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

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:

  • 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
  • Volker Braun (2013-1) port to new Parent, libGAP, extreme refactoring.
  • Sebastian Oehms (2018-8) add _UG, invariant_form(), option for user defined invariant bilinear form, and bug-fix in _check_matrix (see trac ticket #26028)
sage.groups.matrix_gps.unitary.GU(n, R, var='a', invariant_form=None)

Return the general unitary group.

The general unitary group \(GU( d, R )\) consists of all \(d \times d\) matrices that preserve a nondegenerate sesquilinear 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 a prime power q constructs the matrix group over the base ring GF(q^2).

Note

This group is also available via groups.matrix.GU().

INPUT:

  • n – a positive integer
  • R – ring or an integer; if an integer is specified, the corresponding finite field is used
  • var – (optional, 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 hermitian form to be kept invariant by the unitary group; the form is checked to be non-degenerate and hermitian but not to be positive definite

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]
)

Using the invariant_form option:

sage: UCF = UniversalCyclotomicField(); e5=UCF.gen(5)
sage: m=matrix(UCF, 3,3, [[1,e5,0],[e5.conjugate(),2,0],[0,0,1]])
sage: G  = GU(3, UCF)
sage: Gm = GU(3, UCF, invariant_form=m)
sage: G == Gm
False
sage: G.invariant_form()
[1 0 0]
[0 1 0]
[0 0 1]
sage: Gm.invariant_form()
[     1   E(5)      0]
[E(5)^4      2      0]
[     0      0      1]
sage: pm=Permutation((1,2,3)).to_matrix()
sage: g = G(pm); g in G; g
True
[0 0 1]
[1 0 0]
[0 1 0]
sage: Gm(pm)
Traceback (most recent call last):
...
TypeError: matrix must be unitary with respect to the hermitian form
[     1   E(5)      0]
[E(5)^4      2      0]
[     0      0      1]

sage: GU(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: GU(2,QQ, invariant_form=[[1,0],[2,0]])
Traceback (most recent call last):
...
ValueError: invariant_form must be non-degenerate
sage.groups.matrix_gps.unitary.SU(n, R, var='a', invariant_form=None)

The special unitary group \(SU( d, R )\) consists of all \(d \times d\) matrices that preserve a nondegenerate sesquilinear form over the ring \(R\) and have determinant \(1\).

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 a prime power q constructs the matrix group over the base ring GF(q^2).

Note

This group is also available via groups.matrix.SU().

INPUT:

  • n – a positive integer
  • R – ring or an integer; if an integer is specified, the corresponding finite field is used
  • var – (optional, 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 hermitian form to be kept invariant by the unitary group; the form is checked to be non-degenerate and hermitian but not to be positive definite

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

Using the invariant_form option:

sage: CF3 = CyclotomicField(3); e3 = CF3.gen()
sage: m=matrix(CF3, 3,3, [[1,e3,0],[e3.conjugate(),2,0],[0,0,1]])
sage: G  = SU(3, CF3)
sage: Gm = SU(3, CF3, invariant_form=m)
sage: G == Gm
False
sage: G.invariant_form()
[1 0 0]
[0 1 0]
[0 0 1]
sage: Gm.invariant_form()
[         1      zeta3          0]
[-zeta3 - 1          2          0]
[         0          0          1]
sage: pm=Permutation((1,2,3)).to_matrix()
sage: G(pm)
[0 0 1]
[1 0 0]
[0 1 0]
sage: Gm(pm)
Traceback (most recent call last):
...
TypeError: matrix must be unitary with respect to the hermitian form
[         1      zeta3          0]
[-zeta3 - 1          2          0]
[         0          0          1]

sage: SU(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
class sage.groups.matrix_gps.unitary.UnitaryMatrixGroup_gap(degree, base_ring, special, sage_name, latex_string, gap_command_string, category=None)

Bases: sage.groups.matrix_gps.unitary.UnitaryMatrixGroup_generic, sage.groups.matrix_gps.named_group.NamedMatrixGroup_gap, sage.groups.matrix_gps.finitely_generated.FinitelyGeneratedMatrixGroup_gap

The general or special unitary group in GAP.

invariant_form()

Return the hermitian form preserved by the unitary group.

OUTPUT:

A square matrix describing the bilinear form

EXAMPLES:

sage: G32=GU(3,2)
sage: G32.invariant_form()
[0 0 1]
[0 1 0]
[1 0 0]
class sage.groups.matrix_gps.unitary.UnitaryMatrixGroup_generic(degree, base_ring, special, sage_name, latex_string, category=None, invariant_form=None)

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}})

sage: CF3 = CyclotomicField(3); e3 = CF3.gen()
sage: m=matrix(CF3, 3,3, [[1,e3,0],[e3.conjugate(),2,0],[0,0,1]])
sage: G = SU(3, CF3, invariant_form=m)
sage: latex(G)
\text{SU}_{3}(\Bold{Q}(\zeta_{3}))\text{ with respect to positive definite hermitian form }\left(\begin{array}{rrr}
1 & \zeta_{3} & 0 \\
-\zeta_{3} - 1 & 2 & 0 \\
0 & 0 & 1
\end{array}\right)
invariant_form()

Return the hermitian form preserved by the unitary group.

OUTPUT:

A square matrix describing the bilinear form

EXAMPLES:

sage: SU4 = SU(4,QQ)
sage: SU4.invariant_form()
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
sage.groups.matrix_gps.unitary.finite_field_sqrt(ring)

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