Coverage for local/lib/python2.7/site-packages/sage/groups/matrix_gps/symplectic.py : 94%

""" 

Symplectic Linear Groups 

EXAMPLES:: 

sage: G = Sp(4,GF(7)); G 

Symplectic Group of degree 4 over Finite Field of size 7 

sage: g = prod(G.gens()); g 

[3 0 3 0] 

[1 0 0 0] 

[0 1 0 1] 

[0 2 0 0] 

sage: m = g.matrix() 

sage: m * G.invariant_form() * m.transpose() == G.invariant_form() 

True 

sage: G.order() 

276595200 

AUTHORS: 

- David Joyner (2006-03): initial version, modified from 

special_linear (by W. Stein) 

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

""" 

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

# Copyright (C) 2006 David Joyner and William Stein 

# Copyright (C) 2013 Volker Braun <vbraun.name@gmail.com> 

# 

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

# http://www.gnu.org/licenses/ 

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

from sage.misc.latex import latex 

from sage.misc.cachefunc import cached_method 

from sage.groups.matrix_gps.named_group import ( 

normalize_args_vectorspace, NamedMatrixGroup_generic, NamedMatrixGroup_gap ) 

############################################################################### 

# Symplectic Group 

############################################################################### 

def Sp(n, R, var='a'): 

r""" 

Return the symplectic group. 

The special linear group `GL( d, R )` consists of all `d \times d` 

matrices that are invertible over the ring `R` with determinant 

one. 

.. note:: 

This group is also available via ``groups.matrix.Sp()``. 

INPUT: 

- ``n`` -- a positive integer. 

- ``R`` -- ring or an integer. If an integer is specified, the 

corresponding finite field is used. 

- ``var`` -- variable used to represent generator of the finite 

field, if needed. 

EXAMPLES:: 

sage: Sp(4, 5) 

Symplectic Group of degree 4 over Finite Field of size 5 

sage: Sp(4, IntegerModRing(15)) 

Symplectic Group of degree 4 over Ring of integers modulo 15 

sage: Sp(3, GF(7)) 

Traceback (most recent call last): 

... 

ValueError: the degree must be even 

TESTS:: 

sage: groups.matrix.Sp(2, 3) 

Symplectic Group of degree 2 over Finite Field of size 3 

sage: G = Sp(4,5) 

sage: TestSuite(G).run() 

""" 

degree, ring = normalize_args_vectorspace(n, R, var=var) 

if degree % 2 != 0: 

raise ValueError('the degree must be even') 

name = 'Symplectic Group of degree {0} over {1}'.format(degree, ring) 

ltx = r'\text{{Sp}}_{{{0}}}({1})'.format(degree, latex(ring)) 

from sage.libs.gap.libgap import libgap 

try: 

cmd = 'Sp({0}, {1})'.format(degree, ring._gap_init_()) 

return SymplecticMatrixGroup_gap(degree, ring, True, name, ltx, cmd) 

except ValueError: 

return SymplecticMatrixGroup_generic(degree, ring, True, name, ltx) 

class SymplecticMatrixGroup_generic(NamedMatrixGroup_generic): 

@cached_method 

def invariant_form(self): 

""" 

Return the quadratic form preserved by the orthogonal group. 

OUTPUT: 

A matrix. 

EXAMPLES:: 

sage: Sp(4, QQ).invariant_form() 

[0 0 0 1] 

[0 0 1 0] 

[0 1 0 0] 

[1 0 0 0] 

""" 

from sage.matrix.constructor import zero_matrix 

m = zero_matrix(self.base_ring(), self.degree()) 

for i in range(self.degree()): 

m[i, self.degree()-i-1] = 1 

m.set_immutable() 

return m 

def _check_matrix(self, x, *args): 

""" 

Check whether the matrix ``x`` is symplectic. 

See :meth:`~sage.groups.matrix_gps.matrix_group._check_matrix` 

for details. 

EXAMPLES:: 

sage: G = Sp(4,GF(5)) 

sage: G._check_matrix(G.an_element().matrix()) 

""" 

F = self.invariant_form() 

if x * F * x.transpose() != F: 

raise TypeError('matrix must be symplectic') 

class SymplecticMatrixGroup_gap(SymplecticMatrixGroup_generic, NamedMatrixGroup_gap): 

r""" 

Symplectic group in GAP 

EXAMPLES:: 

sage: Sp(2,4) 

Symplectic Group of degree 2 over Finite Field in a of size 2^2 

sage: latex(Sp(4,5)) 

\text{Sp}_{4}(\Bold{F}_{5}) 

""" 

@cached_method 

def invariant_form(self): 

""" 

Return the quadratic form preserved by the orthogonal group. 

OUTPUT: 

A matrix. 

EXAMPLES:: 

sage: Sp(4, GF(3)).invariant_form() 

[0 0 0 1] 

[0 0 1 0] 

[0 2 0 0] 

[2 0 0 0] 

""" 

m = self.gap().InvariantBilinearForm()['matrix'].matrix() 

m.set_immutable() 

return m