CoCalc -- matrix_group

GitHub Repository: sagemath/sagelib
Path: blob/master/sage/groups/matrix_gps/matrix_group_morphism.py
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"""
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Homomorphisms Between Matrix Groups
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AUTHORS:
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- David Joyner and William Stein (2006-03): initial version
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- David Joyner (2006-05): examples
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- Simon King (2011-01): cleaning and improving code
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"""
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#*****************************************************************************
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#       Copyright (C) 2006 David Joyner and William Stein <[email protected]>
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#
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#  Distributed under the terms of the GNU General Public License (GPL)
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#  as published by the Free Software Foundation; either version 2 of
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#  the License, or (at your option) any later version.
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#                  http://www.gnu.org/licenses/
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#*****************************************************************************
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from sage.interfaces.gap import gap
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from sage.categories.morphism import *
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from sage.misc.latex import latex
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class MatrixGroupMap(Morphism):
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    """
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    A set-theoretic map between matrix groups.
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    """
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    def __init__(self, parent):
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        Morphism.__init__(self, parent)
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    def _repr_type(self):
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        return "MatrixGroup"
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class MatrixGroupMorphism(MatrixGroupMap):
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    pass
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class MatrixGroupMorphism_im_gens(MatrixGroupMorphism):
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    """
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    Some python code for wrapping GAP's GroupHomomorphismByImages
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    function but only for matrix groups. Can be expensive if G is
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    large.
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    EXAMPLES::
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        sage: F = GF(5); MS = MatrixSpace(F,2,2)
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        sage: G = MatrixGroup([MS([1,1,0,1])])
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        sage: H = MatrixGroup([MS([1,0,1,1])])
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        sage: phi = G.hom(H.gens())
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        sage: phi
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        Homomorphism : Matrix group over Finite Field of size 5 with 1 generators: 
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         [[[1, 1], [0, 1]]] --> Matrix group over Finite Field of size 5 with 1 generators: 
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         [[[1, 0], [1, 1]]]
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        sage: phi(MS([1,1,0,1]))
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        [1 0]
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        [1 1]
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        sage: F = GF(7); MS = MatrixSpace(F,2,2)
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        sage: F.multiplicative_generator()
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        3
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        sage: G = MatrixGroup([MS([3,0,0,1])])
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        sage: a = G.gens()[0]^2
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        sage: phi = G.hom([a])
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    """
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    def __init__(self, homset, imgsH, check=True):
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        MatrixGroupMorphism.__init__(self, homset)   # sets the parent
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        G = homset.domain()
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        H = homset.codomain()
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        gaplist_gens = [gap(x) for x in G.gens()]
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        gaplist_imgs = [gap(x) for x in imgsH]
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        genss = '[%s]'%(','.join(str(v) for v in gaplist_gens))
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        imgss = '[%s]'%(','.join(str(v) for v in gaplist_imgs))
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        args = '%s, %s, %s, %s'%(G._gap_init_(), H._gap_init_(), genss, imgss)
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        self._gap_str = 'GroupHomomorphismByImages(%s)'%args
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        phi0 = gap(self)
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        if gap.eval("IsGroupHomomorphism(%s)"%phi0.name())!="true":
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            raise ValueError,"The map "+str(gensG)+"-->"+str(imgsH)+" isn't a homomorphism."
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    def _repr_(self):
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        """
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        EXAMPLES::
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            sage: F = GF(5); MS = MatrixSpace(F,2,2)
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            sage: G = MatrixGroup([MS([1,1,0,1])])
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            sage: H = MatrixGroup([MS([1,0,1,1])])
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            sage: phi = G.hom(H.gens())
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            sage: phi
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            Homomorphism : Matrix group over Finite Field of size 5 with 1 generators: 
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             [[[1, 1], [0, 1]]] --> Matrix group over Finite Field of size 5 with 1 generators: 
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             [[[1, 0], [1, 1]]]            
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            sage: phi(MS([1,1,0,1]))
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            [1 0]
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            [1 1]
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        """
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        return "Homomorphism : %s --> %s"%(self.domain(),self.codomain())
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    def _latex_(self):
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        r"""
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        EXAMPLES::
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            sage: F = GF(5); MS = MatrixSpace(F,2,2)
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            sage: G = MatrixGroup([MS([1,1,0,1])])
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            sage: phi = G.hom(G.gens())
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            sage: print latex(phi)
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            \left\langle \left(\begin{array}{rr}
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            1 & 1 \\
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            0 & 1
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            \end{array}\right) \right\rangle \rightarrow{} \left\langle \left(\begin{array}{rr}
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            1 & 1 \\
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            0 & 1
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            \end{array}\right) \right\rangle
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        """
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        return "%s \\rightarrow{} %s"%(latex(self.domain()), latex(self.codomain()))
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    def _gap_init_(self):
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        return self._gap_str
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    def kernel(self):
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        """
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        Return the kernel of ``self``, i.e., a matrix group.
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        EXAMPLES::
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            sage: F = GF(7); MS = MatrixSpace(F,2,2)
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            sage: F.multiplicative_generator()
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            3
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            sage: G = MatrixGroup([MS([3,0,0,1])])
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            sage: a = G.gens()[0]^2
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            sage: phi = G.hom([a])
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            sage: phi.kernel()
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            Matrix group over Finite Field of size 7 with 1 generators: 
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             [[[6, 0], [0, 1]]]
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        """
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        gap_ker = gap(self).Kernel()
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        from sage.all import MatrixGroup
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        F = self.domain().base_ring()
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        return MatrixGroup([x._matrix_(F) for x in gap_ker.GeneratorsOfGroup()])
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    def pushforward(self, J, *args,**kwds):
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        """
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        The image of an element or a subgroup.
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        INPUT:
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        ``J`` -- a subgroup or an element of the domain of ``self``.
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        OUTPUT:
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        The image of ``J`` under ``self``
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        NOTE:
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        ``pushforward`` is the method that is used when a map is called on
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        anything that is not an element of its domain. For historical reasons,
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        we keep the alias ``image()`` for this method.
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        EXAMPLES::
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            sage: F = GF(7); MS = MatrixSpace(F,2,2)
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            sage: F.multiplicative_generator()
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            3
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            sage: G = MatrixGroup([MS([3,0,0,1])])
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            sage: a = G.gens()[0]^2
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            sage: phi = G.hom([a])
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            sage: phi.image(G.gens()[0]) # indirect doctest
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            [2 0]
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            [0 1]
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            sage: H = MatrixGroup([MS(a.list())])
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            sage: H
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            Matrix group over Finite Field of size 7 with 1 generators:
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             [[[2, 0], [0, 1]]]
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        The following tests against trac ticket #10659::
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            sage: phi(H)   # indirect doctestest
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            Matrix group over Finite Field of size 7 with 1 generators: 
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             [[[4, 0], [0, 1]]]
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        """
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        phi = gap(self)
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        F = self.codomain().base_ring()
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        from sage.all import MatrixGroup
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        gapJ = gap(J)
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        if gap.eval("IsGroup(%s)"%gapJ.name()) == "true":
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            return MatrixGroup([x._matrix_(F) for x in phi.Image(gapJ).GeneratorsOfGroup()])
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        return phi.Image(gapJ)._matrix_(F)
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    image = pushforward
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    def _call_( self, g ):
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        """
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        Some python code for wrapping GAP's Images function for a matrix
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        group G. Returns an error if g is not in G.
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        EXAMPLES::
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            sage: F = GF(5); MS = MatrixSpace(F,2,2)
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            sage: g = MS([1,1,0,1])
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            sage: G = MatrixGroup([g])
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            sage: phi = G.hom(G.gens())
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            sage: phi(G.0)
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            [1 1]
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            [0 1]
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            sage: phi(G(g^2))
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            [1 2]
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            [0 1]
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        ::
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            sage: F = GF(5); MS = MatrixSpace(F,2,2)
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            sage: gens = [MS([1,2,  -1,1]),MS([1,1,  0,1])]
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            sage: G = MatrixGroup(gens)
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            sage: phi = G.hom(G.gens())
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            sage: phi(G.0)
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            [1 2]
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            [4 1]
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            sage: phi(G.1)
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            [1 1]
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            [0 1]
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        TEST:
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        The following tests that the call method was successfully
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        improved in trac ticket #10659::
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            sage: O = WeylGroup(['D',6])
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            sage: r = prod(O.gens())
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            sage: r_ = r^-1
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            sage: f = O.hom([r*x*r_ for x in O.gens()])  # long time (19s on sage.math, 2011)
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            sage: [f(x) for x in O.gens()]  # long time
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            [
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            [1 0 0 0 0 0]  [1 0 0 0 0 0]  [1 0 0 0 0 0]  [ 0  0  0  0 -1  0]
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            [0 0 1 0 0 0]  [0 1 0 0 0 0]  [0 1 0 0 0 0]  [ 0  1  0  0  0  0]
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            [0 1 0 0 0 0]  [0 0 0 1 0 0]  [0 0 1 0 0 0]  [ 0  0  1  0  0  0]
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            [0 0 0 1 0 0]  [0 0 1 0 0 0]  [0 0 0 0 1 0]  [ 0  0  0  1  0  0]
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            [0 0 0 0 1 0]  [0 0 0 0 1 0]  [0 0 0 1 0 0]  [-1  0  0  0  0  0]
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            [0 0 0 0 0 1], [0 0 0 0 0 1], [0 0 0 0 0 1], [ 0  0  0  0  0  1],
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            <BLANKLINE>
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            [0 0 0 0 0 1]  [ 0  0  0  0  0 -1]
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            [0 1 0 0 0 0]  [ 0  1  0  0  0  0]
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            [0 0 1 0 0 0]  [ 0  0  1  0  0  0]
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            [0 0 0 1 0 0]  [ 0  0  0  1  0  0]
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            [0 0 0 0 1 0]  [ 0  0  0  0  1  0]
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            [1 0 0 0 0 0], [-1  0  0  0  0  0]
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            ]
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            sage: f(O)  # long time
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            Matrix group over Rational Field with 6 generators: 
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             [[[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]]]
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        """
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        phi = gap(self)
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        G = self.domain()
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        F = G.base_ring()
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        h = gap(g)
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        return phi.Image(h)._matrix_(F)
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