Orthogonal Linear Groups#

The general orthogonal group $G O (n, R)$ 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 $G O (n, R)$ 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 $S O (e, d, q)$ in $G O (e, d, q)$ is $2$ if $q$ is odd, but $S O (e, d, q) = G O (e, d, q)$ 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

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: # needs sage.libs.gap
sage: G.random_element()   # random
[4 3 5 2]
[6 6 4 0]
[0 4 6 0]
[4 4 5 1]

Sage Live

GO(3,7)
G = SO(4, GF(7), 1); G
# needs sage.libs.gap
G.random_element()   # random

Python

>>> from sage.all import *
>>> GO(Integer(3),Integer(7))
General Orthogonal Group of degree 3 over Finite Field of size 7

>>> G = SO(Integer(4), GF(Integer(7)), Integer(1)); G
Special Orthogonal Group of degree 4 and form parameter 1
 over Finite Field of size 7

>>> # needs sage.libs.gap
>>> 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 Issue #26028)

sage.groups.matrix_gps.orthogonal.GO(n, R, e=0, var='a', invariant_form=None)[source]#

Return the general orthogonal group.

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 – (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

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

sage: # needs sage.libs.gap
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]
)

Sage Live

GO(3, GF(7))
# needs sage.libs.gap
GO(3, GF(7)).order()
GO(3, GF(7)).gens()

Python

>>> from sage.all import *
>>> GO(Integer(3), GF(Integer(7)))
General Orthogonal Group of degree 3 over Finite Field of size 7

>>> # needs sage.libs.gap
>>> GO(Integer(3), GF(Integer(7))).order()
672
>>> GO(Integer(3), GF(Integer(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

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]

Sage Live

m = matrix(QQ, 3, 3, [[0, 1, 0], [1, 0, 0], [0, 0, 3]])
GO3  = GO(3, QQ)
GO3m = GO(3, QQ, invariant_form=m)
GO3 == GO3m
GO3.invariant_form()
GO3m.invariant_form()
pm = Permutation([2,3,1]).to_matrix()
g = GO3(pm); g in GO3; g
GO3m(pm)
GO(3,3, invariant_form=[[1,0,0], [0,2,0], [0,0,1]])
5 + 5
R.<x> = ZZ[]
GO(2, R, invariant_form=[[x,0], [0,1]])

Python

>>> from sage.all import *
>>> m = matrix(QQ, Integer(3), Integer(3), [[Integer(0), Integer(1), Integer(0)], [Integer(1), Integer(0), Integer(0)], [Integer(0), Integer(0), Integer(3)]])
>>> GO3  = GO(Integer(3), QQ)
>>> GO3m = GO(Integer(3), QQ, invariant_form=m)
>>> GO3 == GO3m
False
>>> GO3.invariant_form()
[1 0 0]
[0 1 0]
[0 0 1]
>>> GO3m.invariant_form()
[0 1 0]
[1 0 0]
[0 0 3]
>>> pm = Permutation([Integer(2),Integer(3),Integer(1)]).to_matrix()
>>> g = GO3(pm); g in GO3; g
True
[0 0 1]
[1 0 0]
[0 1 0]
>>> 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]

>>> GO(Integer(3),Integer(3), invariant_form=[[Integer(1),Integer(0),Integer(0)], [Integer(0),Integer(2),Integer(0)], [Integer(0),Integer(0),Integer(1)]])
Traceback (most recent call last):
...
NotImplementedError: invariant_form for finite groups is fixed by GAP
>>> Integer(5) + Integer(5)
10
>>> R = ZZ['x']; (x,) = R._first_ngens(1)
>>> GO(Integer(2), R, invariant_form=[[x,Integer(0)], [Integer(0),Integer(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_generic(degree, base_ring, special, sage_name, latex_string, category=None, invariant_form=None)[source]#

Bases: NamedMatrixGroup_generic

General Orthogonal Group over arbitrary rings.

EXAMPLES:

Sage

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: # needs sage.rings.number_field
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)

Sage Live

G = GO(3, GF(7)); G
latex(G)
G = SO(3, GF(5));  G
latex(G)
# needs sage.rings.number_field
CF3 = CyclotomicField(3); e3 = CF3.gen()
m = matrix(CF3, 3,3, [[1,e3,0],[e3,2,0],[0,0,1]])
G = SO(3, CF3, invariant_form=m)
latex(G)

Python

>>> from sage.all import *
>>> G = GO(Integer(3), GF(Integer(7))); G
General Orthogonal Group of degree 3 over Finite Field of size 7
>>> latex(G)
\text{GO}_{3}(\Bold{F}_{7})

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

>>> # needs sage.rings.number_field
>>> CF3 = CyclotomicField(Integer(3)); e3 = CF3.gen()
>>> m = matrix(CF3, Integer(3),Integer(3), [[Integer(1),e3,Integer(0)],[e3,Integer(2),Integer(0)],[Integer(0),Integer(0),Integer(1)]])
>>> G = SO(Integer(3), CF3, invariant_form=m)
>>> 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()[source]#

Return the symmetric bilinear form preserved by self.

OUTPUT: a matrix

EXAMPLES:

Sage

sage: # needs sage.libs.gap
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 Live

# needs sage.libs.gap
GO(2,3,+1).invariant_bilinear_form()
GO(2,3,-1).invariant_bilinear_form()
G = GO(4, QQ)
G.invariant_bilinear_form()
GO3m = GO(3, QQ, invariant_form=(1,0,0, 0,2,0, 0,0,3))
GO3m.invariant_bilinear_form()

Python

>>> from sage.all import *
>>> # needs sage.libs.gap
>>> GO(Integer(2),Integer(3),+Integer(1)).invariant_bilinear_form()
[0 1]
[1 0]
>>> GO(Integer(2),Integer(3),-Integer(1)).invariant_bilinear_form()
[2 1]
[1 1]
>>> G = GO(Integer(4), QQ)
>>> G.invariant_bilinear_form()
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
>>> GO3m = GO(Integer(3), QQ, invariant_form=(Integer(1),Integer(0),Integer(0), Integer(0),Integer(2),Integer(0), Integer(0),Integer(0),Integer(3)))
>>> GO3m.invariant_bilinear_form()
[1 0 0]
[0 2 0]
[0 0 3]

invariant_form()[source]#

Return the symmetric bilinear form preserved by self.

OUTPUT: a matrix

EXAMPLES:

Sage

sage: # needs sage.libs.gap
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 Live

# needs sage.libs.gap
GO(2,3,+1).invariant_bilinear_form()
GO(2,3,-1).invariant_bilinear_form()
G = GO(4, QQ)
G.invariant_bilinear_form()
GO3m = GO(3, QQ, invariant_form=(1,0,0, 0,2,0, 0,0,3))
GO3m.invariant_bilinear_form()

Python

>>> from sage.all import *
>>> # needs sage.libs.gap
>>> GO(Integer(2),Integer(3),+Integer(1)).invariant_bilinear_form()
[0 1]
[1 0]
>>> GO(Integer(2),Integer(3),-Integer(1)).invariant_bilinear_form()
[2 1]
[1 1]
>>> G = GO(Integer(4), QQ)
>>> G.invariant_bilinear_form()
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
>>> GO3m = GO(Integer(3), QQ, invariant_form=(Integer(1),Integer(0),Integer(0), Integer(0),Integer(2),Integer(0), Integer(0),Integer(0),Integer(3)))
>>> GO3m.invariant_bilinear_form()
[1 0 0]
[0 2 0]
[0 0 3]

invariant_quadratic_form()[source]#

Return the symmetric bilinear form preserved by self.

OUTPUT: a matrix

EXAMPLES:

Sage

sage: # needs sage.libs.gap
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 Live

# needs sage.libs.gap
GO(2,3,+1).invariant_bilinear_form()
GO(2,3,-1).invariant_bilinear_form()
G = GO(4, QQ)
G.invariant_bilinear_form()
GO3m = GO(3, QQ, invariant_form=(1,0,0, 0,2,0, 0,0,3))
GO3m.invariant_bilinear_form()

Python

>>> from sage.all import *
>>> # needs sage.libs.gap
>>> GO(Integer(2),Integer(3),+Integer(1)).invariant_bilinear_form()
[0 1]
[1 0]
>>> GO(Integer(2),Integer(3),-Integer(1)).invariant_bilinear_form()
[2 1]
[1 1]
>>> G = GO(Integer(4), QQ)
>>> G.invariant_bilinear_form()
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
>>> GO3m = GO(Integer(3), QQ, invariant_form=(Integer(1),Integer(0),Integer(0), Integer(0),Integer(2),Integer(0), Integer(0),Integer(0),Integer(3)))
>>> 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)[source]#

Return the special orthogonal group.

The special orthogonal group $G O (n, R)$ 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 – (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

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

sage: # needs sage.libs.gap
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.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]
)

Sage Live

G = SO(3,GF(5)); G
# needs sage.libs.gap
G = SO(3,GF(5))
G.gens()
G.as_matrix_group()

Python

>>> from sage.all import *
>>> G = SO(Integer(3),GF(Integer(5))); G
Special Orthogonal Group of degree 3 over Finite Field of size 5

>>> # needs sage.libs.gap
>>> G = SO(Integer(3),GF(Integer(5)))
>>> 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]
)
>>> 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

sage: # needs sage.rings.number_field
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 Live

# needs sage.rings.number_field
CF3 = CyclotomicField(3); e3 = CF3.gen()
m = matrix(CF3, 3, 3, [[1,e3,0], [e3,2,0], [0,0,1]])
SO3  = SO(3, CF3)
SO3m = SO(3, CF3, invariant_form=m)
SO3 == SO3m
SO3.invariant_form()
SO3m.invariant_form()
pm = Permutation([2,3,1]).to_matrix()
g = SO3(pm); g in SO3; g
SO3m(pm)
SO(3, 5, invariant_form=[[1,0,0], [0,2,0], [0,0,3]])
5+5

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> CF3 = CyclotomicField(Integer(3)); e3 = CF3.gen()
>>> m = matrix(CF3, Integer(3), Integer(3), [[Integer(1),e3,Integer(0)], [e3,Integer(2),Integer(0)], [Integer(0),Integer(0),Integer(1)]])
>>> SO3  = SO(Integer(3), CF3)
>>> SO3m = SO(Integer(3), CF3, invariant_form=m)
>>> SO3 == SO3m
False
>>> SO3.invariant_form()
[1 0 0]
[0 1 0]
[0 0 1]
>>> SO3m.invariant_form()
[    1 zeta3     0]
[zeta3     2     0]
[    0     0     1]
>>> pm = Permutation([Integer(2),Integer(3),Integer(1)]).to_matrix()
>>> g = SO3(pm); g in SO3; g
True
[0 0 1]
[1 0 0]
[0 1 0]
>>> 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]

>>> SO(Integer(3), Integer(5), invariant_form=[[Integer(1),Integer(0),Integer(0)], [Integer(0),Integer(2),Integer(0)], [Integer(0),Integer(0),Integer(3)]])
Traceback (most recent call last):
...
NotImplementedError: invariant_form for finite groups is fixed by GAP
>>> Integer(5)+Integer(5)
10

sage.groups.matrix_gps.orthogonal.normalize_args_e(degree, ring, e)[source]#

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