Coverage for local/lib/python2.7/site-packages/sage/groups/matrix_gps/named_group.py : 76%
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""" Base for Classical Matrix Groups
This module implements the base class for matrix groups that have various famous names, like the general linear group.
EXAMPLES::
sage: SL(2, ZZ) Special Linear Group of degree 2 over Integer Ring sage: G = SL(2,GF(3)); G Special Linear Group of degree 2 over Finite Field of size 3 sage: G.is_finite() True sage: G.conjugacy_classes_representatives() ( [1 0] [0 2] [0 1] [2 0] [0 2] [0 1] [0 2] [0 1], [1 1], [2 1], [0 2], [1 2], [2 2], [1 0] ) sage: G = SL(6,GF(5)) sage: G.gens() ( [2 0 0 0 0 0] [4 0 0 0 0 1] [0 3 0 0 0 0] [4 0 0 0 0 0] [0 0 1 0 0 0] [0 4 0 0 0 0] [0 0 0 1 0 0] [0 0 4 0 0 0] [0 0 0 0 1 0] [0 0 0 4 0 0] [0 0 0 0 0 1], [0 0 0 0 4 0] ) """
############################################################################## # Copyright (C) 2006 David Joyner and William Stein <wstein@gmail.com> # Copyright (C) 2013 Volker Braun <vbraun.name@gmail.com> # # Distributed under the terms of the GNU General Public License (GPL) # # The full text of the GPL is available at: # # http://www.gnu.org/licenses/ ##############################################################################
from sage.structure.unique_representation import CachedRepresentation from sage.groups.matrix_gps.matrix_group import ( MatrixGroup_generic, MatrixGroup_gap )
def normalize_args_vectorspace(*args, **kwds): """ Normalize the arguments that relate to a vector space.
INPUT:
Something that defines an affine space. For example
* An affine space itself:
- ``A`` -- affine space
* A vector space:
- ``V`` -- a vector space
* Degree and base ring:
- ``degree`` -- integer. The degree of the affine group, that is, the dimension of the affine space the group is acting on.
- ``ring`` -- a ring or an integer. The base ring of the affine space. If an integer is given, it must be a prime power and the corresponding finite field is constructed.
- ``var='a'`` -- optional keyword argument to specify the finite field generator name in the case where ``ring`` is a prime power.
OUTPUT:
A pair ``(degree, ring)``.
TESTS::
sage: from sage.groups.matrix_gps.named_group import normalize_args_vectorspace sage: A = AffineSpace(2, GF(4,'a')); A Affine Space of dimension 2 over Finite Field in a of size 2^2 sage: normalize_args_vectorspace(A) (2, Finite Field in a of size 2^2)
sage: normalize_args_vectorspace(2,4) # shorthand (2, Finite Field in a of size 2^2)
sage: V = ZZ^3; V Ambient free module of rank 3 over the principal ideal domain Integer Ring sage: normalize_args_vectorspace(V) (3, Integer Ring)
sage: normalize_args_vectorspace(2, QQ) (2, Rational Field) """
class NamedMatrixGroup_generic(CachedRepresentation, MatrixGroup_generic):
def __init__(self, degree, base_ring, special, sage_name, latex_string, category=None): """ Base class for "named" matrix groups
INPUT:
- ``degree`` -- integer. The degree (number of rows/columns of matrices).
- ``base_ring`` -- ring. The base ring of the matrices.
- ``special`` -- boolean. Whether the matrix group is special, that is, elements have determinant one.
- ``latex_string`` -- string. The latex representation.
EXAMPLES::
sage: G = GL(2, QQ) sage: from sage.groups.matrix_gps.named_group import NamedMatrixGroup_generic sage: isinstance(G, NamedMatrixGroup_generic) True """
def _an_element_(self): """ Return an element.
OUTPUT:
A group element.
EXAMPLES::
sage: GL(2, QQ)._an_element_() [1 0] [0 1] """
def _repr_(self): """ Return a string representation.
OUTPUT:
String.
EXAMPLES::
sage: GL(2, QQ)._repr_() 'General Linear Group of degree 2 over Rational Field' """
def _latex_(self): """ Return a LaTeX representation
OUTPUT:
String.
EXAMPLES::
sage: GL(2, QQ)._latex_() 'GL(2, \\Bold{Q})' """
def __richcmp__(self, other, op): """ Override comparison.
We need to override the comparison since the named groups derive from :class:`~sage.structure.unique_representation.UniqueRepresentation`, which compare by identity.
EXAMPLES::
sage: G = GL(2,3) sage: G == MatrixGroup(G.gens()) True
sage: G = groups.matrix.GL(4,2) sage: H = MatrixGroup(G.gens()) sage: G == H True sage: G != H False """
class NamedMatrixGroup_gap(NamedMatrixGroup_generic, MatrixGroup_gap):
def __init__(self, degree, base_ring, special, sage_name, latex_string, gap_command_string, category=None): """ Base class for "named" matrix groups using LibGAP
INPUT:
- ``degree`` -- integer. The degree (number of rows/columns of matrices).
- ``base_ring`` -- ring. The base ring of the matrices.
- ``special`` -- boolean. Whether the matrix group is special, that is, elements have determinant one.
- ``latex_string`` -- string. The latex representation.
- ``gap_command_string`` -- string. The GAP command to construct the matrix group.
EXAMPLES::
sage: G = GL(2, GF(3)) sage: from sage.groups.matrix_gps.named_group import NamedMatrixGroup_gap sage: isinstance(G, NamedMatrixGroup_gap) True """ category=category) |