These are \(n \times n\) unitary matrices with entries in \(GF(q^2)\).
EXAMPLES:
sage: G = SU(3,5)
sage: G.order()
378000
sage: G
Special Unitary Group of degree 3 over Finite Field in a of size 5^2
sage: G.gens()
(
[ a 0 0] [4*a 4 1]
[ 0 2*a + 2 0] [ 4 4 0]
[ 0 0 3*a], [ 1 0 0]
)
sage: G.base_ring()
Finite Field in a of size 5^2
AUTHORS:
Return the general unitary group.
The general unitary group \(GU( d, R )\) consists of all \(d \times d\) matrices that preserve a nondegenerate sequilinear form over the ring \(R\).
Note
For a finite field the matrices that preserve a sesquilinear form over \(F_q\) live over \(F_{q^2}\). So GU(n,q) for integer q constructs the matrix group over the base ring GF(q^2).
Note
This group is also available via groups.matrix.GU().
INPUT:
OUTPUT:
Return the general unitary group.
EXAMPLES:
sage: G = GU(3, 7); G
General Unitary Group of degree 3 over Finite Field in a of size 7^2
sage: G.gens()
(
[ a 0 0] [6*a 6 1]
[ 0 1 0] [ 6 6 0]
[ 0 0 5*a], [ 1 0 0]
)
sage: GU(2,QQ)
General Unitary Group of degree 2 over Rational Field
sage: G = GU(3, 5, var='beta')
sage: G.base_ring()
Finite Field in beta of size 5^2
sage: G.gens()
(
[ beta 0 0] [4*beta 4 1]
[ 0 1 0] [ 4 4 0]
[ 0 0 3*beta], [ 1 0 0]
)
TESTS:
sage: groups.matrix.GU(2, 3)
General Unitary Group of degree 2 over Finite Field in a of size 3^2
The special unitary group \(SU( d, R )\) consists of all \(d imes d\) matrices that preserve a nondegenerate sequilinear form over the ring \(R\) and have determinant one.
Note
For a finite field the matrices that preserve a sesquilinear form over \(F_q\) live over \(F_{q^2}\). So SU(n,q) for integer q constructs the matrix group over the base ring GF(q^2).
Note
This group is also available via groups.matrix.SU().
INPUT:
OUTPUT:
Return the special unitary group.
EXAMPLES:
sage: SU(3,5)
Special Unitary Group of degree 3 over Finite Field in a of size 5^2
sage: SU(3, GF(5))
Special Unitary Group of degree 3 over Finite Field in a of size 5^2
sage: SU(3,QQ)
Special Unitary Group of degree 3 over Rational Field
TESTS:
sage: groups.matrix.SU(2, 3)
Special Unitary Group of degree 2 over Finite Field in a of size 3^2
Bases: sage.groups.matrix_gps.unitary.UnitaryMatrixGroup_generic, sage.groups.matrix_gps.named_group.NamedMatrixGroup_gap
Base class for “named” matrix groups using LibGAP
INPUT:
EXAMPLES:
sage: G = GL(2, GF(3))
sage: from sage.groups.matrix_gps.named_group import NamedMatrixGroup_gap
sage: isinstance(G, NamedMatrixGroup_gap)
True
Bases: sage.groups.matrix_gps.named_group.NamedMatrixGroup_generic
General Unitary Group over arbitrary rings.
EXAMPLES:
sage: G = GU(3, GF(7)); G
General Unitary Group of degree 3 over Finite Field in a of size 7^2
sage: latex(G)
\text{GU}_{3}(\Bold{F}_{7^{2}})
sage: G = SU(3, GF(5)); G
Special Unitary Group of degree 3 over Finite Field in a of size 5^2
sage: latex(G)
\text{SU}_{3}(\Bold{F}_{5^{2}})
Helper function.
INPUT:
A ring.
OUTPUT:
Integer q such that ring is the finite field with \(q^2\) elements.
EXAMPLES:
sage: from sage.groups.matrix_gps.unitary import finite_field_sqrt
sage: finite_field_sqrt(GF(4, 'a'))
2