def central_form(self):
r"""
Return ``self`` expressed in the canonical basis of the center
of the group algebra.
INPUT:
- ``self`` -- an element of the center of the group algebra
OUTPUT:
- A formal linear combination of the conjugacy class
representatives representing its coordinates in the
canonical basis of the center. See
:meth:`Groups.Algebras.ParentMethods.center_basis` for
details.
.. WARNING::
- This method requires the underlying group to
have a method ``conjugacy_classes_representatives``
(every permutation group has one, thanks GAP!).
- This method does not check that the element is
indeed central. Use the method
:meth:`Monoids.Algebras.ElementMethods.is_central`
for this purpose.
- This function has a complexity linear in the
number of conjugacy classes of the group. One
could easily implement a function whose
complexity is linear in the size of the support
of ``self``.
EXAMPLES::
sage: QS3 = SymmetricGroup(3).algebra(QQ)
sage: A = QS3([2,3,1]) + QS3([3,1,2])
sage: A.central_form()
B[(1,2,3)]
sage: QS4 = SymmetricGroup(4).algebra(QQ)
sage: B = sum(len(s.cycle_type())*QS4(s) for s in Permutations(4))
sage: B.central_form()
4*B[()] + 3*B[(1,2)] + 2*B[(1,2)(3,4)] + 2*B[(1,2,3)] + B[(1,2,3,4)]
The following test fails due to a bug involving combinatorial free modules and
the coercion system (see :trac:`28544`)::
sage: QG = GroupAlgebras(QQ).example(PermutationGroup([[(1,2,3),(4,5)],[(3,4)]]))
sage: s = sum(i for i in QG.basis())
sage: s.central_form() # not tested
B[()] + B[(4,5)] + B[(3,4,5)] + B[(2,3)(4,5)] + B[(2,3,4,5)] + B[(1,2)(3,4,5)] + B[(1,2,3,4,5)]
.. SEEALSO::
- :meth:`Groups.Algebras.ParentMethods.center_basis`
- :meth:`Monoids.Algebras.ElementMethods.is_central`
"""
from sage.combinat.free_module import CombinatorialFreeModule
conj_classes_reps = self.parent().basis().keys().conjugacy_classes_representatives()
Z = CombinatorialFreeModule(self.base_ring(), conj_classes_reps)
return sum(self[i] * Z.basis()[i] for i in Z.basis().keys())
def central_form(self):
r"""
Return ``self`` expressed in the canonical basis of the center
of the group algebra.
INPUT:
- ``self`` -- an element of the center of the group algebra
OUTPUT:
- A formal linear combination of the conjugacy class
representatives representing its coordinates in the
canonical basis of the center. See
:meth:`Groups.Algebras.ParentMethods.center_basis` for
details.
.. WARNING::
- This method requires the underlying group to
have a method ``conjugacy_classes_representatives``
(every permutation group has one, thanks GAP!).
- This method does not check that the element is
indeed central. Use the method
:meth:`Monoids.Algebras.ElementMethods.is_central`
for this purpose.
- This function has a complexity linear in the
number of conjugacy classes of the group. One
could easily implement a function whose
complexity is linear in the size of the support
of ``self``.
EXAMPLES::
sage: QS3 = SymmetricGroup(3).algebra(QQ)
sage: A = QS3([2,3,1]) + QS3([3,1,2])
sage: A.central_form()
B[(1,2,3)]
sage: QS4 = SymmetricGroup(4).algebra(QQ)
sage: B = sum(len(s.cycle_type())*QS4(s) for s in Permutations(4))
sage: B.central_form()
4*B[()] + 3*B[(1,2)] + 2*B[(1,2)(3,4)] + 2*B[(1,2,3)] + B[(1,2,3,4)]
sage: QG = GroupAlgebras(QQ).example(PermutationGroup([[(1,2,3),(4,5)],[(3,4)]]))
sage: sum(i for i in QG.basis()).central_form()
B[()] + B[(4,5)] + B[(3,4,5)] + B[(2,3)(4,5)] + B[(2,3,4,5)] + B[(1,2)(3,4,5)] + B[(1,2,3,4,5)]
.. SEEALSO::
- :meth:`Groups.Algebras.ParentMethods.center_basis`
- :meth:`Monoids.Algebras.ElementMethods.is_central`
"""
from sage.combinat.free_module import CombinatorialFreeModule
conj_classes_reps = self.parent().basis().keys().conjugacy_classes_representatives()
Z = CombinatorialFreeModule(self.base_ring(), conj_classes_reps)
return sum(self[i] * Z.basis()[i] for i in Z.basis().keys())
def __call__(self, program, complex, subcomplex=None, **kwds):
"""
Call a CHomP program to compute the homology of a chain
complex, simplicial complex, or cubical complex.
See :class:`CHomP` for full documentation.
EXAMPLES::
sage: from sage.interfaces.chomp import CHomP
sage: T = cubical_complexes.Torus()
sage: CHomP()('homcubes', T) # indirect doctest, optional - CHomP
{0: 0, 1: Z x Z, 2: Z}
"""
from sage.misc.temporary_file import tmp_filename
from sage.homology.all import CubicalComplex, cubical_complexes
from sage.homology.all import SimplicialComplex, Simplex
from sage.homology.chain_complex import HomologyGroup
from subprocess import Popen, PIPE
from sage.rings.all import QQ, ZZ
from sage.modules.all import VectorSpace, vector
from sage.combinat.free_module import CombinatorialFreeModule
if not have_chomp(program):
raise OSError("Program %s not found" % program)
verbose = kwds.get('verbose', False)
generators = kwds.get('generators', False)
extra_opts = kwds.get('extra_opts', '')
base_ring = kwds.get('base_ring', ZZ)
if extra_opts:
extra_opts = extra_opts.split()
else:
extra_opts = []
# type of complex:
cubical = False
simplicial = False
chain = False
# CHomP seems to have problems with cubical complexes if the
# first interval in the first cube defining the complex is
# degenerate. So replace the complex X with [0,1] x X.
if isinstance(complex, CubicalComplex):
cubical = True
edge = cubical_complexes.Cube(1)
original_complex = complex
complex = edge.product(complex)
if verbose:
print("Cubical complex")
elif isinstance(complex, SimplicialComplex):
simplicial = True
if verbose:
print("Simplicial complex")
else:
chain = True
base_ring = kwds.get('base_ring', complex.base_ring())
if verbose:
print("Chain complex over %s" % base_ring)
if base_ring == QQ:
raise ValueError("CHomP doesn't compute over the rationals, only over Z or F_p.")
if base_ring.is_prime_field():
p = base_ring.characteristic()
extra_opts.append('-p%s' % p)
mod_p = True
else:
mod_p = False
#
# complex
#
try:
data = complex._chomp_repr_()
except AttributeError:
raise AttributeError("Complex can not be converted to use with CHomP.")
datafile = tmp_filename()
f = open(datafile, 'w')
f.write(data)
f.close()
#
# subcomplex
#
if subcomplex is None:
if cubical:
subcomplex = CubicalComplex([complex.n_cells(0)[0]])
elif simplicial:
m = re.search(r'\(([^,]*),', data)
v = int(m.group(1))
subcomplex = SimplicialComplex([[v]])
else:
# replace subcomplex with [0,1] x subcomplex.
if cubical:
subcomplex = edge.product(subcomplex)
#
# generators
#
if generators:
genfile = tmp_filename()
extra_opts.append('-g%s' % genfile)
#
# call program
#
if subcomplex is not None:
try:
sub = subcomplex._chomp_repr_()
except AttributeError:
raise AttributeError("Subcomplex can not be converted to use with CHomP.")
subfile = tmp_filename()
f = open(subfile, 'w')
f.write(sub)
f.close()
else:
subfile = ''
if verbose:
print("Popen called with arguments", end="")
print([program, datafile, subfile] + extra_opts)
print("")
print("CHomP output:")
print("")
# output = Popen([program, datafile, subfile, extra_opts],
cmd = [program, datafile]
if subfile:
cmd.append(subfile)
if extra_opts:
cmd.extend(extra_opts)
output = Popen(cmd, stdout=PIPE).communicate()[0]
if verbose:
print(output)
print("End of CHomP output")
print("")
if generators:
gens = open(genfile, 'r').read()
if verbose:
print("Generators:")
print(gens)
#
# process output
#
if output.find('ERROR') != -1:
raise RuntimeError('error inside CHomP')
# output contains substrings of one of the forms
# "H_1 = Z", "H_1 = Z_2 + Z", "H_1 = Z_2 + Z^2",
# "H_1 = Z + Z_2 + Z"
if output.find('trivial') != -1:
if mod_p:
return {0: VectorSpace(base_ring, 0)}
else:
return {0: HomologyGroup(0, ZZ)}
d = {}
h = re.compile("^H_([0-9]*) = (.*)$", re.M)
tors = re.compile("Z_([0-9]*)")
#
# homology groups
#
for m in h.finditer(output):
if verbose:
print(m.groups())
# dim is the dimension of the homology group
dim = int(m.group(1))
# hom_str is the right side of the equation "H_n = Z^r + Z_k + ..."
hom_str = m.group(2)
# need to read off number of summands and their invariants
if hom_str.find("0") == 0:
if mod_p:
hom = VectorSpace(base_ring, 0)
else:
hom = HomologyGroup(0, ZZ)
else:
rk = 0
if hom_str.find("^") != -1:
rk_srch = re.search(r'\^([0-9]*)\s?', hom_str)
rk = int(rk_srch.group(1))
rk += len(re.findall("(Z$)|(Z\s)", hom_str))
if mod_p:
rk = rk if rk != 0 else 1
if verbose:
print("dimension = %s, rank of homology = %s" % (dim, rk))
hom = VectorSpace(base_ring, rk)
else:
n = rk
invts = []
for t in tors.finditer(hom_str):
n += 1
invts.append(int(t.group(1)))
for i in range(rk):
invts.append(0)
if verbose:
print("dimension = %s, number of factors = %s, invariants = %s" % (dim, n, invts))
hom = HomologyGroup(n, ZZ, invts)
#
# generators
#
if generators:
if cubical:
g = process_generators_cubical(gens, dim)
if verbose:
print("raw generators: %s" % g)
if g:
module = CombinatorialFreeModule(base_ring,
original_complex.n_cells(dim),
prefix="",
bracket=True)
basis = module.basis()
output = []
for x in g:
v = module(0)
for term in x:
v += term[0] * basis[term[1]]
output.append(v)
g = output
elif simplicial:
g = process_generators_simplicial(gens, dim, complex)
if verbose:
print("raw generators: %s" % gens)
if g:
module = CombinatorialFreeModule(base_ring,
complex.n_cells(dim),
prefix="",
bracket=False)
basis = module.basis()
output = []
for x in g:
v = module(0)
for term in x:
if complex._is_numeric():
v += term[0] * basis[term[1]]
else:
translate = complex._translation_from_numeric()
simplex = Simplex([translate[a] for a in term[1]])
v += term[0] * basis[simplex]
output.append(v)
g = output
elif chain:
g = process_generators_chain(gens, dim, base_ring)
if verbose:
print("raw generators: %s" % gens)
if g:
if not mod_p:
# sort generators to match up with corresponding invariant
g = [_[1] for _ in sorted(zip(invts, g), key=lambda x: x[0])]
d[dim] = (hom, g)
else:
d[dim] = hom
else:
d[dim] = hom
if chain:
new_d = {}
diff = complex.differential()
if len(diff) == 0:
return {}
bottom = min(diff)
top = max(diff)
for dim in d:
if complex._degree_of_differential == -1: # chain complex
new_dim = bottom + dim
else: # cochain complex
new_dim = top - dim
if isinstance(d[dim], tuple):
# generators included.
group = d[dim][0]
gens = d[dim][1]
new_gens = []
dimension = complex.differential(new_dim).ncols()
# make sure that each vector is embedded in the
# correct ambient space: pad with a zero if
# necessary.
for v in gens:
v_dict = v.dict()
if dimension - 1 not in v.dict():
v_dict[dimension - 1] = 0
new_gens.append(vector(base_ring, v_dict))
else:
new_gens.append(v)
new_d[new_dim] = (group, new_gens)
else:
new_d[new_dim] = d[dim]
d = new_d
return d
def __call__(self, program, complex, subcomplex=None, **kwds):
"""
Call a CHomP program to compute the homology of a chain
complex, simplicial complex, or cubical complex.
See :class:`CHomP` for full documentation.
EXAMPLES::
sage: from sage.interfaces.chomp import CHomP
sage: T = cubical_complexes.Torus()
sage: CHomP()('homcubes', T) # indirect doctest, optional - CHomP
{0: 0, 1: Z x Z, 2: Z}
"""
from sage.misc.temporary_file import tmp_filename
from sage.homology.all import CubicalComplex, cubical_complexes
from sage.homology.all import SimplicialComplex, Simplex
from sage.homology.chain_complex import HomologyGroup
from subprocess import Popen, PIPE
from sage.rings.all import QQ, ZZ
from sage.modules.all import VectorSpace, vector
from sage.combinat.free_module import CombinatorialFreeModule
if not have_chomp(program):
raise OSError("Program %s not found" % program)
verbose = kwds.get('verbose', False)
generators = kwds.get('generators', False)
extra_opts = kwds.get('extra_opts', '')
base_ring = kwds.get('base_ring', ZZ)
if extra_opts:
extra_opts = extra_opts.split()
else:
extra_opts = []
# type of complex:
cubical = False
simplicial = False
chain = False
# CHomP seems to have problems with cubical complexes if the
# first interval in the first cube defining the complex is
# degenerate. So replace the complex X with [0,1] x X.
if isinstance(complex, CubicalComplex):
cubical = True
edge = cubical_complexes.Cube(1)
original_complex = complex
complex = edge.product(complex)
if verbose:
print("Cubical complex")
elif isinstance(complex, SimplicialComplex):
simplicial = True
if verbose:
print("Simplicial complex")
else:
chain = True
base_ring = kwds.get('base_ring', complex.base_ring())
if verbose:
print("Chain complex over %s" % base_ring)
if base_ring == QQ:
raise ValueError(
"CHomP doesn't compute over the rationals, only over Z or F_p."
)
if base_ring.is_prime_field():
p = base_ring.characteristic()
extra_opts.append('-p%s' % p)
mod_p = True
else:
mod_p = False
#
# complex
#
try:
data = complex._chomp_repr_()
except AttributeError:
raise AttributeError(
"Complex cannot be converted to use with CHomP.")
datafile = tmp_filename()
with open(datafile, 'w') as f:
f.write(data)
#
# subcomplex
#
if subcomplex is None:
if cubical:
subcomplex = CubicalComplex([complex.n_cells(0)[0]])
elif simplicial:
m = re.search(r'\(([^,]*),', data)
v = int(m.group(1))
subcomplex = SimplicialComplex([[v]])
else:
# replace subcomplex with [0,1] x subcomplex.
if cubical:
subcomplex = edge.product(subcomplex)
#
# generators
#
if generators:
genfile = tmp_filename()
extra_opts.append('-g%s' % genfile)
#
# call program
#
if subcomplex is not None:
try:
sub = subcomplex._chomp_repr_()
except AttributeError:
raise AttributeError(
"Subcomplex cannot be converted to use with CHomP.")
subfile = tmp_filename()
with open(subfile, 'w') as f:
f.write(sub)
else:
subfile = ''
if verbose:
print("Popen called with arguments", end="")
print([program, datafile, subfile] + extra_opts)
print("")
print("CHomP output:")
print("")
# output = Popen([program, datafile, subfile, extra_opts],
cmd = [program, datafile]
if subfile:
cmd.append(subfile)
if extra_opts:
cmd.extend(extra_opts)
output = Popen(cmd, stdout=PIPE).communicate()[0]
if verbose:
print(output)
print("End of CHomP output")
print("")
if generators:
with open(genfile, 'r') as f:
gens = f.read()
if verbose:
print("Generators:")
print(gens)
#
# process output
#
if output.find('ERROR') != -1:
raise RuntimeError('error inside CHomP')
# output contains substrings of one of the forms
# "H_1 = Z", "H_1 = Z_2 + Z", "H_1 = Z_2 + Z^2",
# "H_1 = Z + Z_2 + Z"
if output.find('trivial') != -1:
if mod_p:
return {0: VectorSpace(base_ring, 0)}
else:
return {0: HomologyGroup(0, ZZ)}
d = {}
h = re.compile("^H_([0-9]*) = (.*)$", re.M)
tors = re.compile("Z_([0-9]*)")
#
# homology groups
#
for m in h.finditer(output):
if verbose:
print(m.groups())
# dim is the dimension of the homology group
dim = int(m.group(1))
# hom_str is the right side of the equation "H_n = Z^r + Z_k + ..."
hom_str = m.group(2)
# need to read off number of summands and their invariants
if hom_str.find("0") == 0:
if mod_p:
hom = VectorSpace(base_ring, 0)
else:
hom = HomologyGroup(0, ZZ)
else:
rk = 0
if hom_str.find("^") != -1:
rk_srch = re.search(r'\^([0-9]*)\s?', hom_str)
rk = int(rk_srch.group(1))
rk += len(re.findall(r"(Z$)|(Z\s)", hom_str))
if mod_p:
rk = rk if rk != 0 else 1
if verbose:
print("dimension = %s, rank of homology = %s" %
(dim, rk))
hom = VectorSpace(base_ring, rk)
else:
n = rk
invts = []
for t in tors.finditer(hom_str):
n += 1
invts.append(int(t.group(1)))
for i in range(rk):
invts.append(0)
if verbose:
print(
"dimension = %s, number of factors = %s, invariants = %s"
% (dim, n, invts))
hom = HomologyGroup(n, ZZ, invts)
#
# generators
#
if generators:
if cubical:
g = process_generators_cubical(gens, dim)
if verbose:
print("raw generators: %s" % g)
if g:
module = CombinatorialFreeModule(
base_ring,
original_complex.n_cells(dim),
prefix="",
bracket=True)
basis = module.basis()
output = []
for x in g:
v = module(0)
for term in x:
v += term[0] * basis[term[1]]
output.append(v)
g = output
elif simplicial:
g = process_generators_simplicial(gens, dim, complex)
if verbose:
print("raw generators: %s" % gens)
if g:
module = CombinatorialFreeModule(base_ring,
complex.n_cells(dim),
prefix="",
bracket=False)
basis = module.basis()
output = []
for x in g:
v = module(0)
for term in x:
if complex._is_numeric():
v += term[0] * basis[term[1]]
else:
translate = complex._translation_from_numeric(
)
simplex = Simplex(
[translate[a] for a in term[1]])
v += term[0] * basis[simplex]
output.append(v)
g = output
elif chain:
g = process_generators_chain(gens, dim, base_ring)
if verbose:
print("raw generators: %s" % gens)
if g:
if not mod_p:
# sort generators to match up with corresponding invariant
g = [
_[1]
for _ in sorted(zip(invts, g), key=lambda x: x[0])
]
d[dim] = (hom, g)
else:
d[dim] = hom
else:
d[dim] = hom
if chain:
new_d = {}
diff = complex.differential()
if len(diff) == 0:
return {}
bottom = min(diff)
top = max(diff)
for dim in d:
if complex._degree_of_differential == -1: # chain complex
new_dim = bottom + dim
else: # cochain complex
new_dim = top - dim
if isinstance(d[dim], tuple):
# generators included.
group = d[dim][0]
gens = d[dim][1]
new_gens = []
dimension = complex.differential(new_dim).ncols()
# make sure that each vector is embedded in the
# correct ambient space: pad with a zero if
# necessary.
for v in gens:
v_dict = v.dict()
if dimension - 1 not in v.dict():
v_dict[dimension - 1] = 0
new_gens.append(vector(base_ring, v_dict))
else:
new_gens.append(v)
new_d[new_dim] = (group, new_gens)
else:
new_d[new_dim] = d[dim]
d = new_d
return d
