def __add__(self, other, **kwargs):
r"""
Direct sum of Hermitian lattices.
"""
if isinstance(other, HermitianWeilRep):
return HermitianWeilRep(block_diagonal_matrix(
[self.complex_gram_matrix(),
other.complex_gram_matrix()],
subdivide=False),
gen=self.__w)
elif isinstance(other, RescaledHyperbolicPlane):
return HermitianRescaledHyperbolicPlane(other._N()).__add__(self)
return NotImplemented
def lift_for_SL(A, N=None):
r"""
Lift a matrix `A` from `SL_m(\ZZ / N\ZZ)` to `SL_m(\ZZ)`.
This follows [Shi1971]_, Lemma 1.38, p. 21.
INPUT:
- ``A`` -- a square matrix with coefficients in `\ZZ / N\ZZ` (or `\ZZ`)
- ``N`` -- the modulus (optional) required only if the matrix ``A``
has coefficients in `\ZZ`
EXAMPLES::
sage: from sage.modular.local_comp.liftings import lift_for_SL
sage: A = matrix(Zmod(11), 4, 4, [6, 0, 0, 9, 1, 6, 9, 4, 4, 4, 8, 0, 4, 0, 0, 8])
sage: A.det()
1
sage: L = lift_for_SL(A)
sage: L.det()
1
sage: (L - A) == 0
True
sage: B = matrix(Zmod(19), 4, 4, [1, 6, 10, 4, 4, 14, 15, 4, 13, 0, 1, 15, 15, 15, 17, 10])
sage: B.det()
1
sage: L = lift_for_SL(B)
sage: L.det()
1
sage: (L - B) == 0
True
TESTS::
sage: lift_for_SL(matrix(3,3,[1,2,0,3,4,0,0,0,1]),3)
[10 14 3]
[ 9 10 3]
[ 3 3 1]
sage: A = matrix(Zmod(7), 2, [1,0,0,1])
sage: L = lift_for_SL(A)
sage: L.parent()
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
sage: A = matrix(Zmod(7), 1, [1])
sage: L = lift_for_SL(A); L
[1]
sage: A = matrix(ZZ, 2, [1,0,0,1])
sage: lift_for_SL(A)
Traceback (most recent call last):
...
ValueError: you must choose the modulus
sage: for _ in range(100):
....: d = randint(0, 10)
....: p = choice([2,3,5,7,11])
....: M = random_matrix(Zmod(p), d, algorithm='unimodular')
....: assert lift_for_SL(M).det() == 1
"""
from sage.matrix.special import (identity_matrix, diagonal_matrix,
block_diagonal_matrix)
from sage.misc.misc_c import prod
ring = A.parent().base_ring()
if N is None:
if ring is ZZ:
raise ValueError('you must choose the modulus')
else:
N = ring.characteristic()
m = A.nrows()
if m <= 1:
return identity_matrix(ZZ, m)
AZZ = A.change_ring(ZZ)
D, U, V = AZZ.smith_form()
diag = diagonal_matrix([-1] + [1] * (m - 1))
if U.det() == -1:
U = diag * U
if V.det() == -1:
V = V * diag
a = [U.row(i) * AZZ * V.column(i) for i in range(m)]
b = prod(a[1:])
Winv = identity_matrix(m)
Winv[1, 0] = 1 - b
Winv[0, 1] = -1
Winv[1, 1] = b
Xinv = identity_matrix(m)
Xinv[0, 1] = a[1]
Cp = diagonal_matrix(a[1:])
Cp[0, 0] *= a[0]
C = lift_for_SL(Cp, N)
Cpp = block_diagonal_matrix(identity_matrix(1), C)
Cpp[1, 0] = 1 - a[0]
return (~U * Winv * Cpp * Xinv * ~V).change_ring(ZZ)
def _direct_sum(self, other):
if not (self._base_ring == other._base_ring and self._domain_poset == other._domain_poset):
raise TypeError("Sheaves are not defined on same poset or not defined over same ring")
direct_sum_stalks = {x:self._stalk_dict[x] + other._stalk_dict[x] for x in self._domain_poset.list()}
direct_sum_res = {tuple(r):block_diagonal_matrix(self.restriction(r[0], r[1]).matrix(), other.restriction(r[0], r[1]).matrix(), subdivide = False) for r in self._domain_poset.cover_relations()}
return LocallyFreeSheafFinitePoset(direct_sum_stalks, direct_sum_res, self._base_ring, self._domain_poset)
def test_one_tensor_m_block_diag():
for m in matrices:
block_diagonal_matrix([m] * one.nrows())