm(Lcor).short_vector_list_up_to_length( a * Lcor * a )[cur_Lcor_length - Lcor_length_inc: cur_Lcor_length], max_level = 1)
for a in short_vectors :
if a * L * a < 0 :
break
else :
continue
break
n = -a * L * a // 2
short_vectors = E8.short_vector_list_up_to_length(n + 1)[-1]
for v in short_vectors :
for w in short_vectors :
if v * E8_gram * w == 2 * n - 1 :
LE8_mat = L.block_sum(E8_gram)
v_form = vector( list(a) + list(v) ) * LE8_mat
w_form = vector( list(a) + list(w) ) * LE8_mat
Lred_basis = matrix(ZZ, [v_form, w_form]).right_kernel().basis_matrix().transpose()
Lred_basis = matrix(ZZ, Lred_basis)
return Lred_basis.transpose() * LE8_mat * Lred_basis
def identity(self):
r"""
Return identity morphism in an endomorphism ring.
EXAMPLE::
sage: V=FreeModule(ZZ,5)
sage: H=V.Hom(V)
sage: H.identity()
Free module morphism defined by the matrix
[1 0 0 0 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]
Domain: Ambient free module of rank 5 over the principal ideal domain ...
Codomain: Ambient free module of rank 5 over the principal ideal domain ...
"""
if self.is_endomorphism_set():
return self(
matrix.identity_matrix(self.base_ring(),
self.domain().rank()))
else:
raise TypeError(
"Identity map only defined for endomorphisms. Try natural_map() instead."
)
def split_off_unimodular_lattice(self, L_basis_matrix, check = True) :
r"""
Split off a unimodular lattice that is an orthogonal summand of self.
INPUT:
- ``L_basis_matrix`` -- A matrix over `\ZZ` whose number of rows equals the
size of the underlying lattice.
- ``check`` -- A boolean (default: ``True``). If ``True`` it will be
checked whether the given sublattice is a direct summand.
OUTPUT:
- A pair of a discriminant group and a homomorphism from self to this group.
"""
if check and L_basis_matrix.base_ring() is not ZZ :
raise ValueError( "L_basis_matrix must define a sublattice" )
pre_bases = matrix(ZZ, [ [ l * self._L * b for l in L_basis_matrix.columns()]
for b in FreeModule(ZZ, self._L.nrows()).gens() ]) \
.augment(identity_matrix(ZZ, self._L.nrows())).echelon_form()
if check and pre_bases[:L_basis_matrix.ncols(),:L_basis_matrix.ncols()] != identity_matrix(ZZ, L_basis_matrix.ncols()) :
raise ValueError( "The sublattice defined by L_basis_matrix must be unimodular" )
K_basis_matrix = pre_bases[L_basis_matrix.ncols():,L_basis_matrix.ncols():].transpose()
if check and L_basis_matrix.column_module() + K_basis_matrix.column_module() != FreeModule(ZZ, self._L.nrows()) :
raise ValueError( "The sublattice defined by L_basis_matrix must be an orthogonal summand" )
K = K_basis_matrix.transpose() * self._L * K_basis_matrix
n_disc = DiscriminantForm(K)
total_basis_matrix = L_basis_matrix.change_ring(QQ).augment(K_basis_matrix * n_disc._dual_basis)
basis_images = [ sum(map(operator.mul, b.lift(), self._dual_basis.columns()))
for b in self.smith_form_gens() ]
basis_images = [ total_basis_matrix.solve_right(b)[-K_basis_matrix.ncols():]
for b in basis_images ]
coercion_hom = self.hom([ sum(map(operator.mul, map(ZZ, b), map(n_disc, FreeModule(ZZ, K.nrows()).gens()) ))
for b in basis_images ])
return (n_disc, coercion_hom)
def P(self, n, names='z+', base_ring=QQ):
r"""
Construct the ``n``-dimensional projective space `\mathbb{P}^n`.
INPUT:
- ``n`` -- positive integer. The dimension of the projective space.
- ``names`` -- string. Names for the homogeneous
coordinates. See
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`CPR-Fano toric variety
<sage.schemes.toric.fano_variety.CPRFanoToricVariety_field>`.
EXAMPLES::
sage: P3 = toric_varieties.P(3)
sage: P3
3-d CPR-Fano toric variety covered by 4 affine patches
sage: P3.fan().rays()
N( 1, 0, 0),
N( 0, 1, 0),
N( 0, 0, 1),
N(-1, -1, -1)
in 3-d lattice N
sage: P3.gens()
(z0, z1, z2, z3)
"""
# We are going to eventually switch off consistency checks, so we need
# to be sure that the input is acceptable.
try:
n = ZZ(n) # make sure that we got a "mathematical" integer
except TypeError:
raise TypeError("dimension of the projective space must be a "
"positive integer!\nGot: %s" % n)
if n <= 0:
raise ValueError("only projective spaces of positive dimension "
"can be constructed!\nGot: %s" % n)
m = identity_matrix(n).augment(matrix(n, 1, [-1] * n))
charts = [
list(range(i)) + list(range(i + 1, n + 1)) for i in range(n + 1)
]
return CPRFanoToricVariety(Delta_polar=LatticePolytope(
m.columns(), lattice=ToricLattice(n)),
charts=charts,
check=self._check,
coordinate_names=names,
base_ring=base_ring)
def P(self, n, names='z+', base_ring=QQ):
r"""
Construct the ``n``-dimensional projective space `\mathbb{P}^n`.
INPUT:
- ``n`` -- positive integer. The dimension of the projective space.
- ``names`` -- string. Names for the homogeneous
coordinates. See
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`CPR-Fano toric variety
<sage.schemes.toric.fano_variety.CPRFanoToricVariety_field>`.
EXAMPLES::
sage: P3 = toric_varieties.P(3)
sage: P3
3-d CPR-Fano toric variety covered by 4 affine patches
sage: P3.fan().rays()
N( 1, 0, 0),
N( 0, 1, 0),
N( 0, 0, 1),
N(-1, -1, -1)
in 3-d lattice N
sage: P3.gens()
(z0, z1, z2, z3)
"""
# We are going to eventually switch off consistency checks, so we need
# to be sure that the input is acceptable.
try:
n = ZZ(n) # make sure that we got a "mathematical" integer
except TypeError:
raise TypeError("dimension of the projective space must be a "
"positive integer!\nGot: %s" % n)
if n <= 0:
raise ValueError("only projective spaces of positive dimension "
"can be constructed!\nGot: %s" % n)
m = identity_matrix(n).augment(matrix(n, 1, [-1]*n))
charts = [list(range(i)) + list(range(i + 1, n + 1))
for i in range(n + 1)]
return CPRFanoToricVariety(
Delta_polar=LatticePolytope(m.columns(), lattice=ToricLattice(n)),
charts=charts, check=self._check, coordinate_names=names,
base_ring=base_ring)
def _split_hyperbolic(L):
cur_cor = 2
Lcor = L + cur_cor * identity_matrix(L.nrows())
while not is_positive_definite(Lcor):
cur_cor += 2
Lcor = L + cur_cor * identity_matrix(L.nrows())
a = FreeModule(ZZ, L.nrows()).gen(0)
if a * L * a >= 0:
Lcor_length_inc = max(3, a * L * a)
cur_Lcor_length = Lcor_length_inc
while True:
short_vectors = flatten(
QuadraticForm(Lcor).short_vector_list_up_to_length(
a * Lcor * a)[cur_Lcor_length -
Lcor_length_inc:cur_Lcor_length],
max_level=1)
for a in short_vectors:
if a * L * a < 0:
break
else:
continue
break
n = -a * L * a // 2
short_vectors = E8.short_vector_list_up_to_length(n + 1)[-1]
for v in short_vectors:
for w in short_vectors:
if v * E8_gram * w == 2 * n - 1:
LE8_mat = L.block_sum(E8_gram)
v_form = vector(list(a) + list(v)) * LE8_mat
w_form = vector(list(a) + list(w)) * LE8_mat
Lred_basis = matrix(
ZZ, [v_form, w_form
]).right_kernel().basis_matrix().transpose()
Lred_basis = matrix(ZZ, Lred_basis)
return Lred_basis.transpose() * LE8_mat * Lred_basis
def A(self, n, names='z+', base_ring=QQ):
r"""
Construct the ``n``-dimensional affine space.
INPUT:
- ``n`` -- positive integer. The dimension of the affine space.
- ``names`` -- string. Names for the homogeneous
coordinates. See
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`toric variety
<sage.schemes.toric.variety.ToricVariety_field>`.
EXAMPLES::
sage: A3 = toric_varieties.A(3)
sage: A3
3-d affine toric variety
sage: A3.fan().rays()
N(1, 0, 0),
N(0, 1, 0),
N(0, 0, 1)
in 3-d lattice N
sage: A3.gens()
(z0, z1, z2)
"""
# We are going to eventually switch off consistency checks, so we need
# to be sure that the input is acceptable.
try:
n = ZZ(n) # make sure that we got a "mathematical" integer
except TypeError:
raise TypeError("dimension of the affine space must be a "
"positive integer!\nGot: %s" % n)
if n <= 0:
raise ValueError("only affine spaces of positive dimension can "
"be constructed!\nGot: %s" % n)
rays = identity_matrix(n).columns()
cones = [list(range(n))]
fan = Fan(cones, rays, check=self._check)
return ToricVariety(fan, coordinate_names=names)
def A(self, n, names='z+', base_ring=QQ):
r"""
Construct the ``n``-dimensional affine space.
INPUT:
- ``n`` -- positive integer. The dimension of the affine space.
- ``names`` -- string. Names for the homogeneous
coordinates. See
:func:`~sage.schemes.toric.variety.normalize_names`
for acceptable formats.
- ``base_ring`` -- a ring (default: `\QQ`). The base ring for
the toric variety.
OUTPUT:
A :class:`toric variety
<sage.schemes.toric.variety.ToricVariety_field>`.
EXAMPLES::
sage: A3 = toric_varieties.A(3)
sage: A3
3-d affine toric variety
sage: A3.fan().rays()
N(1, 0, 0),
N(0, 1, 0),
N(0, 0, 1)
in 3-d lattice N
sage: A3.gens()
(z0, z1, z2)
"""
# We are going to eventually switch off consistency checks, so we need
# to be sure that the input is acceptable.
try:
n = ZZ(n) # make sure that we got a "mathematical" integer
except TypeError:
raise TypeError("dimension of the affine space must be a "
"positive integer!\nGot: %s" % n)
if n <= 0:
raise ValueError("only affine spaces of positive dimension can "
"be constructed!\nGot: %s" % n)
rays = identity_matrix(n).columns()
cones = [ range(0,n) ]
fan = Fan(cones, rays, check=self._check)
return ToricVariety(fan, coordinate_names=names)
def P(self, n, names="z+"):
r"""
Construct the ``n``-dimensional projective space `\mathbb{P}^n`.
INPUT:
- ``n`` -- positive integer. The dimension of the projective space.
- ``names`` -- string. Names for the homogeneous
coordinates. See
:func:`~sage.schemes.generic.toric_variety.normalize_names`
for acceptable formats.
OUTPUT:
A :class:`CPR-Fano toric variety
<sage.schemes.generic.fano_toric_variety.CPRFanoToricVariety_field>`.
EXAMPLES::
sage: P3 = toric_varieties.P(3)
sage: P3
3-d CPR-Fano toric variety covered by 4 affine patches
sage: P3.fan().ray_matrix()
[ 1 0 0 -1]
[ 0 1 0 -1]
[ 0 0 1 -1]
sage: P3.gens()
(z0, z1, z2, z3)
"""
# We are going to eventually switch off consistency checks, so we need
# to be sure that the input is acceptable.
try:
n = ZZ(n) # make sure that we got a "mathematical" integer
except TypeError:
raise TypeError("dimension of the projective space must be a " "positive integer!\nGot: %s" % n)
if n <= 0:
raise ValueError("only projective spaces of positive dimension " "can be constructed!\nGot: %s" % n)
m = identity_matrix(n).augment(matrix(n, 1, [-1] * n))
charts = [range(0, i) + range(i + 1, n + 1) for i in range(0, n + 1)]
return CPRFanoToricVariety(
Delta_polar=LatticePolytope(m), charts=charts, check=self._check, coordinate_names=names
)
def identity(self):
r"""
Return identity morphism in an endomorphism ring.
EXAMPLE::
sage: V=FreeModule(ZZ,5)
sage: H=V.Hom(V)
sage: H.identity()
Free module morphism defined by the matrix
[1 0 0 0 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]
Domain: Ambient free module of rank 5 over the principal ideal domain ...
Codomain: Ambient free module of rank 5 over the principal ideal domain ...
"""
if self.is_endomorphism_set():
return self(matrix.identity_matrix(self.base_ring(),self.domain().rank()))
else:
raise TypeError, "Identity map only defined for endomorphisms. Try natural_map() instead."
def to_matrix(self, side="right", on_space="primal"):
r"""
Return ``self`` as a matrix acting on the underlying vector
space.
- ``side`` -- optional (default: ``"right"``) whether the
action of ``self`` is on the ``"left"`` or on the ``"right"``
- ``on_space`` -- optional (default: ``"primal"``) whether
to act as the reflection representation on the given
basis, or to act on the dual reflection representation
on the dual basis
EXAMPLES::
sage: W = ReflectionGroup(['A',2]) # optional - gap3
sage: for w in W: # optional - gap3
....: w.reduced_word() # optional - gap3
....: [w.to_matrix(), w.to_matrix(on_space="dual")] # optional - gap3
[]
[
[1 0] [1 0]
[0 1], [0 1]
]
[2]
[
[ 1 1] [ 1 0]
[ 0 -1], [ 1 -1]
]
[1]
[
[-1 0] [-1 1]
[ 1 1], [ 0 1]
]
[1, 2]
[
[-1 -1] [ 0 -1]
[ 1 0], [ 1 -1]
]
[2, 1]
[
[ 0 1] [-1 1]
[-1 -1], [-1 0]
]
[1, 2, 1]
[
[ 0 -1] [ 0 -1]
[-1 0], [-1 0]
]
TESTS::
sage: W = ReflectionGroup(['F',4]) # optional - gap3
sage: all(w.to_matrix(side="left") == W.from_reduced_word(reversed(w.reduced_word())).to_matrix(side="right").transpose() for w in W) # optional - gap3
True
sage: all(w.to_matrix(side="right") == W.from_reduced_word(reversed(w.reduced_word())).to_matrix(side="left").transpose() for w in W) # optional - gap3
True
"""
W = self.parent()
if W._reflection_representation is None:
if side == "left":
w = ~self
elif side == "right":
w = self
else:
raise ValueError('side must be "left" or "right"')
Delta = W.independent_roots()
Phi = W.roots()
M = Matrix([Phi[w(Phi.index(alpha) + 1) - 1] for alpha in Delta])
mat = W.base_change_matrix() * M
else:
refl_repr = W._reflection_representation
id_mat = identity_matrix(QQ, refl_repr[W.index_set()[0]].nrows())
mat = prod([refl_repr[i] for i in self.reduced_word()], id_mat)
if on_space == "primal":
if side == "left":
mat = mat.transpose()
elif on_space == "dual":
if side == "left":
mat = mat.inverse()
else:
mat = mat.inverse().transpose()
else:
raise ValueError('on_space must be "primal" or "dual"')
mat.set_immutable()
return mat
def _compute_q_expansion_basis(self, prec=None):
r"""
Compute q-expansion basis using Schaeffer's algorithm.
EXAMPLES::
sage: CuspForms(GammaH(31, [7]), 1).q_expansion_basis() # indirect doctest
[
q - q^2 - q^5 + O(q^6)
]
A more elaborate example (two Galois-conjugate characters each giving a
2-dimensional space)::
sage: CuspForms(GammaH(124, [85]), 1).q_expansion_basis()
[
q - q^4 - q^6 + O(q^7),
q^2 + O(q^7),
q^3 + O(q^7),
q^5 - q^6 + O(q^7)
]
"""
if prec is None:
prec = self.prec()
else:
prec = Integer(prec)
chars=self.group().characters_mod_H(sign=-1, galois_orbits=True)
B = []
dim = 0
for c in chars:
chi = c.minimize_base_ring()
Bchi = [weight1.modular_ratio_to_prec(chi, f, prec)
for f in weight1.hecke_stable_subspace(chi) ]
if Bchi == []:
continue
if chi.base_ring() == QQ:
B += [f.padded_list(prec) for f in Bchi]
dim += len(Bchi)
else:
d = chi.base_ring().degree()
dim += d * len(Bchi)
for f in Bchi:
w = f.padded_list(prec)
for i in range(d):
B.append([x[i] for x in w])
basis_mat = Matrix(QQ, B)
if basis_mat.is_zero():
return []
# Daft thing: "transformation=True" parameter to echelonize
# is ignored for rational matrices!
big_mat = basis_mat.augment(identity_matrix(dim))
big_mat.echelonize()
c = big_mat.pivots()[-1]
echelon_basis_mat = big_mat[:, :prec]
R = self._q_expansion_ring()
if c >= prec:
verbose("Precision %s insufficient to determine basis" % prec, level=1)
else:
verbose("Minimal precision for basis: %s" % (c+1), level=1)
t = big_mat[:, prec:]
assert echelon_basis_mat == t * basis_mat
self.__transformation_matrix = t
self._char_basis = [R(f.list(), c+1) for f in basis_mat.rows()]
return [R(f.list(), prec) for f in echelon_basis_mat.rows() if f != 0]
def to_matrix(self, side="right", on_space="primal"):
r"""
Return ``self`` as a matrix acting on the underlying vector
space.
- ``side`` -- optional (default: ``"right"``) whether the
action of ``self`` is on the ``"left"`` or on the ``"right"``
- ``on_space`` -- optional (default: ``"primal"``) whether
to act as the reflection representation on the given
basis, or to act on the dual reflection representation
on the dual basis
EXAMPLES::
sage: W = ReflectionGroup(['A',2]) # optional - gap3
sage: for w in W: # optional - gap3
....: w.reduced_word() # optional - gap3
....: [w.to_matrix(), w.to_matrix(on_space="dual")] # optional - gap3
[]
[
[1 0] [1 0]
[0 1], [0 1]
]
[2]
[
[ 1 1] [ 1 0]
[ 0 -1], [ 1 -1]
]
[1]
[
[-1 0] [-1 1]
[ 1 1], [ 0 1]
]
[1, 2]
[
[-1 -1] [ 0 -1]
[ 1 0], [ 1 -1]
]
[2, 1]
[
[ 0 1] [-1 1]
[-1 -1], [-1 0]
]
[1, 2, 1]
[
[ 0 -1] [ 0 -1]
[-1 0], [-1 0]
]
TESTS::
sage: W = ReflectionGroup(['F',4]) # optional - gap3
sage: all(w.to_matrix(side="left") == W.from_reduced_word(reversed(w.reduced_word())).to_matrix(side="right").transpose() for w in W) # optional - gap3
True
sage: all(w.to_matrix(side="right") == W.from_reduced_word(reversed(w.reduced_word())).to_matrix(side="left").transpose() for w in W) # optional - gap3
True
"""
W = self.parent()
if W._reflection_representation is None:
if side == "left":
w = ~self
elif side == "right":
w = self
else:
raise ValueError('side must be "left" or "right"')
Delta = W.independent_roots()
Phi = W.roots()
M = Matrix([Phi[w(Phi.index(alpha)+1)-1] for alpha in Delta])
mat = W.base_change_matrix() * M
else:
refl_repr = W._reflection_representation
id_mat = identity_matrix(QQ, refl_repr[W.index_set()[0]].nrows())
mat = prod([refl_repr[i] for i in self.reduced_word()], id_mat)
if on_space == "primal":
if side == "left":
mat = mat.transpose()
elif on_space == "dual":
if side == "left":
mat = mat.inverse()
else:
mat = mat.inverse().transpose()
else:
raise ValueError('on_space must be "primal" or "dual"')
mat.set_immutable()
return mat
def eigenvalue_multiplicity(mat, ev) :
mat = matrix(CC, mat - ev * identity_matrix(subspace_dimension))
return len(filter( lambda row: all( e.contains_zero() for e in row), _qr(mat).rows() ))
def eigenvalue_multiplicity(mat, ev):
mat = matrix(CC, mat - ev * identity_matrix(subspace_dimension))
return len(
filter(lambda row: all(e.contains_zero() for e in row),
_qr(mat).rows()))
def split_off_unimodular_lattice(self, L_basis_matrix, check=True):
r"""
Split off a unimodular lattice that is an orthogonal summand of self.
INPUT:
- ``L_basis_matrix`` -- A matrix over `\ZZ` whose number of rows equals the
size of the underlying lattice.
- ``check`` -- A boolean (default: ``True``). If ``True`` it will be
checked whether the given sublattice is a direct summand.
OUTPUT:
- A pair of a discriminant group and a homomorphism from self to this group.
"""
if check and L_basis_matrix.base_ring() is not ZZ:
raise ValueError("L_basis_matrix must define a sublattice")
pre_bases = matrix(ZZ, [ [ l * self._L * b for l in L_basis_matrix.columns()]
for b in FreeModule(ZZ, self._L.nrows()).gens() ]) \
.augment(identity_matrix(ZZ, self._L.nrows())).echelon_form()
if check and pre_bases[:L_basis_matrix.ncols(), :L_basis_matrix.ncols(
)] != identity_matrix(ZZ, L_basis_matrix.ncols()):
raise ValueError(
"The sublattice defined by L_basis_matrix must be unimodular")
K_basis_matrix = pre_bases[L_basis_matrix.ncols():,
L_basis_matrix.ncols():].transpose()
if check and L_basis_matrix.column_module(
) + K_basis_matrix.column_module() != FreeModule(ZZ, self._L.nrows()):
raise ValueError(
"The sublattice defined by L_basis_matrix must be an orthogonal summand"
)
K = K_basis_matrix.transpose() * self._L * K_basis_matrix
n_disc = DiscriminantForm(K)
total_basis_matrix = L_basis_matrix.change_ring(QQ).augment(
K_basis_matrix * n_disc._dual_basis)
basis_images = [
sum(map(operator.mul, b.lift(), self._dual_basis.columns()))
for b in self.smith_form_gens()
]
basis_images = [
total_basis_matrix.solve_right(b)[-K_basis_matrix.ncols():]
for b in basis_images
]
coercion_hom = self.hom([
sum(
map(operator.mul, map(ZZ, b),
map(n_disc,
FreeModule(ZZ, K.nrows()).gens())))
for b in basis_images
])
return (n_disc, coercion_hom)
