22.19 Unitary Groups $ GU(n,q)$ and $ SU(n,q)$

Module: sage.groups.matrix_gps.unitary

Unitary Groups $ GU(n,q)$ and $ SU(n,q)$

These are $ n \times n$ unitary matrices with entries in $ GF(q^2)$ .

Author Log:

sage: G = SU(3,GF(5))
sage: G.order()
378000
sage: G
Special Unitary Group of degree 3 over Finite Field of size 5
sage: G._gap_init_()
'SU(3, 5)'
sage: G.random_element()
[      1 4*a + 4 4*a + 1]
[2*a + 4 2*a + 1       0]
[      4     3*a 4*a + 2]
sage: G.base_ring()
Finite Field of size 5
sage: G.field_of_definition()
Finite Field in a of size 5^2

Module-level Functions

GU( n, F, [var=a])

Return the general unitary group of degree n over the finite field F.

Input:

n
- a positive integer
F
- finite field
var
- variable used to represent generator of quadratic extension of F, if needed.

sage: G = GU(3,GF(7)); G
General Unitary Group of degree 3 over Finite Field of size 7
sage: G.gens()
[
[  a   0   0]
[  0   1   0]
[  0   0 5*a],
[6*a   6   1]
[  6   6   0]
[  1   0   0]
]
sage: G = GU(2,QQ)
Traceback (most recent call last):
...
NotImplementedError: general unitary group only implemented over finite
fields

sage: G = GU(3,GF(5), var='beta')
sage: G.gens()
[
[  beta      0      0]
[     0      1      0]
[     0      0 3*beta],
[4*beta      4      1]
[     4      4      0]
[     1      0      0]
]

SU( n, F, [var=a])

Return the special unitary group of degree $ n$ over $ F$ .

sage: SU(3,5)
Special Unitary Group of degree 3 over Finite Field of size 5
sage: SU(3,QQ)
Traceback (most recent call last):
...
NotImplementedError: special unitary group only implemented over finite
fields

Class: GeneralUnitaryGroup_finite_field

class GeneralUnitaryGroup_finite_field

Special Functions: _gap_init_,$ \,$ _latex_,$ \,$ _repr_

_gap_init_( self)

Return string that evaluates to creates this group as an element of GAP.

sage: G = GU(3,GF(7)); G
General Unitary Group of degree 3 over Finite Field of size 7
sage: G._gap_init_()
'GU(3, 7)'
sage: gap(G._gap_init_())
GU(3,7)

_latex_( self)

Return LaTeX string representation of this group.

sage: G = GU(3,GF(7)); G
General Unitary Group of degree 3 over Finite Field of size 7
sage: latex(G)
\text{GU}_{3}(\mathbf{F}_{7^{2}})

_repr_( self)

Return text representatin of self.

sage: G = GU(3,GF(5))
sage: G
General Unitary Group of degree 3 over Finite Field of size 5

Class: SpecialUnitaryGroup_finite_field

class SpecialUnitaryGroup_finite_field

Special Functions: _gap_init_,$ \,$ _latex_,$ \,$ _repr_

_gap_init_( self)

Return string that creates this group in GAP.

sage: SU(3,5)._gap_init_()
'SU(3, 5)'

_latex_( self)

Return latex representatin of this group.

sage: G = SU(3,GF(5))
sage: latex(G)
	ext{SU}_{3}(\mathbf{F}_{5^{2}})

_repr_( self)

Return text representation of this special unitary group.

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

Class: UnitaryGroup_finite_field

class UnitaryGroup_finite_field

Functions: field_of_definition

field_of_definition( self)

Return the field of definition of this general unity group.

sage: G = GU(3,GF(5))
sage: G.field_of_definition()
Finite Field in a of size 5^2
sage: G.base_field()
Finite Field of size 5

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