Heisenberg Group#

AUTHORS:

  • Hilder Vitor Lima Pereira (2017-08): initial version

class sage.groups.matrix_gps.heisenberg.HeisenbergGroup(n=1, R=0)[source]#

Bases: UniqueRepresentation, FinitelyGeneratedMatrixGroup_gap

The Heisenberg group of degree \(n\).

Let \(R\) be a ring, and let \(n\) be a positive integer. The Heisenberg group of degree \(n\) over \(R\) is a multiplicative group whose elements are matrices with the following form:

\[\begin{split}\begin{pmatrix} 1 & x^T & z \\ 0 & I_n & y \\ 0 & 0 & 1 \end{pmatrix},\end{split}\]

where \(x\) and \(y\) are column vectors in \(R^n\), \(z\) is a scalar in \(R\), and \(I_n\) is the identity matrix of size \(n\).

INPUT:

  • n – the degree of the Heisenberg group

  • R – (default: \(\ZZ\)) the ring \(R\) or a positive integer as a shorthand for the ring \(\ZZ/R\ZZ\)

EXAMPLES:

sage: H = groups.matrix.Heisenberg(); H
Heisenberg group of degree 1 over Integer Ring
sage: H.gens()
(
[1 1 0]  [1 0 0]  [1 0 1]
[0 1 0]  [0 1 1]  [0 1 0]
[0 0 1], [0 0 1], [0 0 1]
)
sage: X, Y, Z = H.gens()
sage: Z * X * Y**-1
[ 1  1  0]
[ 0  1 -1]
[ 0  0  1]
sage: X * Y * X**-1 * Y**-1 == Z
True

sage: H = groups.matrix.Heisenberg(R=5); H
Heisenberg group of degree 1 over Ring of integers modulo 5
sage: H = groups.matrix.Heisenberg(n=3, R=13); H
Heisenberg group of degree 3 over Ring of integers modulo 13

>>> from sage.all import *
>>> H = groups.matrix.Heisenberg(); H
Heisenberg group of degree 1 over Integer Ring
>>> H.gens()
(
[1 1 0]  [1 0 0]  [1 0 1]
[0 1 0]  [0 1 1]  [0 1 0]
[0 0 1], [0 0 1], [0 0 1]
)
>>> X, Y, Z = H.gens()
>>> Z * X * Y**-Integer(1)
[ 1  1  0]
[ 0  1 -1]
[ 0  0  1]
>>> X * Y * X**-Integer(1) * Y**-Integer(1) == Z
True

>>> H = groups.matrix.Heisenberg(R=Integer(5)); H
Heisenberg group of degree 1 over Ring of integers modulo 5
>>> H = groups.matrix.Heisenberg(n=Integer(3), R=Integer(13)); H
Heisenberg group of degree 3 over Ring of integers modulo 13

REFERENCES:

cardinality()[source]#

Return the order of self.

EXAMPLES:

sage: H = groups.matrix.Heisenberg()
sage: H.order()
+Infinity
sage: H = groups.matrix.Heisenberg(n=4)
sage: H.order()
+Infinity
sage: H = groups.matrix.Heisenberg(R=3)
sage: H.order()
27
sage: H = groups.matrix.Heisenberg(n=2, R=3)
sage: H.order()
243
sage: H = groups.matrix.Heisenberg(n=2, R=GF(4))
sage: H.order()
1024

>>> from sage.all import *
>>> H = groups.matrix.Heisenberg()
>>> H.order()
+Infinity
>>> H = groups.matrix.Heisenberg(n=Integer(4))
>>> H.order()
+Infinity
>>> H = groups.matrix.Heisenberg(R=Integer(3))
>>> H.order()
27
>>> H = groups.matrix.Heisenberg(n=Integer(2), R=Integer(3))
>>> H.order()
243
>>> H = groups.matrix.Heisenberg(n=Integer(2), R=GF(Integer(4)))
>>> H.order()
1024
center()[source]#

Return the center of self.

This is the subgroup generated by the \(z\), the matrix with a \(1\) in the upper right corner and along the diagonal.

EXAMPLES:

sage: H = groups.matrix.Heisenberg(2)
sage: H.center()
Subgroup with 1 generators (
[1 0 0 1]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
) of Heisenberg group of degree 2 over Integer Ring

sage: H = groups.matrix.Heisenberg(3, 4)
sage: H.center()
Subgroup with 1 generators (
[1 0 0 0 1]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]
) of Heisenberg group of degree 3 over Ring of integers modulo 4

>>> from sage.all import *
>>> H = groups.matrix.Heisenberg(Integer(2))
>>> H.center()
Subgroup with 1 generators (
[1 0 0 1]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
) of Heisenberg group of degree 2 over Integer Ring

>>> H = groups.matrix.Heisenberg(Integer(3), Integer(4))
>>> H.center()
Subgroup with 1 generators (
[1 0 0 0 1]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]
) of Heisenberg group of degree 3 over Ring of integers modulo 4
order()[source]#

Return the order of self.

EXAMPLES:

sage: H = groups.matrix.Heisenberg()
sage: H.order()
+Infinity
sage: H = groups.matrix.Heisenberg(n=4)
sage: H.order()
+Infinity
sage: H = groups.matrix.Heisenberg(R=3)
sage: H.order()
27
sage: H = groups.matrix.Heisenberg(n=2, R=3)
sage: H.order()
243
sage: H = groups.matrix.Heisenberg(n=2, R=GF(4))
sage: H.order()
1024

>>> from sage.all import *
>>> H = groups.matrix.Heisenberg()
>>> H.order()
+Infinity
>>> H = groups.matrix.Heisenberg(n=Integer(4))
>>> H.order()
+Infinity
>>> H = groups.matrix.Heisenberg(R=Integer(3))
>>> H.order()
27
>>> H = groups.matrix.Heisenberg(n=Integer(2), R=Integer(3))
>>> H.order()
243
>>> H = groups.matrix.Heisenberg(n=Integer(2), R=GF(Integer(4)))
>>> H.order()
1024