Symplectic Linear Groups#
EXAMPLES:
sage: G = Sp(4, GF(7)); G
Symplectic Group of degree 4 over Finite Field of size 7
sage: # needs sage.libs.gap
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.
Sebastian Oehms (2018-8) add option for user defined invariant bilinear form and bug-fix in
invariant_form()
(see github issue #26028)
- sage.groups.matrix_gps.symplectic.Sp(n, R, var='a', invariant_form=None)#
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 integerR
– ring or an integer; if an integer is specified, the corresponding finite field is usedvar
– (optional, default:'a'
) variable used to represent generator of the finite field, if neededinvariant_form
– (optional) instances being accepted by the matrix-constructor which define a \(n \times n\) square matrix overR
describing the alternating form to be kept invariant by the symplectic group
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
Using the
invariant_form
option:sage: m = matrix(QQ, 4,4, [[0, 0, 1, 0], [0, 0, 0, 2], [-1, 0, 0, 0], [0, -2, 0, 0]]) sage: Sp4m = Sp(4, QQ, invariant_form=m) sage: Sp4 = Sp(4, QQ) sage: Sp4 == Sp4m False sage: Sp4.invariant_form() [ 0 0 0 1] [ 0 0 1 0] [ 0 -1 0 0] [-1 0 0 0] sage: Sp4m.invariant_form() [ 0 0 1 0] [ 0 0 0 2] [-1 0 0 0] [ 0 -2 0 0] sage: pm = Permutation([2,1,4,3]).to_matrix() sage: g = Sp4(pm); g in Sp4; g True [0 1 0 0] [1 0 0 0] [0 0 0 1] [0 0 1 0] sage: Sp4m(pm) Traceback (most recent call last): ... TypeError: matrix must be symplectic with respect to the alternating form [ 0 0 1 0] [ 0 0 0 2] [-1 0 0 0] [ 0 -2 0 0] sage: Sp(4,3, invariant_form=[[0,0,0,1],[0,0,1,0],[0,2,0,0], [2,0,0,0]]) Traceback (most recent call last): ... NotImplementedError: invariant_form for finite groups is fixed by GAP
- class sage.groups.matrix_gps.symplectic.SymplecticMatrixGroup_generic(degree, base_ring, special, sage_name, latex_string, category=None, invariant_form=None)#
Bases:
NamedMatrixGroup_generic
Symplectic Group over arbitrary rings.
EXAMPLES:
sage: Sp43 = Sp(4,3); Sp43 Symplectic Group of degree 4 over Finite Field of size 3 sage: latex(Sp43) \text{Sp}_{4}(\Bold{F}_{3}) sage: Sp4m = Sp(4, QQ, invariant_form=(0, 0, 1, 0, 0, 0, 0, 2, ....: -1, 0, 0, 0, 0, -2, 0, 0)); Sp4m Symplectic Group of degree 4 over Rational Field with respect to alternating bilinear form [ 0 0 1 0] [ 0 0 0 2] [-1 0 0 0] [ 0 -2 0 0] sage: latex(Sp4m) \text{Sp}_{4}(\Bold{Q})\text{ with respect to alternating bilinear form}\left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 2 \\ -1 & 0 & 0 & 0 \\ 0 & -2 & 0 & 0 \end{array}\right)
- invariant_form()#
Return the quadratic form preserved by the symplectic 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]