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"""

Homomorphisms Between Matrix Groups

AUTHORS:

- David Joyner and William Stein (2006-03): initial version

- David Joyner (2006-05): examples

- Simon King (2011-01): cleaning and improving code

- Volker Braun (2013-1) port to new Parent, libGAP.

"""

#*****************************************************************************

# Distributed under the terms of the GNU General Public License (GPL)

# as published by the Free Software Foundation; either version 2 of

# the License, or (at your option) any later version.

# http://www.gnu.org/licenses/

#*****************************************************************************

from __future__ import print_function

from sage.categories.morphism import Morphism

from sage.misc.latex import latex

def to_libgap(x):

"""

Helper to convert ``x`` to a LibGAP matrix or matrix group

element.

EXAMPLES::

sage: from sage.groups.matrix_gps.morphism import to_libgap

sage: to_libgap(GL(2,3).gen(0))

[ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0 ] ]

sage: to_libgap(matrix(QQ, [[1,2],[3,4]]))

[ [ 1, 2 ], [ 3, 4 ] ]

"""

try:

return x.gap()

except AttributeError:

from sage.libs.gap.libgap import libgap

return libgap(x)

class MatrixGroupMap(Morphism):

def __init__(self, parent):

"""

Set-theoretic map between matrix groups.

EXAMPLES::

sage: from sage.groups.matrix_gps.morphism import MatrixGroupMap

sage: MatrixGroupMap(ZZ.Hom(ZZ)) # mathematical nonsense

MatrixGroup endomorphism of Integer Ring

"""

Morphism.__init__(self, parent)

def _repr_type(self):

"""

Part of the implementation of :meth:`_repr_`

EXAMPLES::

sage: from sage.groups.matrix_gps.morphism import MatrixGroupMap

sage: MatrixGroupMap(ZZ.Hom(ZZ))._repr_type()

'MatrixGroup'

"""

return "MatrixGroup"

class MatrixGroupMorphism(MatrixGroupMap):

pass

class MatrixGroupMorphism_im_gens(MatrixGroupMorphism):

def __init__(self, homset, imgsH, check=True):

"""

Group morphism specified by images of generators.

Some Python code for wrapping GAP's GroupHomomorphismByImages

function but only for matrix groups. Can be expensive if G is

large.

EXAMPLES::

sage: F = GF(5); MS = MatrixSpace(F,2,2)

sage: G = MatrixGroup([MS([1,1,0,1])])

sage: H = MatrixGroup([MS([1,0,1,1])])

sage: phi = G.hom(H.gens())

sage: phi

Homomorphism : Matrix group over Finite Field of size 5 with 1 generators (

[1 1]

[0 1]

) --> Matrix group over Finite Field of size 5 with 1 generators (

[1 0]

[1 1]

)

sage: phi(MS([1,1,0,1]))

[1 0]

[1 1]

sage: F = GF(7); MS = MatrixSpace(F,2,2)

sage: F.multiplicative_generator()

sage: G = MatrixGroup([MS([3,0,0,1])])

sage: a = G.gens()[0]^2

sage: phi = G.hom([a])

TESTS:

Check that :trac:`19406` is fixed::

sage: G = GL(2, GF(3))

sage: H = GL(3, GF(2))

sage: mat1 = H([[-1,0,0],[0,0,-1],[0,-1,0]])

sage: mat2 = H([[1,1,1],[0,0,-1],[-1,0,0]])

sage: phi = G.hom([mat1, mat2])

Traceback (most recent call last):

...

TypeError: images do not define a group homomorphism

"""

MatrixGroupMorphism.__init__(self, homset) # sets the parent

from sage.libs.gap.libgap import libgap

G = homset.domain()

H = homset.codomain()

gens = [x.gap() for x in G.gens()]

imgs = [to_libgap(x) for x in imgsH]

self._phi = phi = libgap.GroupHomomorphismByImages(G.gap(), H.gap(), gens, imgs)

if not phi.IsGroupHomomorphism():

raise ValueError('the map {}-->{} is not a homomorphism'.format(G.gens(), imgsH))

def gap(self):

"""

Return the underlying LibGAP group homomorphism

OUTPUT:

A LibGAP element.

EXAMPLES::

sage: F = GF(5); MS = MatrixSpace(F,2,2)

sage: G = MatrixGroup([MS([1,1,0,1])])

sage: H = MatrixGroup([MS([1,0,1,1])])

sage: phi = G.hom(H.gens())

sage: phi.gap()

CompositionMapping( [ (6,7,8,10,9)(11,13,14,12,15)(16,19,20,18,17)(21,25,22,24,23) ]

-> [ [ [ Z(5)^0, 0*Z(5) ], [ Z(5)^0, Z(5)^0 ] ] ], <action isomorphism> )

sage: type(_)

"""

return self._phi

def _repr_(self):

"""

EXAMPLES::

sage: F = GF(5); MS = MatrixSpace(F,2,2)

sage: G = MatrixGroup([MS([1,1,0,1])])

sage: H = MatrixGroup([MS([1,0,1,1])])

sage: phi = G.hom(H.gens())

sage: phi

Homomorphism : Matrix group over Finite Field of size 5 with 1 generators (

[1 1]

[0 1]

) --> Matrix group over Finite Field of size 5 with 1 generators (

[1 0]

[1 1]

)

sage: phi(MS([1,1,0,1]))

[1 0]

[1 1]

"""

return "Homomorphism : %s --> %s"%(self.domain(),self.codomain())

def _latex_(self):

r"""

EXAMPLES::

sage: F = GF(5); MS = MatrixSpace(F,2,2)

sage: G = MatrixGroup([MS([1,1,0,1])])

sage: phi = G.hom(G.gens())

sage: print(latex(phi))

\left\langle \left(\begin{array}{rr}

1 & 1 \\

0 & 1

\end{array}\right) \right\rangle \rightarrow{} \left\langle \left(\begin{array}{rr}

1 & 1 \\

0 & 1

\end{array}\right) \right\rangle

"""

return "%s \\rightarrow{} %s"%(latex(self.domain()), latex(self.codomain()))

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()

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 = self.gap().Kernel()

F = self.domain().base_ring()

from sage.groups.matrix_gps.all import MatrixGroup

return MatrixGroup([x.matrix(F) for x in gap_ker.GeneratorsOfGroup()])

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()

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:`10659`::

sage: phi(H) # indirect doctest

Matrix group over Finite Field of size 7 with 1 generators (

[4 0]

[0 1]

)

"""

phi = self.gap()

F = self.codomain().base_ring()

gapJ = to_libgap(J)

if gapJ.IsGroup():

from sage.groups.matrix_gps.all import MatrixGroup

img_gens = [x.matrix(F) for x in phi.Image(gapJ).GeneratorsOfGroup()]

return MatrixGroup(img_gens)

C = self.codomain()

return C(phi.Image(gapJ).matrix(F))

image = pushforward

def _call_(self, g):

"""

Call syntax for morphisms.

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]

TESTS:

The following tests that the call method was successfully

improved in :trac:`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],

[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

sage: f(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],

[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]

)

We check that :trac:`19780` is fixed::

sage: G = groups.matrix.SO(3, 3)

sage: H = groups.matrix.GL(3, 3)

sage: phi = G.hom([H(x) for x in G.gens()])

sage: phi(G.one()).parent()

General Linear Group of degree 3 over Finite Field of size 3

"""

phi = self.gap()

G = self.domain()

C = self.codomain()

F = C.base_ring()

h = g.gap()

return C(phi.Image(h).matrix(F))

Coverage for local/lib/python2.7/site-packages/sage/groups/matrix_gps/morphism.py : 100%

56 statements 56 run 0 missing 0 excluded