sage.groups.matrix

Module unitary

Unitary Groups $GU(n,q)$ and $SU(n,q)$ These are $n \times n$ unitary matrices with entries in $GF(q^2)$. AUTHOR: -- David Joyner (2006-03): initial version, modified from special_linear (by W. Stein) -- David Joyner (2006-05): minor additions (examples, _latex_, __str__, gens) -- William Stein (2006-12) -- rewrite EXAMPLES: 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

Functions

[hide private]

GU(n, F, var='a')
Return the general unitary group of degree n over the finite field F. source code

SU(n, F, var='a')
Return the special unitary group of degree $n$ over $F$. source code

Function Details

[hide private]

GU(n, F, var=`'a'`)

source code


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.

EXAMPLES:
    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'`)

source code


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

EXAMPLES:
    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

Module unitary

GU(n, F, var='a')

SU(n, F, var='a')

GU(n, F, var=`'a'`)

SU(n, F, var=`'a'`)