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 muliple 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]

TESTS:

sage: G = GO(3, GF(5))
sage: latex(G)
\text{GO}_{3}(\Bold{F}_{5})
sage: G = SO(3, GF(5))
sage: latex(G)
\text{SO}_{3}(\Bold{F}_{5})
sage: G = SO(4, GF(5), 1)
sage: latex(G)
\text{SO}_{4}(\Bold{F}_{5}, +)

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.

sage.groups.matrix_gps.orthogonal.GO(n, R, e=0, var='a')

Return the general orthogonal group.

The general orthogonal group \(GO(n,R)\) consists of all \(n imes n\) matrices over the ring \(R\) preserving an \(n\)-ary positive definite quadratic form. In cases where there are muliple 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.

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]
)

TESTS:

sage: groups.matrix.GO(2, 3, e=-1)
General Orthogonal Group of degree 2 and form parameter -1 over Finite Field of size 3

class sage.groups.matrix_gps.orthogonal.OrthogonalMatrixGroup_gap(degree, base_ring, special, sage_name, latex_string, gap_command_string)

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

Base class for “named” matrix groups using LibGAP

INPUT:

• degree – integer. The degree (number of rows/columns of matrices).
• base_ring – rinrg. 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

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 = 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: 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)

Bases: sage.groups.matrix_gps.named_group.NamedMatrixGroup_generic

Base class for “named” matrix groups

INPUT:

• degree – integer. The degree (number of rows/columns of matrices).
• base_ring – rinrg. 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

invariant_bilinear_form()

Return the symmetric bilinear form preserved by the orthogonal group.

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]

invariant_quadratic_form()

Return the quadratic form preserved by the orthogonal group.

OUTPUT:

A matrix.

EXAMPLES:

sage: GO(2,3,+1).invariant_quadratic_form()
[0 1]
[0 0]
sage: GO(2,3,-1).invariant_quadratic_form()
[1 1]
[0 2]

sage.groups.matrix_gps.orthogonal.SO(n, R, e=None, var='a')

Return the special orthogonal group.

The special orthogonal group \(GO(n,R)\) consists of all \(n imes n\) matrices with determint one over the ring \(R\) preserving an \(n\)-ary positive definite quadratic form. In cases where there are muliple 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.

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]
)

TESTS:

sage: groups.matrix.SO(2, 3, e=1)
Special Orthogonal Group of degree 2 and form parameter 1 over Finite Field of size 3

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.

TESTS:

sage: from sage.groups.matrix_gps.orthogonal import normalize_args_e
sage: normalize_args_e(2, GF(3), +1)
1
sage: normalize_args_e(3, GF(3), 0)
0
sage: normalize_args_e(3, GF(3), +1)
0
sage: normalize_args_e(2, GF(3), 0)
Traceback (most recent call last):
...
ValueError: must have e=-1 or e=1 for even degree