Orthogonal Linear Groups

The general orthogonal group \(GO(n,R)\) consists of all \(n \times n\) matrices over the ring \(R\) preserving an \(n\)-ary positive definite quadratic form. In cases where there are multiple non-isomorphic quadratic forms, additional data needs to be specified to disambiguate. The special orthogonal group is the normal subgroup of matrices of determinant one.

In characteristics different from 2, a quadratic form is equivalent to a bilinear symmetric form. Furthermore, over the real numbers a positive definite quadratic form is equivalent to the diagonal quadratic form, equivalent to the bilinear symmetric form defined by the identity matrix. Hence, the orthogonal group \(GO(n,\RR)\) is the group of orthogonal matrices in the usual sense.

In the case of a finite field and if the degree \(n\) is even, then there are two inequivalent quadratic forms and a third parameter e must be specified to disambiguate these two possibilities. The index of \(SO(e,d,q)\) in \(GO(e,d,q)\) is \(2\) if \(q\) is odd, but \(SO(e,d,q) = GO(e,d,q)\) if \(q\) is even.)

Warning

GAP and Sage use different notations:

  • GAP notation: The optional e comes first, that is, GO([e,] d, q), SO([e,] d, q).
  • Sage notation: The optional e comes last, the standard Python convention: GO(d, GF(q), e=0), SO(d, GF(q), e=0).

EXAMPLES:

sage: GO(3,7)
General Orthogonal Group of degree 3 over Finite Field of size 7

sage: G = SO( 4, GF(7), 1); G
Special Orthogonal Group of degree 4 and form parameter 1 over Finite Field of size 7
sage: G.random_element()   # random
[4 3 5 2]
[6 6 4 0]
[0 4 6 0]
[4 4 5 1]

AUTHORS:

  • David Joyner (2006-03): initial version
  • David Joyner (2006-05): added examples, _latex_, __str__, gens, as_matrix_group
  • William Stein (2006-12-09): rewrite
  • Volker Braun (2013-1) port to new Parent, libGAP, extreme refactoring.
  • Sebastian Oehms (2018-8) add invariant_form() (as alias), _OG, option for user defined invariant bilinear form, and bug-fix in cmd-string for calling GAP (see trac ticket #26028)
sage.groups.matrix_gps.orthogonal.GO(n, R, e=0, var='a', invariant_form=None)

Return the general orthogonal group.

The general orthogonal group \(GO(n,R)\) consists of all \(n \times n\) matrices over the ring \(R\) preserving an \(n\)-ary positive definite quadratic form. In cases where there are multiple non-isomorphic quadratic forms, additional data needs to be specified to disambiguate.

In the case of a finite field and if the degree \(n\) is even, then there are two inequivalent quadratic forms and a third parameter e must be specified to disambiguate these two possibilities.

Note

This group is also available via groups.matrix.GO().

INPUT:

  • n – integer; the degree
  • R – ring or an integer; if an integer is specified, the corresponding finite field is used
  • e+1 or -1, and ignored by default; only relevant for finite fields and if the degree is even: a parameter that distinguishes inequivalent invariant forms
  • var – (optional, default: 'a') variable used to represent generator of the finite field, if needed
  • invariant_form – (optional) instances being accepted by the matrix-constructor which define a \(n \times n\) square matrix over R describing the symmetric form to be kept invariant by the orthogonal group; the form is checked to be non-degenerate and symmetric but not to be positive definite

OUTPUT:

The general orthogonal group of given degree, base ring, and choice of invariant form.

EXAMPLES:

sage: GO( 3, GF(7))
General Orthogonal Group of degree 3 over Finite Field of size 7
sage: GO( 3, GF(7)).order()
672
sage: GO( 3, GF(7)).gens()
(
[3 0 0]  [0 1 0]
[0 5 0]  [1 6 6]
[0 0 1], [0 2 1]
)

Using the invariant_form option:

sage: m = matrix(QQ, 3,3, [[0, 1, 0], [1, 0, 0], [0, 0, 3]])
sage: GO3  = GO(3,QQ)
sage: GO3m = GO(3,QQ, invariant_form=m)
sage: GO3 == GO3m
False
sage: GO3.invariant_form()
[1 0 0]
[0 1 0]
[0 0 1]
sage: GO3m.invariant_form()
[0 1 0]
[1 0 0]
[0 0 3]
sage: pm = Permutation([2,3,1]).to_matrix()
sage: g = GO3(pm); g in GO3; g
True
[0 0 1]
[1 0 0]
[0 1 0]
sage: GO3m(pm)
Traceback (most recent call last):
...
TypeError: matrix must be orthogonal with respect to the symmetric form
[0 1 0]
[1 0 0]
[0 0 3]

sage: GO(3,3, invariant_form=[[1,0,0],[0,2,0],[0,0,1]])
Traceback (most recent call last):
...
NotImplementedError: invariant_form for finite groups is fixed by GAP
sage: 5+5
10
sage: R.<x> = ZZ[]
sage: GO(2, R, invariant_form=[[x,0],[0,1]])
General Orthogonal Group of degree 2 over Univariate Polynomial Ring in x over Integer Ring with respect to symmetric form
[x 0]
[0 1]
class sage.groups.matrix_gps.orthogonal.OrthogonalMatrixGroup_gap(degree, base_ring, special, sage_name, latex_string, gap_command_string, category=None)

Bases: sage.groups.matrix_gps.orthogonal.OrthogonalMatrixGroup_generic, sage.groups.matrix_gps.named_group.NamedMatrixGroup_gap, sage.groups.matrix_gps.finitely_generated.FinitelyGeneratedMatrixGroup_gap

The general or special orthogonal group in GAP.

invariant_bilinear_form()

Return the symmetric bilinear form preserved by the orthogonal group.

OUTPUT:

A matrix \(M\) such that, for every group element g, the identity \(g m g^T = m\) holds. In characteristic different from two, this uniquely determines the orthogonal group.

EXAMPLES:

sage: G = GO(4, GF(7), -1)
sage: G.invariant_bilinear_form()
[0 1 0 0]
[1 0 0 0]
[0 0 2 0]
[0 0 0 2]

sage: G = GO(4, GF(7), +1)
sage: G.invariant_bilinear_form()
[0 1 0 0]
[1 0 0 0]
[0 0 6 0]
[0 0 0 2]

sage: G = SO(4, GF(7), -1)
sage: G.invariant_bilinear_form()
[0 1 0 0]
[1 0 0 0]
[0 0 2 0]
[0 0 0 2]
invariant_form()

Return the symmetric bilinear form preserved by the orthogonal group.

OUTPUT:

A matrix \(M\) such that, for every group element g, the identity \(g m g^T = m\) holds. In characteristic different from two, this uniquely determines the orthogonal group.

EXAMPLES:

sage: G = GO(4, GF(7), -1)
sage: G.invariant_bilinear_form()
[0 1 0 0]
[1 0 0 0]
[0 0 2 0]
[0 0 0 2]

sage: G = GO(4, GF(7), +1)
sage: G.invariant_bilinear_form()
[0 1 0 0]
[1 0 0 0]
[0 0 6 0]
[0 0 0 2]

sage: G = SO(4, GF(7), -1)
sage: G.invariant_bilinear_form()
[0 1 0 0]
[1 0 0 0]
[0 0 2 0]
[0 0 0 2]
invariant_quadratic_form()

Return the quadratic form preserved by the orthogonal group.

OUTPUT:

The matrix \(Q\) defining “orthogonal” as follows. The matrix determines a quadratic form \(q\) on the natural vector space \(V\), on which \(G\) acts, by \(q(v) = v Q v^t\). A matrix \(M\) is an element of the orthogonal group if \(q(v) = q(v M)\) for all \(v \in V\).

EXAMPLES:

sage: G = GO(4, GF(7), -1)
sage: G.invariant_quadratic_form()
[0 1 0 0]
[0 0 0 0]
[0 0 1 0]
[0 0 0 1]

sage: G = GO(4, GF(7), +1)
sage: G.invariant_quadratic_form()
[0 1 0 0]
[0 0 0 0]
[0 0 3 0]
[0 0 0 1]

sage: G = GO(4, QQ)
sage: G.invariant_quadratic_form()
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]

sage: G = SO(4, GF(7), -1)
sage: G.invariant_quadratic_form()
[0 1 0 0]
[0 0 0 0]
[0 0 1 0]
[0 0 0 1]
class sage.groups.matrix_gps.orthogonal.OrthogonalMatrixGroup_generic(degree, base_ring, special, sage_name, latex_string, category=None, invariant_form=None)

Bases: sage.groups.matrix_gps.named_group.NamedMatrixGroup_generic

General Orthogonal Group over arbitrary rings.

EXAMPLES:

sage: G = GO(3, GF(7)); G
General Orthogonal Group of degree 3 over Finite Field of size 7
sage: latex(G)
\text{GO}_{3}(\Bold{F}_{7})

sage: G = SO(3, GF(5));  G
Special Orthogonal Group of degree 3 over Finite Field of size 5
sage: latex(G)
\text{SO}_{3}(\Bold{F}_{5})

sage: CF3 = CyclotomicField(3); e3 = CF3.gen()
sage: m=matrix(CF3, 3,3, [[1,e3,0],[e3,2,0],[0,0,1]])
sage: G = SO(3, CF3, invariant_form=m)
sage: latex(G)
\text{SO}_{3}(\Bold{Q}(\zeta_{3}))\text{ with respect to non positive definite symmetric form }\left(\begin{array}{rrr}
1 & \zeta_{3} & 0 \\
\zeta_{3} & 2 & 0 \\
0 & 0 & 1
\end{array}\right)
invariant_bilinear_form()

Return the symmetric bilinear form preserved by self.

OUTPUT:

A matrix.

EXAMPLES:

sage: GO(2,3,+1).invariant_bilinear_form()
[0 1]
[1 0]
sage: GO(2,3,-1).invariant_bilinear_form()
[2 1]
[1 1]
sage: G = GO(4, QQ)
sage: G.invariant_bilinear_form()
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
sage: GO3m = GO(3,QQ, invariant_form=(1,0,0,0,2,0,0,0,3))
sage: GO3m.invariant_bilinear_form()
[1 0 0]
[0 2 0]
[0 0 3]
invariant_form()

Return the symmetric bilinear form preserved by self.

OUTPUT:

A matrix.

EXAMPLES:

sage: GO(2,3,+1).invariant_bilinear_form()
[0 1]
[1 0]
sage: GO(2,3,-1).invariant_bilinear_form()
[2 1]
[1 1]
sage: G = GO(4, QQ)
sage: G.invariant_bilinear_form()
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
sage: GO3m = GO(3,QQ, invariant_form=(1,0,0,0,2,0,0,0,3))
sage: GO3m.invariant_bilinear_form()
[1 0 0]
[0 2 0]
[0 0 3]
invariant_quadratic_form()

Return the symmetric bilinear form preserved by self.

OUTPUT:

A matrix.

EXAMPLES:

sage: GO(2,3,+1).invariant_bilinear_form()
[0 1]
[1 0]
sage: GO(2,3,-1).invariant_bilinear_form()
[2 1]
[1 1]
sage: G = GO(4, QQ)
sage: G.invariant_bilinear_form()
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
sage: GO3m = GO(3,QQ, invariant_form=(1,0,0,0,2,0,0,0,3))
sage: GO3m.invariant_bilinear_form()
[1 0 0]
[0 2 0]
[0 0 3]
sage.groups.matrix_gps.orthogonal.SO(n, R, e=None, var='a', invariant_form=None)

Return the special orthogonal group.

The special orthogonal group \(GO(n,R)\) consists of all \(n \times n\) matrices with determinant one over the ring \(R\) preserving an \(n\)-ary positive definite quadratic form. In cases where there are multiple non-isomorphic quadratic forms, additional data needs to be specified to disambiguate.

Note

This group is also available via groups.matrix.SO().

INPUT:

  • n – integer; the degree
  • R – ring or an integer; if an integer is specified, the corresponding finite field is used
  • e+1 or -1, and ignored by default; only relevant for finite fields and if the degree is even: a parameter that distinguishes inequivalent invariant forms
  • var – (optional, default: 'a') variable used to represent generator of the finite field, if needed
  • invariant_form – (optional) instances being accepted by the matrix-constructor which define a \(n \times n\) square matrix over R describing the symmetric form to be kept invariant by the orthogonal group; the form is checked to be non-degenerate and symmetric but not to be positive definite

OUTPUT:

The special orthogonal group of given degree, base ring, and choice of invariant form.

EXAMPLES:

sage: G = SO(3,GF(5))
sage: G
Special Orthogonal Group of degree 3 over Finite Field of size 5

sage: G = SO(3,GF(5))
sage: G.gens()
(
[2 0 0]  [3 2 3]  [1 4 4]
[0 3 0]  [0 2 0]  [4 0 0]
[0 0 1], [0 3 1], [2 0 4]
)
sage: G = SO(3,GF(5))
sage: G.as_matrix_group()
Matrix group over Finite Field of size 5 with 3 generators (
[2 0 0]  [3 2 3]  [1 4 4]
[0 3 0]  [0 2 0]  [4 0 0]
[0 0 1], [0 3 1], [2 0 4]
)

Using the invariant_form option:

sage: CF3 = CyclotomicField(3); e3 = CF3.gen()
sage: m=matrix(CF3, 3,3, [[1,e3,0],[e3,2,0],[0,0,1]])
sage: SO3  = SO(3, CF3)
sage: SO3m = SO(3, CF3, invariant_form=m)
sage: SO3 == SO3m
False
sage: SO3.invariant_form()
[1 0 0]
[0 1 0]
[0 0 1]
sage: SO3m.invariant_form()
[    1 zeta3     0]
[zeta3     2     0]
[    0     0     1]
sage: pm = Permutation([2,3,1]).to_matrix()
sage: g = SO3(pm); g in SO3; g
True
[0 0 1]
[1 0 0]
[0 1 0]
sage: SO3m(pm)
Traceback (most recent call last):
...
TypeError: matrix must be orthogonal with respect to the symmetric form
[    1 zeta3     0]
[zeta3     2     0]
[    0     0     1]

sage: SO(3,5, invariant_form=[[1,0,0],[0,2,0],[0,0,3]])
Traceback (most recent call last):
...
NotImplementedError: invariant_form for finite groups is fixed by GAP
sage: 5+5
10
sage.groups.matrix_gps.orthogonal.normalize_args_e(degree, ring, e)

Normalize the arguments that relate the choice of quadratic form for special orthogonal groups over finite fields.

INPUT:

  • degree – integer. The degree of the affine group, that is, the dimension of the affine space the group is acting on.
  • ring – a ring. The base ring of the affine space.
  • e – integer, one of \(+1\), \(0\), \(-1\). Only relevant for finite fields and if the degree is even. A parameter that distinguishes inequivalent invariant forms.

OUTPUT:

The integer e with values required by GAP.