def galois_action_on_embeddings(G_K):
K = G_K.number_field()
Kgal = G_K.splitting_field()
embeddings = K.embeddings(Kgal)
# first a shortcut in the case where G_K is normal in S_d since then it doesn't
# matter for our application since we only care about the image
# of the galois group in S_d
d = K.absolute_degree()
G_K_roots = TransitiveGroup(d, G_K.transitive_number())
if G_K_roots.is_normal(SymmetricGroup(d)):
id_G_K = G_K.Hom(G_K).identity()
return G_K, id_G_K, id_G_K, Kgal, embeddings
# now for the actual computation
permutations = []
for g in G_K.gens():
phi = g.as_hom()
g_perm = Permutation(
[embeddings.index(phi * emb) + 1 for emb in embeddings]).inverse()
permutations.append(g_perm)
G_K_emb = PermutationGroup(permutations, canonicalize=False)
to_emb = G_K.hom(G_K_emb.gens())
from_emb = G_K_emb.hom(G_K.gens())
return G_K_emb, to_emb, from_emb, Kgal, embeddings
def permute_indices(self, permutation):
r"""
Return a tensor with indices with permuted indices.
INPUT:
- ``permutation`` -- permutation that has to be applied to the indices
the input should be a ``list`` containing the second line of the permutation
in Cauchy notation.
OUTPUT:
- an instance of ``TensorWithIndices`` whose indices names and place
are those of ``self`` but whose components have been permuted with
``permutation``.
EXAMPLES::
sage: M = FiniteRankFreeModule(QQ, 3, name='M')
sage: e = M.basis('e')
sage: a = M.tensor((2,0), name='a')
sage: a[:] = [[1,2,3], [4,5,6], [7,8,9]]
sage: b = M.tensor((2,0), name='b')
sage: b[:] = [[-1,2,-3], [-4,5,6], [7,-8,9]]
sage: identity = [0,1]
sage: transposition = [1,0]
sage: a["ij"].permute_indices(identity) == a["ij"]
True
sage: a["ij"].permute_indices(transposition)[:] == a[:].transpose()
True
sage: cycle = [1,2,3,0] # the cyclic permutation sending 0 to 1
sage: (a*b)[0,1,2,0] == (a*b)["ijkl"].permute_indices(cycle)[1,2,0,0]
True
TESTS::
sage: M = FiniteRankFreeModule(QQ, 3, name='M')
sage: e = M.basis('e')
sage: a = M.tensor((2,0), name='a')
sage: a[:] = [[1,2,3], [4,5,6], [7,8,9]]
sage: identity = [0,1]
sage: transposition = [1,0]
sage: a["ij"].permute_indices(identity) == a["ij"]
True
sage: a["ij"].permute_indices(transposition)[:] == a[:].transpose()
True
sage: (a*a)["ijkl"].permute_indices([1,2,3,0])[0,1,2,1] == (a*a)[1,2,1,0]
True
"""
# Decomposition of the permutation of the components of self
# into product of swaps given by the method
# sage.tensor.modules.comp.Components.swap_adjacent_indices
# A swap is determined by 3 distinct integers
swap_params = list(
combinations(range(self._tensor.tensor_rank() + 1), 3))
# The associated permutation is as follows
def swap(param, N):
i, j, k = param
L = list(range(1, N + 1))
L = L[:i] + L[j:k] + L[i:j] + L[k:]
return L
# Construction of the permutation group generated by swaps
perm_group = PermutationGroup(
[swap(param, self._tensor.tensor_rank()) for param in swap_params],
canonicalize=False)
# Compute a decomposition of the permutation as a product of swaps
decomposition_as_string = perm_group(
[x + 1 for x in permutation]).word_problem(perm_group.gens(),
display=False)[0]
if decomposition_as_string != "<identity ...>":
decomposition_as_string = [
# Two cases wether the term appear with an exponent or not
("^" in term) * term.split("^") + ("^" not in term) *
(term.split("^") + ['1'])
for term in decomposition_as_string.replace("x", "").split("*")
]
decomposition = [(swap_params[int(x) - 1], int(y))
for x, y in decomposition_as_string]
decomposition.reverse(
) # /!\ The symetric group acts on the right by default /!\.
else:
decomposition = []
# Choice of a basis
basis = self._tensor._fmodule._def_basis
# Swap of components
swaped_components = self._tensor.comp(basis)
for swap_param, exponent in decomposition:
if exponent > 0:
for i in range(exponent):
# Apply the swap given by swap_param
swaped_components = swaped_components\
.swap_adjacent_indices(*swap_param)
elif exponent < 0:
for i in range(-exponent):
# Apply the opposite of the swap given by swap_param
swaped_components = swaped_components\
.swap_adjacent_indices(
swap_param[0],
swap_param[0] + swap_param[2] - swap_param[1],
swap_param[2]
)
else:
pass
result = self.__pos__()
result._tensor = self._tensor._fmodule.tensor_from_comp(
self._tensor.tensor_type(), swaped_components)
return result
