def __call__(self, x):
"""
EXAMPLES::
sage: W = WeylGroup(['A',2])
sage: W(1)
[1 0 0]
[0 1 0]
[0 0 1]
::
sage: W(2)
Traceback (most recent call last):
...
TypeError: no way to coerce element into self.
sage: W2 = WeylGroup(['A',3])
sage: W(1) in W2 # indirect doctest
False
"""
if isinstance(x, self.element_class) and x.parent() is self:
return x
from sage.matrix.matrix import is_Matrix
if not (x in ZZ or is_Matrix(x)): # this should be handled by self.matrix_space()(x)
raise TypeError, "no way to coerce element into self"
M = self.matrix_space()(x)
# This is really bad, especially for infinite groups!
# TODO: compute the image of rho, compose by s_i until back in
# the fundamental chamber. Return True iff the matrix is the identity
g = self._element_constructor_(M)
if not gap(g) in gap(self):
raise TypeError, "no way to coerce element into self."
return g
def __call__(self, x):
"""
EXAMPLES::
sage: W = WeylGroup(['A',2])
sage: W(1)
[1 0 0]
[0 1 0]
[0 0 1]
::
sage: W(2)
Traceback (most recent call last):
...
TypeError: no way to coerce element into self.
sage: W2 = WeylGroup(['A',3])
sage: W(1) in W2 # indirect doctest
False
"""
if isinstance(x, self.element_class) and x.parent() is self:
return x
from sage.matrix.matrix import is_Matrix
if not (x in ZZ or is_Matrix(x)): # this should be handled by self.matrix_space()(x)
raise TypeError, "no way to coerce element into self"
M = self.matrix_space()(x)
# This is really bad, especially for infinite groups!
# TODO: compute the image of rho, compose by s_i until back in
# the fundamental chamber. Return True iff the matrix is the identity
g = self._element_constructor_(M)
if not gap(g) in gap(self):
raise TypeError, "no way to coerce element into self."
return g
def __init__(self, homset, imgsH, check=True):
MatrixGroupMorphism.__init__(self, homset) # sets the parent
G = homset.domain()
H = homset.codomain()
gaplist_gens = [gap(x) for x in G.gens()]
gaplist_imgs = [gap(x) for x in imgsH]
genss = '[%s]'%(','.join(str(v) for v in gaplist_gens))
imgss = '[%s]'%(','.join(str(v) for v in gaplist_imgs))
args = '%s, %s, %s, %s'%(G._gap_init_(), H._gap_init_(), genss, imgss)
self._gap_str = 'GroupHomomorphismByImages(%s)'%args
phi0 = gap(self)
if gap.eval("IsGroupHomomorphism(%s)"%phi0.name())!="true":
raise ValueError,"The map "+str(gensG)+"-->"+str(imgsH)+" isn't a homomorphism."
def __init__(self, homset, imgsH, check=True):
MatrixGroupMorphism.__init__(self, homset) # sets the parent
G = homset.domain()
H = homset.codomain()
gaplist_gens = [gap(x) for x in G.gens()]
gaplist_imgs = [gap(x) for x in imgsH]
genss = '[%s]' % (','.join(str(v) for v in gaplist_gens))
imgss = '[%s]' % (','.join(str(v) for v in gaplist_imgs))
args = '%s, %s, %s, %s' % (G._gap_init_(), H._gap_init_(), genss,
imgss)
self._gap_str = 'GroupHomomorphismByImages(%s)' % args
phi0 = gap(self)
if gap.eval("IsGroupHomomorphism(%s)" % phi0.name()) != "true":
raise ValueError, "The map " + str(gensG) + "-->" + str(
imgsH) + " isn't a homomorphism."
def pushforward(self, J, *args, **kwds):
"""
The image of an element or a subgroup.
INPUT:
``J`` -- a subgroup or an element of the domain of ``self``.
OUTPUT:
The image of ``J`` under ``self``
NOTE:
``pushforward`` is the method that is used when a map is called on
anything that is not an element of its domain. For historical reasons,
we keep the alias ``image()`` for this method.
EXAMPLES::
sage: F = GF(7); MS = MatrixSpace(F,2,2)
sage: F.multiplicative_generator()
3
sage: G = MatrixGroup([MS([3,0,0,1])])
sage: a = G.gens()[0]^2
sage: phi = G.hom([a])
sage: phi.image(G.gens()[0]) # indirect doctest
[2 0]
[0 1]
sage: H = MatrixGroup([MS(a.list())])
sage: H
Matrix group over Finite Field of size 7 with 1 generators:
[[[2, 0], [0, 1]]]
The following tests against trac ticket #10659::
sage: phi(H) # indirect doctestest
Matrix group over Finite Field of size 7 with 1 generators:
[[[4, 0], [0, 1]]]
"""
phi = gap(self)
F = self.codomain().base_ring()
from sage.all import MatrixGroup
gapJ = gap(J)
if gap.eval("IsGroup(%s)" % gapJ.name()) == "true":
return MatrixGroup(
[x._matrix_(F) for x in phi.Image(gapJ).GeneratorsOfGroup()])
return phi.Image(gapJ)._matrix_(F)
def invariant_quadratic_form(self):
r"""
Return the quadratic form `q(v) = v Q v^t` on the space on
which this group `G` that satisfies the equation
`q(v) = q(v M)` for all `v \in V` and
`M \in G`.
.. note::
Uses GAP's command InvariantQuadraticForm.
OUTPUT:
- ``Q`` - matrix that defines the invariant quadratic
form.
EXAMPLES::
sage: G = SO( 4, GF(7), 1)
sage: G.invariant_quadratic_form()
[0 1 0 0]
[0 0 0 0]
[0 0 3 0]
[0 0 0 1]
"""
F = self.base_ring()
I = gap(self).InvariantQuadraticForm()
Q = I.attribute("matrix")
return Q._matrix_(F)
def invariant_quadratic_form(self):
"""
This wraps GAP's command "InvariantQuadraticForm". From the GAP
documentation:
INPUT:
- ``self`` - a matrix group G
OUTPUT:
- ``Q`` - the matrix satisfying the property: The
quadratic form q on the natural vector space V on which G acts is
given by `q(v) = v Q v^t`, and the invariance under G is
given by the equation `q(v) = q(v M)` for all
`v \in V` and `M \in G`.
EXAMPLES::
sage: G = GO( 4, GF(7), 1)
sage: G.invariant_quadratic_form()
[0 1 0 0]
[0 0 0 0]
[0 0 3 0]
[0 0 0 1]
"""
F = self.base_ring()
G = self._gap_init_()
cmd = "r := InvariantQuadraticForm("+G+")"
gap.eval(cmd)
cmd = "r.matrix"
Q = gap(cmd)
return Q._matrix_(F)
def __call__(self, g):
"""
Evaluate the character on the group element `g`.
Return an error if `g` is not in `G`.
EXAMPLES::
sage: G = GL(2,7)
sage: values = G.gap().CharacterTable().Irr()[2].List().sage()
sage: chi = ClassFunction(G, values)
sage: z = G([[3,0],[0,3]]); z
[3 0]
[0 3]
sage: chi(z)
zeta3
sage: G = GL(2,3)
sage: chi = G.irreducible_characters()[3]
sage: g = G.conjugacy_classes_representatives()[6]
sage: chi(g)
zeta8^3 + zeta8
sage: G = SymmetricGroup(3)
sage: h = G((2,3))
sage: triv = G.trivial_character()
sage: triv(h)
1
"""
return self._base_ring(gap(g)._operation("^", self._gap_classfunction))
def character_table(self):
"""
Returns the character table as a matrix
Each row is an irreducible character. For larger tables you
may preface this with a command such as
gap.eval("SizeScreen([120,40])") in order to widen the screen.
EXAMPLES::
sage: WeylGroup(['A',3]).character_table()
CT1
<BLANKLINE>
2 3 2 2 . 3
3 1 . . 1 .
<BLANKLINE>
1a 4a 2a 3a 2b
<BLANKLINE>
X.1 1 -1 -1 1 1
X.2 3 1 -1 . -1
X.3 2 . . -1 2
X.4 3 -1 1 . -1
X.5 1 1 1 1 1
"""
gens_str = ', '.join(str(g.gap()) for g in self.gens())
ctbl = gap('CharacterTable(Group({0}))'.format(gens_str))
return ctbl.Display()
def cardinality(self):
"""
Implements :meth:`EnumeratedSets.ParentMethods.cardinality`.
EXAMPLES::
sage: G = Sp(4,GF(3))
sage: G.cardinality()
51840
sage: G = SL(4,GF(3))
sage: G.cardinality()
12130560
sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[1,2],[-1,1]]),MS([[1,1],[0,1]])]
sage: G = MatrixGroup(gens)
sage: G.cardinality()
480
sage: G = MatrixGroup([matrix(ZZ,2,[1,1,0,1])])
sage: G.cardinality()
+Infinity
"""
g = self._gap_()
if g.IsFinite().bool():
return integer.Integer(gap(self).Size())
return infinity
def random_element(self):
"""
Return a random element of this group.
EXAMPLES::
sage: G = Sp(4,GF(3))
sage: G.random_element() # random
[2 1 1 1]
[1 0 2 1]
[0 1 1 0]
[1 0 0 1]
sage: G.random_element() in G
True
::
sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[1,2],[-1,1]]),MS([[1,1],[0,1]])]
sage: G = MatrixGroup(gens)
sage: G.random_element() # random
[1 3]
[0 3]
sage: G.random_element() in G
True
"""
# Note: even with fixed random seed, the Random() element
# returned by gap does depend on execution order and
# architecture. Presumably due to different memory loctions.
current_randstate().set_seed_gap()
F = self.field_of_definition()
return self.element_class(gap(self).Random()._matrix_(F), self, check=False)
def gens(self):
"""
Return generators for this matrix group.
EXAMPLES::
sage: G = GO(3,GF(5))
sage: G.gens()
[
[2 0 0]
[0 3 0]
[0 0 1],
[0 1 0]
[1 4 4]
[0 2 1]
]
"""
try:
return self.__gens
except AttributeError:
pass
F = self.field_of_definition()
gap_gens = list(gap(self).GeneratorsOfGroup())
gens = Sequence([self.element_class(g._matrix_(F), self, check=False) for g in gap_gens],
cr=True, universe=self, check=False)
self.__gens = gens
return gens
def random_element(self):
"""
Return a random element of this group.
EXAMPLES::
sage: G = Sp(4,GF(3))
sage: G.random_element() # random
[2 1 1 1]
[1 0 2 1]
[0 1 1 0]
[1 0 0 1]
sage: G.random_element() in G
True
::
sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[1,2],[-1,1]]),MS([[1,1],[0,1]])]
sage: G = MatrixGroup(gens)
sage: G.random_element() # random
[1 3]
[0 3]
sage: G.random_element() in G
True
"""
# Note: even with fixed random seed, the Random() element
# returned by gap does depend on execution order and
# architecture. Presumably due to different memory loctions.
current_randstate().set_seed_gap()
F = self.field_of_definition()
return self.element_class(gap(self).Random()._matrix_(F),
self,
check=False)
def gens(self):
"""
Return generators for this matrix group.
EXAMPLES::
sage: G = GO(3,GF(5))
sage: G.gens()
[
[2 0 0]
[0 3 0]
[0 0 1],
[0 1 0]
[1 4 4]
[0 2 1]
]
"""
try:
return self.__gens
except AttributeError:
pass
F = self.field_of_definition()
gap_gens = list(gap(self).GeneratorsOfGroup())
gens = Sequence([
self.element_class(g._matrix_(F), self, check=False)
for g in gap_gens
],
cr=True,
universe=self,
check=False)
self.__gens = gens
return gens
def character_table(self):
"""
Returns the character table as a matrix
Each row is an irreducible character. For larger tables you
may preface this with a command such as
gap.eval("SizeScreen([120,40])") in order to widen the screen.
EXAMPLES::
sage: WeylGroup(['A',3]).character_table()
CT1
<BLANKLINE>
2 3 2 2 . 3
3 1 . . 1 .
<BLANKLINE>
1a 4a 2a 3a 2b
<BLANKLINE>
X.1 1 -1 -1 1 1
X.2 3 1 -1 . -1
X.3 2 . . -1 2
X.4 3 -1 1 . -1
X.5 1 1 1 1 1
"""
gens_str = ', '.join(str(g.gap()) for g in self.gens())
ctbl = gap('CharacterTable(Group({0}))'.format(gens_str))
return ctbl.Display()
def __call__(self, g):
"""
Evaluate the character on the group element `g`.
Return an error if `g` is not in `G`.
EXAMPLES::
sage: G = GL(2,7)
sage: values = G.gap().CharacterTable().Irr()[2].List().sage()
sage: chi = ClassFunction(G, values)
sage: z = G([[3,0],[0,3]]); z
[3 0]
[0 3]
sage: chi(z)
zeta3
sage: G = GL(2,3)
sage: chi = G.irreducible_characters()[3]
sage: g = G.conjugacy_classes_representatives()[6]
sage: chi(g)
zeta8^3 + zeta8
sage: G = SymmetricGroup(3)
sage: h = G((2,3))
sage: triv = G.trivial_character()
sage: triv(h)
1
"""
return self._base_ring(gap(g)._operation("^", self._gap_classfunction))
def invariant_quadratic_form(self):
r"""
Return the quadratic form `q(v) = v Q v^t` on the space on
which this group `G` that satisfies the equation
`q(v) = q(v M)` for all `v \in V` and
`M \in G`.
.. note::
Uses GAP's command InvariantQuadraticForm.
OUTPUT:
- ``Q`` - matrix that defines the invariant quadratic
form.
EXAMPLES::
sage: G = SO( 4, GF(7), 1)
sage: G.invariant_quadratic_form()
[0 1 0 0]
[0 0 0 0]
[0 0 3 0]
[0 0 0 1]
"""
F = self.base_ring()
I = gap(self).InvariantQuadraticForm()
Q = I.attribute('matrix')
return Q._matrix_(F)
def cardinality(self):
"""
Implements :meth:`EnumeratedSets.ParentMethods.cardinality`.
EXAMPLES::
sage: G = Sp(4,GF(3))
sage: G.cardinality()
51840
sage: G = SL(4,GF(3))
sage: G.cardinality()
12130560
sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[1,2],[-1,1]]),MS([[1,1],[0,1]])]
sage: G = MatrixGroup(gens)
sage: G.cardinality()
480
sage: G = MatrixGroup([matrix(ZZ,2,[1,1,0,1])])
sage: G.cardinality()
+Infinity
"""
g = self._gap_()
if g.IsFinite().bool():
return integer.Integer(gap(self).Size())
return infinity
def pushforward(self, J, *args,**kwds):
"""
The image of an element or a subgroup.
INPUT:
``J`` -- a subgroup or an element of the domain of ``self``.
OUTPUT:
The image of ``J`` under ``self``
NOTE:
``pushforward`` is the method that is used when a map is called on
anything that is not an element of its domain. For historical reasons,
we keep the alias ``image()`` for this method.
EXAMPLES::
sage: F = GF(7); MS = MatrixSpace(F,2,2)
sage: F.multiplicative_generator()
3
sage: G = MatrixGroup([MS([3,0,0,1])])
sage: a = G.gens()[0]^2
sage: phi = G.hom([a])
sage: phi.image(G.gens()[0]) # indirect doctest
[2 0]
[0 1]
sage: H = MatrixGroup([MS(a.list())])
sage: H
Matrix group over Finite Field of size 7 with 1 generators:
[[[2, 0], [0, 1]]]
The following tests against trac ticket #10659::
sage: phi(H) # indirect doctestest
Matrix group over Finite Field of size 7 with 1 generators:
[[[4, 0], [0, 1]]]
"""
phi = gap(self)
F = self.codomain().base_ring()
from sage.all import MatrixGroup
gapJ = gap(J)
if gap.eval("IsGroup(%s)"%gapJ.name()) == "true":
return MatrixGroup([x._matrix_(F) for x in phi.Image(gapJ).GeneratorsOfGroup()])
return phi.Image(gapJ)._matrix_(F)
def module_composition_factors(self, algorithm=None):
r"""
Returns a list of triples consisting of [base field, dimension,
irreducibility], for each of the Meataxe composition factors
modules. The algorithm="verbose" option returns more information, but
in Meataxe notation.
EXAMPLES::
sage: F=GF(3);MS=MatrixSpace(F,4,4)
sage: M=MS(0)
sage: M[0,1]=1;M[1,2]=1;M[2,3]=1;M[3,0]=1
sage: G = MatrixGroup([M])
sage: G.module_composition_factors()
[(Finite Field of size 3, 1, True),
(Finite Field of size 3, 1, True),
(Finite Field of size 3, 2, True)]
sage: F = GF(7); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[0,1],[-1,0]]),MS([[1,1],[2,3]])]
sage: G = MatrixGroup(gens)
sage: G.module_composition_factors()
[(Finite Field of size 7, 2, True)]
Type "G.module_composition_factors(algorithm='verbose')" to get a
more verbose version.
For more on MeatAxe notation, see
http://www.gap-system.org/Manuals/doc/htm/ref/CHAP067.htm
"""
from sage.misc.sage_eval import sage_eval
F = self.base_ring()
if not (F.is_finite()):
raise NotImplementedError, "Base ring must be finite."
q = F.cardinality()
gens = self.gens()
n = self.degree()
MS = MatrixSpace(F, n, n)
mats = [] # initializing list of mats by which the gens act on self
W = self.matrix_space().row_space()
for g in gens:
p = MS(g.matrix())
m = p.rows()
mats.append(m)
mats_str = str(gap([[list(r) for r in m] for m in mats]))
gap.eval("M:=GModuleByMats(" + mats_str + ", GF(" + str(q) + "))")
gap.eval("MCFs := MTX.CompositionFactors( M )")
N = eval(gap.eval("Length(MCFs)"))
if algorithm == "verbose":
print gap.eval('MCFs') + "\n"
L = []
for i in range(1, N + 1):
gap.eval("MCF := MCFs[%s]" % i)
L.append(
tuple([
sage_eval(gap.eval("MCF.field")),
eval(gap.eval("MCF.dimension")),
sage_eval(gap.eval("MCF.IsIrreducible"))
]))
return sorted(L)
def module_composition_factors(self, algorithm=None):
r"""
Return a list of triples consisting of [base field, dimension,
irreducibility], for each of the Meataxe composition factors
modules. The ``algorithm="verbose"`` option returns more information,
but in Meataxe notation.
EXAMPLES::
sage: F=GF(3);MS=MatrixSpace(F,4,4)
sage: M=MS(0)
sage: M[0,1]=1;M[1,2]=1;M[2,3]=1;M[3,0]=1
sage: G = MatrixGroup([M])
sage: G.module_composition_factors()
[(Finite Field of size 3, 1, True),
(Finite Field of size 3, 1, True),
(Finite Field of size 3, 2, True)]
sage: F = GF(7); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[0,1],[-1,0]]),MS([[1,1],[2,3]])]
sage: G = MatrixGroup(gens)
sage: G.module_composition_factors()
[(Finite Field of size 7, 2, True)]
Type ``G.module_composition_factors(algorithm='verbose')`` to get a
more verbose version.
For more on MeatAxe notation, see
http://www.gap-system.org/Manuals/doc/ref/chap69.html
"""
from sage.misc.sage_eval import sage_eval
F = self.base_ring()
if not(F.is_finite()):
raise NotImplementedError("Base ring must be finite.")
q = F.cardinality()
gens = self.gens()
n = self.degree()
MS = MatrixSpace(F,n,n)
mats = [] # initializing list of mats by which the gens act on self
W = self.matrix_space().row_space()
for g in gens:
p = MS(g.matrix())
m = p.rows()
mats.append(m)
mats_str = str(gap([[list(r) for r in m] for m in mats]))
gap.eval("M:=GModuleByMats("+mats_str+", GF("+str(q)+"))")
gap.eval("MCFs := MTX.CompositionFactors( M )")
N = eval(gap.eval("Length(MCFs)"))
if algorithm == "verbose":
print(gap.eval('MCFs') + "\n")
L = []
for i in range(1,N+1):
gap.eval("MCF := MCFs[%s]"%i)
L.append(tuple([sage_eval(gap.eval("MCF.field")),
eval(gap.eval("MCF.dimension")),
sage_eval(gap.eval("MCF.IsIrreducible")) ]))
return sorted(L)
def __call__(self, x):
"""
EXAMPLES::
sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: G = MatrixGroup([MS(1), MS([1,2,3,4])])
sage: G.matrix_space()
Full MatrixSpace of 2 by 2 dense matrices over Finite Field of size 5
sage: G(1)
[1 0]
[0 1]
"""
if isinstance(x, self.element_class) and x.parent() is self:
return x
M = self.matrix_space()(x)
g = self.element_class(M, self)
if not gap(g) in gap(self):
raise TypeError, "no way to coerce element to self."
return g
def __call__(self, x):
"""
EXAMPLES::
sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: G = MatrixGroup([MS(1), MS([1,2,3,4])])
sage: G.matrix_space()
Full MatrixSpace of 2 by 2 dense matrices over Finite Field of size 5
sage: G(1)
[1 0]
[0 1]
"""
if isinstance(x, self.element_class) and x.parent() is self:
return x
M = self.matrix_space()(x)
g = self.element_class(M, self)
if not gap(g) in gap(self):
raise TypeError, "no way to coerce element to self."
return g
def __contains__(self, x):
"""
Return True if `x` is an element of this abelian group.
EXAMPLES::
sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: G = MatrixGroup([MS(1), MS([1,2,3,4])])
sage: G
Matrix group over Finite Field of size 5 with 2 generators:
[[[1, 0], [0, 1]], [[1, 2], [3, 4]]]
sage: G.cardinality()
8
sage: G(1)
[1 0]
[0 1]
sage: G.1 in G
True
sage: 1 in G
True
sage: [1,2,3,4] in G
True
sage: matrix(GF(5),2,[1,2,3,5]) in G
False
sage: G(matrix(GF(5),2,[1,2,3,5]))
Traceback (most recent call last):
...
TypeError: no way to coerce element to self.
"""
if isinstance(x, self.element_class):
if x.parent() == self:
return True
return gap(x) in gap(self)
try:
self(x)
return True
except TypeError:
return False
def __contains__(self, x):
"""
Return True if `x` is an element of this abelian group.
EXAMPLES::
sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: G = MatrixGroup([MS(1), MS([1,2,3,4])])
sage: G
Matrix group over Finite Field of size 5 with 2 generators:
[[[1, 0], [0, 1]], [[1, 2], [3, 4]]]
sage: G.cardinality()
8
sage: G(1)
[1 0]
[0 1]
sage: G.1 in G
True
sage: 1 in G
True
sage: [1,2,3,4] in G
True
sage: matrix(GF(5),2,[1,2,3,5]) in G
False
sage: G(matrix(GF(5),2,[1,2,3,5]))
Traceback (most recent call last):
...
TypeError: no way to coerce element to self.
"""
if isinstance(x, self.element_class):
if x.parent() == self:
return True
return gap(x) in gap(self)
try:
self(x)
return True
except TypeError:
return False
def cardinality(self):
"""
EXAMPLES::
sage: G = Sp(4,GF(3))
sage: G.cardinality()
51840
sage: G = SL(4,GF(3))
sage: G.cardinality()
12130560
sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[1,2],[-1,1]]),MS([[1,1],[0,1]])]
sage: G = MatrixGroup(gens)
sage: G.cardinality()
480
sage: G = MatrixGroup([matrix(ZZ,2,[1,1,0,1])])
sage: G.cardinality()
+Infinity
"""
return integer.Integer(gap(self).Size())
def cardinality(self):
"""
EXAMPLES::
sage: G = Sp(4,GF(3))
sage: G.cardinality()
51840
sage: G = SL(4,GF(3))
sage: G.cardinality()
12130560
sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[1,2],[-1,1]]),MS([[1,1],[0,1]])]
sage: G = MatrixGroup(gens)
sage: G.cardinality()
480
sage: G = MatrixGroup([matrix(ZZ,2,[1,1,0,1])])
sage: G.cardinality()
+Infinity
"""
return integer.Integer(gap(self).Size())
def kernel(self):
"""
Return the kernel of ``self``, i.e., a matrix group.
EXAMPLES::
sage: F = GF(7); MS = MatrixSpace(F,2,2)
sage: F.multiplicative_generator()
3
sage: G = MatrixGroup([MS([3,0,0,1])])
sage: a = G.gens()[0]^2
sage: phi = G.hom([a])
sage: phi.kernel()
Matrix group over Finite Field of size 7 with 1 generators:
[[[6, 0], [0, 1]]]
"""
gap_ker = gap(self).Kernel()
from sage.all import MatrixGroup
F = self.domain().base_ring()
return MatrixGroup([x._matrix_(F) for x in gap_ker.GeneratorsOfGroup()])
def kernel(self):
"""
Return the kernel of ``self``, i.e., a matrix group.
EXAMPLES::
sage: F = GF(7); MS = MatrixSpace(F,2,2)
sage: F.multiplicative_generator()
3
sage: G = MatrixGroup([MS([3,0,0,1])])
sage: a = G.gens()[0]^2
sage: phi = G.hom([a])
sage: phi.kernel()
Matrix group over Finite Field of size 7 with 1 generators:
[[[6, 0], [0, 1]]]
"""
gap_ker = gap(self).Kernel()
from sage.all import MatrixGroup
F = self.domain().base_ring()
return MatrixGroup(
[x._matrix_(F) for x in gap_ker.GeneratorsOfGroup()])
def character_table(self):
"""
Returns the GAP character table as a string. For larger tables you
may preface this with a command such as
gap.eval("SizeScreen([120,40])") in order to widen the screen.
EXAMPLES::
sage: print WeylGroup(['A',3]).character_table()
CT1
<BLANKLINE>
2 3 2 2 . 3
3 1 . . 1 .
<BLANKLINE>
1a 4a 2a 3a 2b
<BLANKLINE>
X.1 1 -1 -1 1 1
X.2 3 1 -1 . -1
X.3 2 . . -1 2
X.4 3 -1 1 . -1
X.5 1 1 1 1 1
"""
return gap.eval("Display(CharacterTable(%s))" % gap(self).name())
def character_table(self):
"""
Returns the GAP character table as a string. For larger tables you
may preface this with a command such as
gap.eval("SizeScreen([120,40])") in order to widen the screen.
EXAMPLES::
sage: print WeylGroup(['A',3]).character_table()
CT1
<BLANKLINE>
2 3 2 2 . 3
3 1 . . 1 .
<BLANKLINE>
1a 4a 2a 3a 2b
<BLANKLINE>
X.1 1 -1 -1 1 1
X.2 3 1 -1 . -1
X.3 2 . . -1 2
X.4 3 -1 1 . -1
X.5 1 1 1 1 1
"""
return gap.eval("Display(CharacterTable(%s))" % gap(self).name())
def as_permutation_group(self, algorithm=None):
r"""
Return a permutation group representation for the group.
In most cases occurring in practice, this is a permutation
group of minimal degree (the degree begin determined from
orbits under the group action). When these orbits are hard to
compute, the procedure can be time-consuming and the degree
may not be minimal.
INPUT:
- ``algorithm`` -- ``None`` or ``'smaller'``. In the latter
case, try harder to find a permutation representation of
small degree.
OUTPUT:
A permutation group isomorphic to ``self``. The
``algorithm='smaller'`` option tries to return an isomorphic
group of low degree, but is not guaranteed to find the
smallest one.
EXAMPLES::
sage: MS = MatrixSpace(GF(2), 5, 5)
sage: A = MS([[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]])
sage: G = MatrixGroup([A])
sage: G.as_permutation_group()
Permutation Group with generators [(1,2)]
sage: MS = MatrixSpace( GF(7), 12, 12)
sage: GG = gap("ImfMatrixGroup( 12, 3 )")
sage: GG.GeneratorsOfGroup().Length()
3
sage: g1 = MS(eval(str(GG.GeneratorsOfGroup()[1]).replace("\n","")))
sage: g2 = MS(eval(str(GG.GeneratorsOfGroup()[2]).replace("\n","")))
sage: g3 = MS(eval(str(GG.GeneratorsOfGroup()[3]).replace("\n","")))
sage: G = MatrixGroup([g1, g2, g3])
sage: G.cardinality()
21499084800
sage: set_random_seed(0); current_randstate().set_seed_gap()
sage: P = G.as_permutation_group()
sage: P.cardinality()
21499084800
sage: P.degree() # random output
144
sage: set_random_seed(3); current_randstate().set_seed_gap()
sage: Psmaller = G.as_permutation_group(algorithm="smaller")
sage: Psmaller.cardinality()
21499084800
sage: Psmaller.degree() # random output
108
In this case, the "smaller" option returned an isomorphic group of
lower degree. The above example used GAP's library of irreducible
maximal finite ("imf") integer matrix groups to construct the
MatrixGroup G over GF(7). The section "Irreducible Maximal Finite
Integral Matrix Groups" in the GAP reference manual has more
details.
"""
# Note that the output of IsomorphismPermGroup() depends on
# memory locations and will change if you change the order of
# doctests and/or architecture
from sage.groups.perm_gps.permgroup import PermutationGroup
if not self.is_finite():
raise NotImplementedError, "Group must be finite."
n = self.degree()
MS = MatrixSpace(self.base_ring(), n, n)
mats = [] # initializing list of mats by which the gens act on self
for g in self.gens():
p = MS(g.matrix())
m = p.rows()
mats.append(m)
mats_str = str(gap([[list(r) for r in m] for m in mats]))
gap.eval("iso:=IsomorphismPermGroup(Group(" + mats_str + "))")
if algorithm == "smaller":
gap.eval(
"small:= SmallerDegreePermutationRepresentation( Image( iso ) );"
)
C = gap("Image( small )")
else:
C = gap("Image( iso )")
return PermutationGroup(gap_group=C)
def as_permutation_group(self, algorithm=None):
r"""
Return a permutation group representation for the group.
In most cases occurring in practice, this is a permutation
group of minimal degree (the degree begin determined from
orbits under the group action). When these orbits are hard to
compute, the procedure can be time-consuming and the degree
may not be minimal.
INPUT:
- ``algorithm`` -- ``None`` or ``'smaller'``. In the latter
case, try harder to find a permutation representation of
small degree.
OUTPUT:
A permutation group isomorphic to ``self``. The
``algorithm='smaller'`` option tries to return an isomorphic
group of low degree, but is not guaranteed to find the
smallest one.
EXAMPLES::
sage: MS = MatrixSpace(GF(2), 5, 5)
sage: A = MS([[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]])
sage: G = MatrixGroup([A])
sage: G.as_permutation_group()
Permutation Group with generators [(1,2)]
sage: MS = MatrixSpace( GF(7), 12, 12)
sage: GG = gap("ImfMatrixGroup( 12, 3 )")
sage: GG.GeneratorsOfGroup().Length()
3
sage: g1 = MS(eval(str(GG.GeneratorsOfGroup()[1]).replace("\n","")))
sage: g2 = MS(eval(str(GG.GeneratorsOfGroup()[2]).replace("\n","")))
sage: g3 = MS(eval(str(GG.GeneratorsOfGroup()[3]).replace("\n","")))
sage: G = MatrixGroup([g1, g2, g3])
sage: G.cardinality()
21499084800
sage: set_random_seed(0); current_randstate().set_seed_gap()
sage: P = G.as_permutation_group()
sage: P.cardinality()
21499084800
sage: P.degree() # random output
144
sage: set_random_seed(3); current_randstate().set_seed_gap()
sage: Psmaller = G.as_permutation_group(algorithm="smaller")
sage: Psmaller.cardinality()
21499084800
sage: Psmaller.degree() # random output
108
In this case, the "smaller" option returned an isomorphic group of
lower degree. The above example used GAP's library of irreducible
maximal finite ("imf") integer matrix groups to construct the
MatrixGroup G over GF(7). The section "Irreducible Maximal Finite
Integral Matrix Groups" in the GAP reference manual has more
details.
TESTS::
sage: A= matrix(QQ, 2, [0, 1, 1, 0])
sage: B= matrix(QQ, 2, [1, 0, 0, 1])
sage: a, b= MatrixGroup([A, B]).as_permutation_group().gens()
sage: a.order(), b.order()
(2, 1)
"""
# Note that the output of IsomorphismPermGroup() depends on
# memory locations and will change if you change the order of
# doctests and/or architecture
from sage.groups.perm_gps.permgroup import PermutationGroup
if not self.is_finite():
raise NotImplementedError("Group must be finite.")
n = self.degree()
MS = MatrixSpace(self.base_ring(), n, n)
mats = [] # initializing list of mats by which the gens act on self
for g in self.gens():
p = MS(g.matrix())
m = p.rows()
mats.append(m)
mats_str = str(gap([[list(r) for r in m] for m in mats]))
gap.eval("iso:=IsomorphismPermGroup(Group("+mats_str+"))")
if algorithm == "smaller":
gap.eval("small:= SmallerDegreePermutationRepresentation( Image( iso ) );")
C = gap("Image( small )")
else:
C = gap("Image( iso )")
return PermutationGroup(gap_group=C, canonicalize=False)
def word_problem(self, gens=None):
r"""
Write this group element in terms of the elements of the list
``gens``, or the standard generators of the parent of self if
``gens`` is None.
INPUT:
- ``gens`` - a list of elements that can be coerced into the parent of
self, or None.
ALGORITHM: Use GAP, which has optimized algorithms for solving the
word problem (the GAP functions EpimorphismFromFreeGroup and
PreImagesRepresentative).
EXAMPLE::
sage: G = GL(2,5); G
General Linear Group of degree 2 over Finite Field of size 5
sage: G.gens()
[
[2 0]
[0 1],
[4 1]
[4 0]
]
sage: G(1).word_problem([G.1, G.0])
1
Next we construct a more complicated element of the group from the
generators::
sage: s,t = G.0, G.1
sage: a = (s * t * s); b = a.word_problem(); b
([2 0]
[0 1]) *
([4 1]
[4 0]) *
([2 0]
[0 1])
sage: b.prod() == a
True
We solve the word problem using some different generators::
sage: s = G([2,0,0,1]); t = G([1,1,0,1]); u = G([0,-1,1,0])
sage: a.word_problem([s,t,u])
([2 0]
[0 1])^-1 *
([1 1]
[0 1])^-1 *
([0 4]
[1 0]) *
([2 0]
[0 1])^-1
We try some elements that don't actually generate the group::
sage: a.word_problem([t,u])
Traceback (most recent call last):
...
ValueError: Could not solve word problem
AUTHORS:
- David Joyner and William Stein
- David Loeffler (2010): fixed some bugs
"""
G = self.parent()
gg = gap(G)
if not gens:
gens = gg.GeneratorsOfGroup()
else:
gens = gap([G(x) for x in gens])
H = gens.Group()
hom = H.EpimorphismFromFreeGroup()
in_terms_of_gens = str(hom.PreImagesRepresentative(self))
if 'identity' in in_terms_of_gens:
return Factorization([])
elif in_terms_of_gens == 'fail':
raise ValueError, "Could not solve word problem"
v = list(H.GeneratorsOfGroup())
F = G.field_of_definition()
w = [G(x._matrix_(F)) for x in v]
factors = [Factorization([(x,1)]) for x in w]
d = dict([('x%s'%(i+1),factors[i]) for i in range(len(factors))])
from sage.misc.sage_eval import sage_eval
F = sage_eval(str(in_terms_of_gens), d)
F._set_cr(True)
return F
def _call_(self, g):
"""
Some python code for wrapping GAP's Images function for a matrix
group G. Returns an error if g is not in G.
EXAMPLES::
sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: g = MS([1,1,0,1])
sage: G = MatrixGroup([g])
sage: phi = G.hom(G.gens())
sage: phi(G.0)
[1 1]
[0 1]
sage: phi(G(g^2))
[1 2]
[0 1]
::
sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: gens = [MS([1,2, -1,1]),MS([1,1, 0,1])]
sage: G = MatrixGroup(gens)
sage: phi = G.hom(G.gens())
sage: phi(G.0)
[1 2]
[4 1]
sage: phi(G.1)
[1 1]
[0 1]
TEST:
The following tests that the call method was successfully
improved in trac ticket #10659::
sage: O = WeylGroup(['D',6])
sage: r = prod(O.gens())
sage: r_ = r^-1
sage: f = O.hom([r*x*r_ for x in O.gens()]) # long time (19s on sage.math, 2011)
sage: [f(x) for x in O.gens()] # long time
[
[1 0 0 0 0 0] [1 0 0 0 0 0] [1 0 0 0 0 0] [ 0 0 0 0 -1 0]
[0 0 1 0 0 0] [0 1 0 0 0 0] [0 1 0 0 0 0] [ 0 1 0 0 0 0]
[0 1 0 0 0 0] [0 0 0 1 0 0] [0 0 1 0 0 0] [ 0 0 1 0 0 0]
[0 0 0 1 0 0] [0 0 1 0 0 0] [0 0 0 0 1 0] [ 0 0 0 1 0 0]
[0 0 0 0 1 0] [0 0 0 0 1 0] [0 0 0 1 0 0] [-1 0 0 0 0 0]
[0 0 0 0 0 1], [0 0 0 0 0 1], [0 0 0 0 0 1], [ 0 0 0 0 0 1],
<BLANKLINE>
[0 0 0 0 0 1] [ 0 0 0 0 0 -1]
[0 1 0 0 0 0] [ 0 1 0 0 0 0]
[0 0 1 0 0 0] [ 0 0 1 0 0 0]
[0 0 0 1 0 0] [ 0 0 0 1 0 0]
[0 0 0 0 1 0] [ 0 0 0 0 1 0]
[1 0 0 0 0 0], [-1 0 0 0 0 0]
]
sage: f(O) # long time
Matrix group over Rational Field with 6 generators:
[[[1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]], [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]], [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1]], [[0, 0, 0, 0, -1, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [-1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1]], [[0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [1, 0, 0, 0, 0, 0]], [[0, 0, 0, 0, 0, -1], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [-1, 0, 0, 0, 0, 0]]]
"""
phi = gap(self)
G = self.domain()
F = G.base_ring()
h = gap(g)
return phi.Image(h)._matrix_(F)
def __cmp__(self, H):
if not isinstance(H, MatrixGroup_gap):
return cmp(type(self), type(H))
return cmp(gap(self), gap(H))
def invariant_generators(self):
"""
Wraps Singular's invariant_algebra_reynolds and invariant_ring
in finvar.lib, with help from Simon King and Martin Albrecht.
Computes generators for the polynomial ring
`F[x_1,\ldots,x_n]^G`, where G in GL(n,F) is a finite
matrix group.
In the "good characteristic" case the polynomials returned form a
minimal generating set for the algebra of G-invariant polynomials.
In the "bad" case, the polynomials returned are primary and
secondary invariants, forming a not necessarily minimal generating
set for the algebra of G-invariant polynomials.
EXAMPLES::
sage: F = GF(7); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[0,1],[-1,0]]),MS([[1,1],[2,3]])]
sage: G = MatrixGroup(gens)
sage: G.invariant_generators()
[x1^7*x2 - x1*x2^7, x1^12 - 2*x1^9*x2^3 - x1^6*x2^6 + 2*x1^3*x2^9 + x2^12, x1^18 + 2*x1^15*x2^3 + 3*x1^12*x2^6 + 3*x1^6*x2^12 - 2*x1^3*x2^15 + x2^18]
sage: q = 4; a = 2
sage: MS = MatrixSpace(QQ, 2, 2)
sage: gen1 = [[1/a,(q-1)/a],[1/a, -1/a]]; gen2 = [[1,0],[0,-1]]; gen3 = [[-1,0],[0,1]]
sage: G = MatrixGroup([MS(gen1),MS(gen2),MS(gen3)])
sage: G.cardinality()
12
sage: G.invariant_generators()
[x1^2 + 3*x2^2, x1^6 + 15*x1^4*x2^2 + 15*x1^2*x2^4 + 33*x2^6]
sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[1,2],[-1,1]]),MS([[1,1],[-1,1]])]
sage: G = MatrixGroup(gens)
sage: G.invariant_generators() # long time (67s on sage.math, 2012)
[x1^20 + x1^16*x2^4 + x1^12*x2^8 + x1^8*x2^12 + x1^4*x2^16 + x2^20, x1^20*x2^4 + x1^16*x2^8 + x1^12*x2^12 + x1^8*x2^16 + x1^4*x2^20]
sage: F=CyclotomicField(8)
sage: z=F.gen()
sage: a=z+1/z
sage: b=z^2
sage: MS=MatrixSpace(F,2,2)
sage: g1=MS([[1/a,1/a],[1/a,-1/a]])
sage: g2=MS([[1,0],[0,b]])
sage: g3=MS([[b,0],[0,1]])
sage: G=MatrixGroup([g1,g2,g3])
sage: G.invariant_generators() # long time (12s on sage.math, 2011)
[x1^8 + 14*x1^4*x2^4 + x2^8,
x1^24 + 10626/1025*x1^20*x2^4 + 735471/1025*x1^16*x2^8 + 2704156/1025*x1^12*x2^12 + 735471/1025*x1^8*x2^16 + 10626/1025*x1^4*x2^20 + x2^24]
AUTHORS:
- David Joyner, Simon King and Martin Albrecht.
REFERENCES:
- Singular reference manual
- B. Sturmfels, "Algorithms in invariant theory", Springer-Verlag, 1993.
- S. King, "Minimal Generating Sets of non-modular invariant rings of
finite groups", arXiv:math.AC/0703035
"""
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.interfaces.singular import singular
gens = self.gens()
singular.LIB("finvar.lib")
n = self.degree() #len((gens[0].matrix()).rows())
F = self.base_ring()
q = F.characteristic()
## test if the field is admissible
if F.gen() == 1: # we got the rationals or GF(prime)
FieldStr = str(F.characteristic())
elif hasattr(F, 'polynomial'): # we got an algebraic extension
if len(F.gens()) > 1:
raise NotImplementedError, "can only deal with finite fields and (simple algebraic extensions of) the rationals"
FieldStr = '(%d,%s)' % (F.characteristic(), str(F.gen()))
else: # we have a transcendental extension
FieldStr = '(%d,%s)' % (F.characteristic(), ','.join(
[str(p) for p in F.gens()]))
## Setting Singular's variable names
## We need to make sure that field generator and variables get different names.
if str(F.gen())[0] == 'x':
VarStr = 'y'
else:
VarStr = 'x'
VarNames = '(' + ','.join(
(VarStr + str(i + 1) for i in range(n))) + ')'
R = singular.ring(FieldStr, VarNames, 'dp')
if hasattr(F,
'polynomial') and F.gen() != 1: # we have to define minpoly
singular.eval('minpoly = ' +
str(F.polynomial()).replace('x', str(F.gen())))
A = [singular.matrix(n, n, str((x.matrix()).list())) for x in gens]
Lgens = ','.join((x.name() for x in A))
PR = PolynomialRing(F, n, [VarStr + str(i) for i in range(1, n + 1)])
if q == 0 or (q > 0 and self.cardinality() % q != 0):
from sage.all import Integer, Matrix
try:
gapself = gap(self)
# test whether the backwards transformation works as well:
for i in range(self.ngens()):
if Matrix(gapself.gen(i + 1), F) != self.gen(i).matrix():
raise ValueError
except (TypeError, ValueError):
gapself is None
if gapself is not None:
ReyName = 't' + singular._next_var_name()
singular.eval('matrix %s[%d][%d]' %
(ReyName, self.cardinality(), n))
En = gapself.Enumerator()
for i in range(1, self.cardinality() + 1):
M = Matrix(En[i], F)
D = [{} for foobar in range(self.degree())]
for x, y in M.dict().items():
D[x[0]][x[1]] = y
for row in range(self.degree()):
for t in D[row].items():
singular.eval('%s[%d,%d]=%s[%d,%d]+(%s)*var(%d)' %
(ReyName, i, row + 1, ReyName, i,
row + 1, repr(t[1]), t[0] + 1))
foobar = singular(ReyName)
IRName = 't' + singular._next_var_name()
singular.eval('matrix %s = invariant_algebra_reynolds(%s)' %
(IRName, ReyName))
else:
ReyName = 't' + singular._next_var_name()
singular.eval('list %s=group_reynolds((%s))' %
(ReyName, Lgens))
IRName = 't' + singular._next_var_name()
singular.eval('matrix %s = invariant_algebra_reynolds(%s[1])' %
(IRName, ReyName))
OUT = [
singular.eval(IRName + '[1,%d]' % (j))
for j in range(1, 1 + singular('ncols(' + IRName + ')'))
]
return [PR(gen) for gen in OUT]
if self.cardinality() % q == 0:
PName = 't' + singular._next_var_name()
SName = 't' + singular._next_var_name()
singular.eval('matrix %s,%s=invariant_ring(%s)' %
(PName, SName, Lgens))
OUT = [
singular.eval(PName + '[1,%d]' % (j))
for j in range(1, 1 + singular('ncols(' + PName + ')'))
] + [
singular.eval(SName + '[1,%d]' % (j))
for j in range(2, 1 + singular('ncols(' + SName + ')'))
]
return [PR(gen) for gen in OUT]
def word_problem(self, gens=None):
r"""
Write this group element in terms of the elements of the list
``gens``, or the standard generators of the parent of self if
``gens`` is None.
INPUT:
- ``gens`` - a list of elements that can be coerced into the parent of
self, or None.
ALGORITHM: Use GAP, which has optimized algorithms for solving the
word problem (the GAP functions EpimorphismFromFreeGroup and
PreImagesRepresentative).
EXAMPLE::
sage: G = GL(2,5); G
General Linear Group of degree 2 over Finite Field of size 5
sage: G.gens()
[
[2 0]
[0 1],
[4 1]
[4 0]
]
sage: G(1).word_problem([G.1, G.0])
1
Next we construct a more complicated element of the group from the
generators::
sage: s,t = G.0, G.1
sage: a = (s * t * s); b = a.word_problem(); b
([2 0]
[0 1]) *
([4 1]
[4 0]) *
([2 0]
[0 1])
sage: b.prod() == a
True
We solve the word problem using some different generators::
sage: s = G([2,0,0,1]); t = G([1,1,0,1]); u = G([0,-1,1,0])
sage: a.word_problem([s,t,u])
([2 0]
[0 1])^-1 *
([1 1]
[0 1])^-1 *
([0 4]
[1 0]) *
([2 0]
[0 1])^-1
We try some elements that don't actually generate the group::
sage: a.word_problem([t,u])
Traceback (most recent call last):
...
ValueError: Could not solve word problem
AUTHORS:
- David Joyner and William Stein
- David Loeffler (2010): fixed some bugs
"""
G = self.parent()
gg = gap(G)
if not gens:
gens = gg.GeneratorsOfGroup()
else:
gens = gap([G(x) for x in gens])
H = gens.Group()
hom = H.EpimorphismFromFreeGroup()
in_terms_of_gens = str(hom.PreImagesRepresentative(self))
if "identity" in in_terms_of_gens:
return Factorization([])
elif in_terms_of_gens == "fail":
raise ValueError, "Could not solve word problem"
v = list(H.GeneratorsOfGroup())
F = G.field_of_definition()
w = [G(x._matrix_(F)) for x in v]
factors = [Factorization([(x, 1)]) for x in w]
d = dict([("x%s" % (i + 1), factors[i]) for i in range(len(factors))])
from sage.misc.sage_eval import sage_eval
F = sage_eval(str(in_terms_of_gens), d)
F._set_cr(True)
return F
def cycle_index(self, parent = None):
r"""
INPUT:
- ``self`` - a permutation group `G`
- ``parent`` -- a free module with basis indexed by partitions,
or behave as such, with a ``term`` and ``sum`` method
(default: the symmetric functions over the rational field in the p basis)
Returns the *cycle index* of `G`, which is a gadget counting
the elements of `G` by cycle type, averaged over the group:
.. math::
P = \frac{1}{|G|} \sum_{g\in G} p_{ \operatorname{cycle\ type}(g) }
EXAMPLES:
Among the permutations of the symmetric group `S_4`, there is
the identity, 6 cycles of length 2, 3 products of two cycles
of length 2, 8 cycles of length 3, and 6 cycles of length 4::
sage: S4 = SymmetricGroup(4)
sage: P = S4.cycle_index()
sage: 24 * P
p[1, 1, 1, 1] + 6*p[2, 1, 1] + 3*p[2, 2] + 8*p[3, 1] + 6*p[4]
If `l = (l_1,\dots,l_k)` is a partition, ``|G| P[l]`` is the number
of elements of `G` with cycles of length `(p_1,\dots,p_k)`::
sage: 24 * P[ Partition([3,1]) ]
8
The cycle index plays an important role in the enumeration of
objects modulo the action of a group (Polya enumeration), via
the use of symmetric functions and plethysms. It is therefore
encoded as a symmetric function, expressed in the powersum
basis::
sage: P.parent()
Symmetric Functions over Rational Field in the powersum basis
This symmetric function can have some nice properties; for
example, for the symmetric group `S_n`, we get the complete
symmetric function `h_n`::
sage: S = SymmetricFunctions(QQ); h = S.h()
sage: h( P )
h[4]
TODO: add some simple examples of Polya enumeration, once it
will be easy to expand symmetric functions on any alphabet.
Here are the cycle indices of some permutation groups::
sage: 6 * CyclicPermutationGroup(6).cycle_index()
p[1, 1, 1, 1, 1, 1] + p[2, 2, 2] + 2*p[3, 3] + 2*p[6]
sage: 60 * AlternatingGroup(5).cycle_index()
p[1, 1, 1, 1, 1] + 15*p[2, 2, 1] + 20*p[3, 1, 1] + 24*p[5]
sage: for G in TransitiveGroups(5): # optional - database_gap # long time
... G.cardinality() * G.cycle_index()
p[1, 1, 1, 1, 1] + 4*p[5]
p[1, 1, 1, 1, 1] + 5*p[2, 2, 1] + 4*p[5]
p[1, 1, 1, 1, 1] + 5*p[2, 2, 1] + 10*p[4, 1] + 4*p[5]
p[1, 1, 1, 1, 1] + 15*p[2, 2, 1] + 20*p[3, 1, 1] + 24*p[5]
p[1, 1, 1, 1, 1] + 10*p[2, 1, 1, 1] + 15*p[2, 2, 1] + 20*p[3, 1, 1] + 20*p[3, 2] + 30*p[4, 1] + 24*p[5]
One may specify another parent for the result::
sage: F = CombinatorialFreeModule(QQ, Partitions())
sage: P = CyclicPermutationGroup(6).cycle_index(parent = F)
sage: 6 * P
B[[1, 1, 1, 1, 1, 1]] + B[[2, 2, 2]] + 2*B[[3, 3]] + 2*B[[6]]
sage: P.parent() is F
True
This parent should have a ``term`` and ``sum`` method::
sage: CyclicPermutationGroup(6).cycle_index(parent = QQ)
Traceback (most recent call last):
...
AssertionError: `parent` should be (or behave as) a free module with basis indexed by partitions
REFERENCES:
.. [Ker1991] A. Kerber. Algebraic combinatorics via finite group actions, 2.2 p. 70.
BI-Wissenschaftsverlag, Mannheim, 1991.
AUTHORS:
- Nicolas Borie and Nicolas M. Thiery
TESTS::
sage: P = PermutationGroup([]); P
Permutation Group with generators [()]
sage: P.cycle_index()
p[1]
sage: P = PermutationGroup([[(1)]]); P
Permutation Group with generators [()]
sage: P.cycle_index()
p[1]
"""
from sage.combinat.permutation import Permutation
if parent is None:
from sage.rings.rational_field import QQ
from sage.combinat.sf.sf import SymmetricFunctions
parent = SymmetricFunctions(QQ).powersum()
else:
assert hasattr(parent, "term") and hasattr(parent, "sum"), \
"`parent` should be (or behave as) a free module with basis indexed by partitions"
base_ring = parent.base_ring()
# TODO: use self.conjugacy_classes() once available
from sage.interfaces.gap import gap
CC = ([Permutation(self(C.Representative())).cycle_type(), base_ring(C.Size())] for C in gap(self).ConjugacyClasses())
return parent.sum( parent.term( partition, coeff ) for (partition, coeff) in CC)/self.cardinality()
def invariant_generators(self):
"""
Wraps Singular's invariant_algebra_reynolds and invariant_ring
in finvar.lib, with help from Simon King and Martin Albrecht.
Computes generators for the polynomial ring
`F[x_1,\ldots,x_n]^G`, where G in GL(n,F) is a finite
matrix group.
In the "good characteristic" case the polynomials returned form a
minimal generating set for the algebra of G-invariant polynomials.
In the "bad" case, the polynomials returned are primary and
secondary invariants, forming a not necessarily minimal generating
set for the algebra of G-invariant polynomials.
EXAMPLES::
sage: F = GF(7); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[0,1],[-1,0]]),MS([[1,1],[2,3]])]
sage: G = MatrixGroup(gens)
sage: G.invariant_generators()
[x1^7*x2 - x1*x2^7, x1^12 - 2*x1^9*x2^3 - x1^6*x2^6 + 2*x1^3*x2^9 + x2^12, x1^18 + 2*x1^15*x2^3 + 3*x1^12*x2^6 + 3*x1^6*x2^12 - 2*x1^3*x2^15 + x2^18]
sage: q = 4; a = 2
sage: MS = MatrixSpace(QQ, 2, 2)
sage: gen1 = [[1/a,(q-1)/a],[1/a, -1/a]]; gen2 = [[1,0],[0,-1]]; gen3 = [[-1,0],[0,1]]
sage: G = MatrixGroup([MS(gen1),MS(gen2),MS(gen3)])
sage: G.cardinality()
12
sage: G.invariant_generators()
[x1^2 + 3*x2^2, x1^6 + 15*x1^4*x2^2 + 15*x1^2*x2^4 + 33*x2^6]
sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[1,2],[-1,1]]),MS([[1,1],[-1,1]])]
sage: G = MatrixGroup(gens)
sage: G.invariant_generators() # long time (64s on sage.math, 2011)
[x1^20 + x1^16*x2^4 + x1^12*x2^8 + x1^8*x2^12 + x1^4*x2^16 + x2^20, x1^20*x2^4 + x1^16*x2^8 + x1^12*x2^12 + x1^8*x2^16 + x1^4*x2^20]
sage: F=CyclotomicField(8)
sage: z=F.gen()
sage: a=z+1/z
sage: b=z^2
sage: MS=MatrixSpace(F,2,2)
sage: g1=MS([[1/a,1/a],[1/a,-1/a]])
sage: g2=MS([[1,0],[0,b]])
sage: g3=MS([[b,0],[0,1]])
sage: G=MatrixGroup([g1,g2,g3])
sage: G.invariant_generators() # long time (12s on sage.math, 2011)
[x1^8 + 14*x1^4*x2^4 + x2^8,
x1^24 + 10626/1025*x1^20*x2^4 + 735471/1025*x1^16*x2^8 + 2704156/1025*x1^12*x2^12 + 735471/1025*x1^8*x2^16 + 10626/1025*x1^4*x2^20 + x2^24]
AUTHORS:
- David Joyner, Simon King and Martin Albrecht.
REFERENCES:
- Singular reference manual
- B. Sturmfels, "Algorithms in invariant theory", Springer-Verlag, 1993.
- S. King, "Minimal Generating Sets of non-modular invariant rings of
finite groups", arXiv:math.AC/0703035
"""
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.interfaces.singular import singular
gens = self.gens()
singular.LIB("finvar.lib")
n = self.degree() #len((gens[0].matrix()).rows())
F = self.base_ring()
q = F.characteristic()
## test if the field is admissible
if F.gen()==1: # we got the rationals or GF(prime)
FieldStr = str(F.characteristic())
elif hasattr(F,'polynomial'): # we got an algebraic extension
if len(F.gens())>1:
raise NotImplementedError, "can only deal with finite fields and (simple algebraic extensions of) the rationals"
FieldStr = '(%d,%s)'%(F.characteristic(),str(F.gen()))
else: # we have a transcendental extension
FieldStr = '(%d,%s)'%(F.characteristic(),','.join([str(p) for p in F.gens()]))
## Setting Singular's variable names
## We need to make sure that field generator and variables get different names.
if str(F.gen())[0]=='x':
VarStr = 'y'
else:
VarStr = 'x'
VarNames='('+','.join((VarStr+str(i+1) for i in range(n)))+')'
R=singular.ring(FieldStr,VarNames,'dp')
if hasattr(F,'polynomial') and F.gen()!=1: # we have to define minpoly
singular.eval('minpoly = '+str(F.polynomial()).replace('x',str(F.gen())))
A = [singular.matrix(n,n,str((x.matrix()).list())) for x in gens]
Lgens = ','.join((x.name() for x in A))
PR = PolynomialRing(F,n,[VarStr+str(i) for i in range(1,n+1)])
if q == 0 or (q > 0 and self.cardinality()%q != 0):
from sage.all import Integer, Matrix
try:
gapself = gap(self)
# test whether the backwards transformation works as well:
for i in range(self.ngens()):
if Matrix(gapself.gen(i+1),F) != self.gen(i).matrix():
raise ValueError
except (TypeError,ValueError):
gapself is None
if gapself is not None:
ReyName = 't'+singular._next_var_name()
singular.eval('matrix %s[%d][%d]'%(ReyName,self.cardinality(),n))
En = gapself.Enumerator()
for i in range(1,self.cardinality()+1):
M = Matrix(En[i],F)
D = [{} for foobar in range(self.degree())]
for x,y in M.dict().items():
D[x[0]][x[1]] = y
for row in range(self.degree()):
for t in D[row].items():
singular.eval('%s[%d,%d]=%s[%d,%d]+(%s)*var(%d)'%(ReyName,i,row+1,ReyName,i,row+1, repr(t[1]),t[0]+1))
foobar = singular(ReyName)
IRName = 't'+singular._next_var_name()
singular.eval('matrix %s = invariant_algebra_reynolds(%s)'%(IRName,ReyName))
else:
ReyName = 't'+singular._next_var_name()
singular.eval('list %s=group_reynolds((%s))'%(ReyName,Lgens))
IRName = 't'+singular._next_var_name()
singular.eval('matrix %s = invariant_algebra_reynolds(%s[1])'%(IRName,ReyName))
OUT = [singular.eval(IRName+'[1,%d]'%(j)) for j in range(1,1+singular('ncols('+IRName+')'))]
return [PR(gen) for gen in OUT]
if self.cardinality()%q == 0:
PName = 't'+singular._next_var_name()
SName = 't'+singular._next_var_name()
singular.eval('matrix %s,%s=invariant_ring(%s)'%(PName,SName,Lgens))
OUT = [singular.eval(PName+'[1,%d]'%(j)) for j in range(1,1+singular('ncols('+PName+')'))] + [singular.eval(SName+'[1,%d]'%(j)) for j in range(2,1+singular('ncols('+SName+')'))]
return [PR(gen) for gen in OUT]
def as_permutation_group(self, algorithm=None):
r"""
Return a permutation group representation for the group.
In most cases occurring in practice, this is a permutation
group of minimal degree (the degree being determined from
orbits under the group action). When these orbits are hard to
compute, the procedure can be time-consuming and the degree
may not be minimal.
INPUT:
- ``algorithm`` -- ``None`` or ``'smaller'``. In the latter
case, try harder to find a permutation representation of
small degree.
OUTPUT:
A permutation group isomorphic to ``self``. The
``algorithm='smaller'`` option tries to return an isomorphic
group of low degree, but is not guaranteed to find the
smallest one.
EXAMPLES::
sage: MS = MatrixSpace(GF(2), 5, 5)
sage: A = MS([[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]])
sage: G = MatrixGroup([A])
sage: G.as_permutation_group()
Permutation Group with generators [(1,2)]
sage: MS = MatrixSpace( GF(7), 12, 12)
sage: GG = gap("ImfMatrixGroup( 12, 3 )")
sage: GG.GeneratorsOfGroup().Length()
3
sage: g1 = MS(eval(str(GG.GeneratorsOfGroup()[1]).replace("\n","")))
sage: g2 = MS(eval(str(GG.GeneratorsOfGroup()[2]).replace("\n","")))
sage: g3 = MS(eval(str(GG.GeneratorsOfGroup()[3]).replace("\n","")))
sage: G = MatrixGroup([g1, g2, g3])
sage: G.cardinality()
21499084800
sage: set_random_seed(0); current_randstate().set_seed_gap()
sage: P = G.as_permutation_group()
sage: P.cardinality()
21499084800
sage: P.degree() # random output
144
sage: set_random_seed(3); current_randstate().set_seed_gap()
sage: Psmaller = G.as_permutation_group(algorithm="smaller")
sage: Psmaller.cardinality()
21499084800
sage: Psmaller.degree() # random output
108
In this case, the "smaller" option returned an isomorphic group of
lower degree. The above example used GAP's library of irreducible
maximal finite ("imf") integer matrix groups to construct the
MatrixGroup G over GF(7). The section "Irreducible Maximal Finite
Integral Matrix Groups" in the GAP reference manual has more
details.
TESTS::
sage: A= matrix(QQ, 2, [0, 1, 1, 0])
sage: B= matrix(QQ, 2, [1, 0, 0, 1])
sage: a, b= MatrixGroup([A, B]).as_permutation_group().gens()
sage: a.order(), b.order()
(2, 1)
Check that ``_permutation_group_morphism`` works (:trac:`25706`)::
sage: MG = GU(3,2).as_matrix_group()
sage: PG = MG.as_permutation_group() # this constructs the morphism
sage: mg = MG.an_element()
sage: MG._permutation_group_morphism(mg)
(1,2,6,19,35,33)(3,9,26,14,31,23)(4,13,5)(7,22,17)(8,24,12)(10,16,32,27,20,28)(11,30,18)(15,25,36,34,29,21)
"""
# Note that the output of IsomorphismPermGroup() depends on
# memory locations and will change if you change the order of
# doctests and/or architecture
from sage.groups.perm_gps.permgroup import PermutationGroup
if not self.is_finite():
raise NotImplementedError("Group must be finite.")
n = self.degree()
MS = MatrixSpace(self.base_ring(), n, n)
mats = [] # initializing list of mats by which the gens act on self
for g in self.gens():
p = MS(g.matrix())
m = p.rows()
mats.append(m)
mats_str = str(gap([[list(r) for r in m] for m in mats]))
gap.eval("iso:=IsomorphismPermGroup(Group("+mats_str+"))")
gap_permutation_map = gap("iso;")
if algorithm == "smaller":
gap.eval("small:= SmallerDegreePermutationRepresentation( Image( iso ) );")
C = gap("Image( small )")
else:
C = gap("Image( iso )")
PG = PermutationGroup(gap_group=C, canonicalize=False)
def permutation_group_map(element):
return PG(gap_permutation_map.ImageElm(element.gap()))
from sage.categories.homset import Hom
self._permutation_group_morphism = Hom(self, PG)(permutation_group_map)
return PG
def __cmp__(self, H):
if not isinstance(H, MatrixGroup_gap):
return cmp(type(self), type(H))
return cmp(gap(self), gap(H))
def cycle_index(self, parent = None):
r"""
INPUT:
- ``self`` - a permutation group `G`
- ``parent`` -- a free module with basis indexed by partitions,
or behave as such, with a ``term`` and ``sum`` method
(default: the symmetric functions over the rational field in the p basis)
Returns the *cycle index* of `G`, which is a gadget counting
the elements of `G` by cycle type, averaged over the group:
.. math::
P = \frac{1}{|G|} \sum_{g\in G} p_{ \operatorname{cycle\ type}(g) }
EXAMPLES:
Among the permutations of the symmetric group `S_4`, there is
the identity, 6 cycles of length 2, 3 products of two cycles
of length 2, 8 cycles of length 3, and 6 cycles of length 4::
sage: S4 = SymmetricGroup(4)
sage: P = S4.cycle_index()
sage: 24 * P
p[1, 1, 1, 1] + 6*p[2, 1, 1] + 3*p[2, 2] + 8*p[3, 1] + 6*p[4]
If `l = (l_1,\dots,l_k)` is a partition, ``|G| P[l]`` is the number
of elements of `G` with cycles of length `(p_1,\dots,p_k)`::
sage: 24 * P[ Partition([3,1]) ]
8
The cycle index plays an important role in the enumeration of
objects modulo the action of a group (Polya enumeration), via
the use of symmetric functions and plethysms. It is therefore
encoded as a symmetric function, expressed in the powersum
basis::
sage: P.parent()
Symmetric Functions over Rational Field in the powersum basis
This symmetric function can have some nice properties; for
example, for the symmetric group `S_n`, we get the complete
symmetric function `h_n`::
sage: S = SymmetricFunctions(QQ); h = S.h()
sage: h( P )
h[4]
TODO: add some simple examples of Polya enumeration, once it
will be easy to expand symmetric functions on any alphabet.
Here are the cycle indices of some permutation groups::
sage: 6 * CyclicPermutationGroup(6).cycle_index()
p[1, 1, 1, 1, 1, 1] + p[2, 2, 2] + 2*p[3, 3] + 2*p[6]
sage: 60 * AlternatingGroup(5).cycle_index()
p[1, 1, 1, 1, 1] + 15*p[2, 2, 1] + 20*p[3, 1, 1] + 24*p[5]
sage: for G in TransitiveGroups(5): # optional - database_gap # long time
... G.cardinality() * G.cycle_index()
p[1, 1, 1, 1, 1] + 4*p[5]
p[1, 1, 1, 1, 1] + 5*p[2, 2, 1] + 4*p[5]
p[1, 1, 1, 1, 1] + 5*p[2, 2, 1] + 10*p[4, 1] + 4*p[5]
p[1, 1, 1, 1, 1] + 15*p[2, 2, 1] + 20*p[3, 1, 1] + 24*p[5]
p[1, 1, 1, 1, 1] + 10*p[2, 1, 1, 1] + 15*p[2, 2, 1] + 20*p[3, 1, 1] + 20*p[3, 2] + 30*p[4, 1] + 24*p[5]
One may specify another parent for the result::
sage: F = CombinatorialFreeModule(QQ, Partitions())
sage: P = CyclicPermutationGroup(6).cycle_index(parent = F)
sage: 6 * P
B[[1, 1, 1, 1, 1, 1]] + B[[2, 2, 2]] + 2*B[[3, 3]] + 2*B[[6]]
sage: P.parent() is F
True
This parent should have a ``term`` and ``sum`` method::
sage: CyclicPermutationGroup(6).cycle_index(parent = QQ)
Traceback (most recent call last):
...
AssertionError: `parent` should be (or behave as) a free module with basis indexed by partitions
REFERENCES:
.. [Ker1991] A. Kerber. Algebraic combinatorics via finite group actions, 2.2 p. 70.
BI-Wissenschaftsverlag, Mannheim, 1991.
AUTHORS:
- Nicolas Borie and Nicolas M. Thiery
TESTS::
sage: P = PermutationGroup([]); P
Permutation Group with generators [()]
sage: P.cycle_index()
p[1]
sage: P = PermutationGroup([[(1)]]); P
Permutation Group with generators [()]
sage: P.cycle_index()
p[1]
"""
from sage.combinat.permutation import Permutation
if parent is None:
from sage.rings.rational_field import QQ
from sage.combinat.sf.sf import SymmetricFunctions
parent = SymmetricFunctions(QQ).powersum()
else:
assert hasattr(parent, "term") and hasattr(parent, "sum"), \
"`parent` should be (or behave as) a free module with basis indexed by partitions"
base_ring = parent.base_ring()
# TODO: use self.conjugacy_classes() once available
from sage.interfaces.gap import gap
CC = ([Permutation(self(C.Representative())).cycle_type(), base_ring(C.Size())] for C in gap(self).ConjugacyClasses())
return parent.sum( parent.term( partition, coeff ) for (partition, coeff) in CC)/self.cardinality()
def _call_( self, g ):
"""
Some python code for wrapping GAP's Images function for a matrix
group G. Returns an error if g is not in G.
EXAMPLES::
sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: g = MS([1,1,0,1])
sage: G = MatrixGroup([g])
sage: phi = G.hom(G.gens())
sage: phi(G.0)
[1 1]
[0 1]
sage: phi(G(g^2))
[1 2]
[0 1]
::
sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: gens = [MS([1,2, -1,1]),MS([1,1, 0,1])]
sage: G = MatrixGroup(gens)
sage: phi = G.hom(G.gens())
sage: phi(G.0)
[1 2]
[4 1]
sage: phi(G.1)
[1 1]
[0 1]
TEST:
The following tests that the call method was successfully
improved in trac ticket #10659::
sage: O = WeylGroup(['D',6])
sage: r = prod(O.gens())
sage: r_ = r^-1
sage: f = O.hom([r*x*r_ for x in O.gens()]) # long time (19s on sage.math, 2011)
sage: [f(x) for x in O.gens()] # long time
[
[1 0 0 0 0 0] [1 0 0 0 0 0] [1 0 0 0 0 0] [ 0 0 0 0 -1 0]
[0 0 1 0 0 0] [0 1 0 0 0 0] [0 1 0 0 0 0] [ 0 1 0 0 0 0]
[0 1 0 0 0 0] [0 0 0 1 0 0] [0 0 1 0 0 0] [ 0 0 1 0 0 0]
[0 0 0 1 0 0] [0 0 1 0 0 0] [0 0 0 0 1 0] [ 0 0 0 1 0 0]
[0 0 0 0 1 0] [0 0 0 0 1 0] [0 0 0 1 0 0] [-1 0 0 0 0 0]
[0 0 0 0 0 1], [0 0 0 0 0 1], [0 0 0 0 0 1], [ 0 0 0 0 0 1],
<BLANKLINE>
[0 0 0 0 0 1] [ 0 0 0 0 0 -1]
[0 1 0 0 0 0] [ 0 1 0 0 0 0]
[0 0 1 0 0 0] [ 0 0 1 0 0 0]
[0 0 0 1 0 0] [ 0 0 0 1 0 0]
[0 0 0 0 1 0] [ 0 0 0 0 1 0]
[1 0 0 0 0 0], [-1 0 0 0 0 0]
]
sage: f(O) # long time
Matrix group over Rational Field with 6 generators:
[[[1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]], [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]], [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1]], [[0, 0, 0, 0, -1, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [-1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1]], [[0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [1, 0, 0, 0, 0, 0]], [[0, 0, 0, 0, 0, -1], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [-1, 0, 0, 0, 0, 0]]]
"""
phi = gap(self)
G = self.domain()
F = G.base_ring()
h = gap(g)
return phi.Image(h)._matrix_(F)