44 Matrix Groups

44.6 Matrix Groups in Characteristic 0

44.6-1 IsCyclotomicMatrixGroup

44.6-2 IsRationalMatrixGroup

44.6-3 IsIntegerMatrixGroup

44.6-4 IsNaturalGLnZ

44.6-5 IsNaturalSLnZ

44.6-6 InvariantLattice

44.6-7 NormalizerInGLnZ

44.6-8 CentralizerInGLnZ

44.6-9 ZClassRepsQClass

44.6-10 IsBravaisGroup

44.6-11 BravaisGroup

44.6-12 BravaisSubgroups

44.6-13 BravaisSupergroups

44.6-14 NormalizerInGLnZBravaisGroup

44.6-1 IsCyclotomicMatrixGroup

44.6-2 IsRationalMatrixGroup

44.6-3 IsIntegerMatrixGroup

44.6-4 IsNaturalGLnZ

44.6-5 IsNaturalSLnZ

44.6-6 InvariantLattice

44.6-7 NormalizerInGLnZ

44.6-8 CentralizerInGLnZ

44.6-9 ZClassRepsQClass

44.6-10 IsBravaisGroup

44.6-11 BravaisGroup

44.6-12 BravaisSubgroups

44.6-13 BravaisSupergroups

44.6-14 NormalizerInGLnZBravaisGroup

Matrix groups are groups generated by invertible square matrices.

In the following example we temporarily increase the line length limit from its default value 80 to 83 in order to get a nicer output format.

gap> m1 := [ [ Z(3)^0, Z(3)^0, Z(3) ], > [ Z(3), 0*Z(3), Z(3) ], > [ 0*Z(3), Z(3), 0*Z(3) ] ];; gap> m2 := [ [ Z(3), Z(3), Z(3)^0 ], > [ Z(3), 0*Z(3), Z(3) ], > [ Z(3)^0, 0*Z(3), Z(3) ] ];; gap> m := Group( m1, m2 ); Group( [ [ [ Z(3)^0, Z(3)^0, Z(3) ], [ Z(3), 0*Z(3), Z(3) ], [ 0*Z(3), Z(3), 0*Z(3) ] ], [ [ Z(3), Z(3), Z(3)^0 ], [ Z(3), 0*Z(3), Z(3) ], [ Z(3)^0, 0*Z(3), Z(3) ] ] ])

For most operations, **GAP** only provides methods for finite matrix groups. Many calculations in finite matrix groups are done via so-called "nice monomorphisms" (see Section 40.5) that represent a faithful action on vectors.

`‣ IsMatrixGroup` ( grp ) | ( category ) |

The category of matrix groups.

`‣ DimensionOfMatrixGroup` ( mat-grp ) | ( attribute ) |

The dimension of the matrix group.

`‣ DefaultFieldOfMatrixGroup` ( mat-grp ) | ( attribute ) |

Is a field containing all the matrix entries. It is not guaranteed to be the smallest field with this property.

`‣ FieldOfMatrixGroup` ( matgrp ) | ( attribute ) |

The smallest field containing all the matrix entries of all elements of the matrix group `matgrp`. As the calculation of this can be hard, this should only be used if one *really* needs the smallest field, use `DefaultFieldOfMatrixGroup`

(44.2-2) to get (for example) the characteristic.

gap> DimensionOfMatrixGroup(m); 3 gap> DefaultFieldOfMatrixGroup(m); GF(3)

`‣ TransposedMatrixGroup` ( matgrp ) | ( attribute ) |

returns the transpose of the matrix group `matgrp`. The transpose of the transpose of `matgrp` is identical to `matgrp`.

gap> G := Group( [[0,-1],[1,0]] ); Group([ [ [ 0, -1 ], [ 1, 0 ] ] ]) gap> T := TransposedMatrixGroup( G ); Group([ [ [ 0, 1 ], [ -1, 0 ] ] ]) gap> IsIdenticalObj( G, TransposedMatrixGroup( T ) ); true

`‣ IsFFEMatrixGroup` ( G ) | ( category ) |

tests whether all matrices in `G` have finite field element entries.

The basic operations for groups are described in Chapter 41, special actions for *matrix* groups mentioned there are `OnLines`

(41.2-12), `OnRight`

(41.2-2), and `OnSubspacesByCanonicalBasis`

(41.2-15).

For subtleties concerning multiplication from the left or from the right, see 44.7.

`‣ ProjectiveActionOnFullSpace` ( G, F, n ) | ( function ) |

Let `G` be a group of `n` by `n` matrices over a field contained in the finite field `F`. `ProjectiveActionOnFullSpace`

returns the image of the projective action of `G` on the full row space \(\textit{F}^{\textit{n}}\).

`‣ ProjectiveActionHomomorphismMatrixGroup` ( G ) | ( function ) |

returns an action homomorphism for a faithful projective action of `G` on the underlying vector space. (Note: The action is not necessarily on the full space, if a smaller subset can be found on which the action is faithful.)

`‣ BlowUpIsomorphism` ( matgrp, B ) | ( function ) |

For a matrix group `matgrp` and a basis `B` of a field extension \(L / K\) such that the entries of all matrices in `matgrp` lie in \(L\), `BlowUpIsomorphism`

returns the isomorphism with source `matgrp` that is defined by mapping the matrix \(A\) to `BlownUpMat`

\(( A, \textit{B} )\), see `BlownUpMat`

(24.13-4).

gap> g:= GL(2,4);; gap> B:= CanonicalBasis( GF(4) );; BasisVectors( B ); [ Z(2)^0, Z(2^2) ] gap> iso:= BlowUpIsomorphism( g, B );; gap> Display( Image( iso, [ [ Z(4), Z(2) ], [ 0*Z(2), Z(4)^2 ] ] ) ); . 1 1 . 1 1 . 1 . . 1 1 . . 1 . gap> img:= Image( iso, g ); <matrix group with 2 generators> gap> Index( GL(4,2), img ); 112

(See also section 50.2.)

`‣ IsGeneralLinearGroup` ( grp ) | ( property ) |

`‣ IsGL` ( grp ) | ( property ) |

The General Linear group is the group of all invertible matrices over a ring. This property tests, whether a group is isomorphic to a General Linear group. (Note that currently only a few trivial methods are available for this operation. We hope to improve this in the future.)

`‣ IsNaturalGL` ( matgrp ) | ( property ) |

This property tests, whether a matrix group is the General Linear group in the right dimension over the (smallest) ring which contains all entries of its elements. (Currently, only a trivial test that computes the order of the group is available.)

`‣ IsSpecialLinearGroup` ( grp ) | ( property ) |

`‣ IsSL` ( grp ) | ( property ) |

The Special Linear group is the group of all invertible matrices over a ring, whose determinant is equal to 1. This property tests, whether a group is isomorphic to a Special Linear group. (Note that currently only a few trivial methods are available for this operation. We hope to improve this in the future.)

`‣ IsNaturalSL` ( matgrp ) | ( property ) |

This property tests, whether a matrix group is the Special Linear group in the right dimension over the (smallest) ring which contains all entries of its elements. (Currently, only a trivial test that computes the order of the group is available.)

gap> IsNaturalGL(m); false

`‣ IsSubgroupSL` ( matgrp ) | ( property ) |

This property tests, whether a matrix group is a subgroup of the Special Linear group in the right dimension over the (smallest) ring which contains all entries of its elements.

`‣ InvariantBilinearForm` ( matgrp ) | ( attribute ) |

This attribute describes a bilinear form that is invariant under the matrix group `matgrp`. The form is given by a record with the component `matrix`

which is a matrix \(F\) such that for every generator \(g\) of `matgrp` the equation \(g \cdot F \cdot g^{tr} = F\) holds.

`‣ IsFullSubgroupGLorSLRespectingBilinearForm` ( matgrp ) | ( property ) |

This property tests, whether a matrix group `matgrp` is the full subgroup of GL or SL (the property `IsSubgroupSL`

(44.4-5) determines which it is) respecting the form stored as the value of `InvariantBilinearForm`

(44.5-1) for `matgrp`.

`‣ InvariantSesquilinearForm` ( matgrp ) | ( attribute ) |

This attribute describes a sesquilinear form that is invariant under the matrix group `matgrp` over the field \(F\) with \(q^2\) elements. The form is given by a record with the component `matrix`

which is a matrix \(M\) such that for every generator \(g\) of `matgrp` the equation \(g \cdot M \cdot (g^{tr})^f = M\) holds, where \(f\) is the automorphism of \(F\) that raises each element to its \(q\)-th power. (\(f\) can be obtained as a power of the `FrobeniusAutomorphism`

(59.4-1) value of \(F\).)

`‣ IsFullSubgroupGLorSLRespectingSesquilinearForm` ( matgrp ) | ( property ) |

This property tests, whether a matrix group `matgrp` is the full subgroup of GL or SL (the property `IsSubgroupSL`

(44.4-5) determines which it is) respecting the form stored as the value of `InvariantSesquilinearForm`

(44.5-3) for `matgrp`.

`‣ InvariantQuadraticForm` ( matgrp ) | ( attribute ) |

For a matrix group `matgrp`, `InvariantQuadraticForm`

returns a record containing at least the component `matrix`

whose value is a matrix \(Q\). The quadratic form \(q\) on the natural vector space \(V\) on which `matgrp` acts is given by \(q(v) = v Q v^{tr}\), and the invariance under `matgrp` is given by the equation \(q(v) = q(v M)\) for all \(v \in V\) and \(M\) in `matgrp`. (Note that the invariance of \(q\) does *not* imply that the matrix \(Q\) is invariant under `matgrp`.)

\(q\) is defined relative to an invariant symmetric bilinear form \(f\) (see `InvariantBilinearForm`

(44.5-1)), via the equation \(q(\lambda x + \mu y) = \lambda^2 q(x) + \lambda \mu f(x,y) + \mu^2 q(y)\), see [CCN+85, Chapter 3.4]. If \(f\) is represented by the matrix \(F\) then this implies \(F = Q + Q^{tr}\). In characteristic different from \(2\), we have \(q(x) = f(x,x)/2\), so \(Q\) can be chosen as the strictly upper triangular part of \(F\) plus half of the diagonal part of \(F\). In characteristic \(2\), \(F\) does not determine \(Q\) but still \(Q\) can be chosen as an upper (or lower) triangular matrix.

Whenever the `InvariantQuadraticForm`

value is set in a matrix group then also the `InvariantBilinearForm`

(44.5-1) value can be accessed, and the two values are compatible in the above sense.

`‣ IsFullSubgroupGLorSLRespectingQuadraticForm` ( matgrp ) | ( property ) |

This property tests, whether the matrix group `matgrp` is the full subgroup of GL or SL (the property `IsSubgroupSL`

(44.4-5) determines which it is) respecting the `InvariantQuadraticForm`

(44.5-5) value of `matgrp`.

gap> g:= Sp( 2, 3 );; gap> m:= InvariantBilinearForm( g ).matrix; [ [ 0*Z(3), Z(3)^0 ], [ Z(3), 0*Z(3) ] ] gap> [ 0, 1 ] * m * [ 1, -1 ]; # evaluate the bilinear form Z(3) gap> IsFullSubgroupGLorSLRespectingBilinearForm( g ); true gap> g:= SU( 2, 4 );; gap> m:= InvariantSesquilinearForm( g ).matrix; [ [ 0*Z(2), Z(2)^0 ], [ Z(2)^0, 0*Z(2) ] ] gap> [ 0, 1 ] * m * [ 1, 1 ]; # evaluate the bilinear form Z(2)^0 gap> IsFullSubgroupGLorSLRespectingSesquilinearForm( g ); true gap> g:= GO( 1, 2, 3 );; gap> m:= InvariantBilinearForm( g ).matrix; [ [ 0*Z(3), Z(3)^0 ], [ Z(3)^0, 0*Z(3) ] ] gap> [ 0, 1 ] * m * [ 1, 1 ]; # evaluate the bilinear form Z(3)^0 gap> q:= InvariantQuadraticForm( g ).matrix; [ [ 0*Z(3), Z(3)^0 ], [ 0*Z(3), 0*Z(3) ] ] gap> [ 0, 1 ] * q * [ 0, 1 ]; # evaluate the quadratic form 0*Z(3) gap> IsFullSubgroupGLorSLRespectingQuadraticForm( g ); true

Most of the functions described in this and the following section have implementations which use functions from the **GAP** package **Carat**. If **Carat** is not installed or not compiled, no suitable methods are available.

`‣ IsCyclotomicMatrixGroup` ( G ) | ( category ) |

tests whether all matrices in `G` have cyclotomic entries.

`‣ IsRationalMatrixGroup` ( G ) | ( property ) |

tests whether all matrices in `G` have rational entries.

`‣ IsIntegerMatrixGroup` ( G ) | ( property ) |

tests whether all matrices in `G` have integer entries.

`‣ IsNaturalGLnZ` ( G ) | ( property ) |

tests whether `G` is \(GL_n(ℤ)\) in its natural representation by \(n \times n\) integer matrices. (The dimension \(n\) will be read off the generating matrices.)

gap> IsNaturalGLnZ( GL( 2, Integers ) ); true

`‣ IsNaturalSLnZ` ( G ) | ( property ) |

tests whether `G` is \(SL_n(ℤ)\) in its natural representation by \(n \times n\) integer matrices. (The dimension \(n\) will be read off the generating matrices.)

gap> IsNaturalSLnZ( SL( 2, Integers ) ); true

`‣ InvariantLattice` ( G ) | ( attribute ) |

returns a matrix \(B\), whose rows form a basis of a \(ℤ\)-lattice that is invariant under the rational matrix group `G` acting from the right. It returns `fail`

if the group is not unimodular. The columns of the inverse of \(B\) span a \(ℤ\)-lattice invariant under `G` acting from the left.

`‣ NormalizerInGLnZ` ( G ) | ( attribute ) |

is an attribute used to store the normalizer of `G` in \(GL_n(ℤ)\), where `G` is an integer matrix group of dimension `n`. This attribute is used by `Normalizer( GL( n, Integers ), G )`

.

`‣ CentralizerInGLnZ` ( G ) | ( attribute ) |

is an attribute used to store the centralizer of `G` in \(GL_n(ℤ)\), where `G` is an integer matrix group of dimension `n`. This attribute is used by `Centralizer( GL( n, Integers ), G )`

.

`‣ ZClassRepsQClass` ( G ) | ( attribute ) |

The conjugacy class in \(GL_n(ℚ)\) of the finite integer matrix group `G` splits into finitely many conjugacy classes in \(GL_n(ℤ)\). `ZClassRepsQClass( `

returns representative groups for these.`G` )

`‣ IsBravaisGroup` ( G ) | ( property ) |

test whether `G` coincides with its Bravais group (see `BravaisGroup`

(44.6-11)).

`‣ BravaisGroup` ( G ) | ( attribute ) |

returns the Bravais group of a finite integer matrix group `G`. If \(C\) is the cone of positive definite quadratic forms \(Q\) invariant under \(g \mapsto g Q g^{tr}\) for all \(g \in \textit{G}\), then the Bravais group of `G` is the maximal subgroup of \(GL_n(ℤ)\) leaving the forms in that same cone invariant. Alternatively, the Bravais group of `G` can also be defined with respect to the action \(g \mapsto g^{tr} Q g\) on positive definite quadratic forms \(Q\). This latter definition is appropriate for groups `G` acting from the right on row vectors, whereas the former definition is appropriate for groups acting from the left on column vectors. Both definitions yield the same Bravais group.

`‣ BravaisSubgroups` ( G ) | ( attribute ) |

returns the subgroups of the Bravais group of `G`, which are themselves Bravais groups.

`‣ BravaisSupergroups` ( G ) | ( attribute ) |

returns the subgroups of \(GL_n(ℤ)\) that contain the Bravais group of `G` and are Bravais groups themselves.

`‣ NormalizerInGLnZBravaisGroup` ( G ) | ( attribute ) |

returns the normalizer of the Bravais group of `G` in the appropriate \(GL_n(ℤ)\).

In **GAP**, matrices by convention act on row vectors from the right, whereas in crystallography the convention is to act on column vectors from the left. The definition of certain algebraic objects important in crystallography implicitly depends on which action is assumed. This holds true in particular for quadratic forms invariant under a matrix group. In a similar way, the representation of affine crystallographic groups, as they are provided by the **GAP** package **CrystGap**, depends on which action is assumed. Crystallographers are used to the action from the left, whereas the action from the right is the natural one for **GAP**. For this reason, a number of functions which are important in crystallography, and whose result depends on which action is assumed, are provided in two versions, one for the usual action from the right, and one for the crystallographic action from the left.

For every such function, this fact is explicitly mentioned. The naming scheme is as follows: If `SomeThing`

is such a function, there will be functions `SomeThingOnRight`

and `SomeThingOnLeft`

, assuming action from the right and from the left, respectively. In addition, there is a generic function `SomeThing`

, which returns either the result of `SomeThingOnRight`

or `SomeThingOnLeft`

, depending on the global variable `CrystGroupDefaultAction`

(44.7-1).

`‣ CrystGroupDefaultAction` | ( global variable ) |

can have either of the two values `RightAction`

and `LeftAction`

. The initial value is `RightAction`

. For functions which have variants OnRight and OnLeft, this variable determines which variant is returned by the generic form. The value of `CrystGroupDefaultAction`

can be changed with the function `SetCrystGroupDefaultAction`

(44.7-2).

`‣ SetCrystGroupDefaultAction` ( action ) | ( function ) |

allows one to set the value of the global variable `CrystGroupDefaultAction`

(44.7-1). Only the arguments `RightAction`

and `LeftAction`

are allowed. Initially, the value of `CrystGroupDefaultAction`

(44.7-1) is `RightAction`

.

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