Most of the functions described in this and the following section have implementations which use functions from the share package Carat. If Carat is not installed or not compiled, no suitable methods are available.
IsCyclotomicMatrixGroup( G ) P
tests whether all matrices in G have cyclotomic entries.
IsRationalMatrixGroup( G ) P
tests whether all matrices in G have rational entries.
IsIntegerMatrixGroup( G ) P
tests whether all matrices in G have integer entries.
IsNaturalGLnZ( G ) P
tests whether G is GLn(Z) in its natural representation by n×n integer matrices. (The dimension n will be read off the generating matrices.)
gap> IsNaturalGLnZ( GL( 2, Integers ) ); true
InvariantLattice( G ) A
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 ) A
is an attribute used to store the normalizer of G in GLn(Z),
where G is an integer matrix group of dimension n. This attribute
is used by Normalizer( GL( n, Integers ), G ).
CentralizerInGLnZ( G ) A
is an attribute used to store the centralizer of G in GLn(Z),
where G is an integer matrix group of dimension n. This attribute
is used by Centralizer( GL( n, Integers ), G ).
ZClassRepsQClass( G ) A
The conjugacy class in GLn(Q) of the finite integer matrix
group G splits into finitely many conjugacy classes in GLn(Z).
ZClassRepsQClass( G ) returns representative groups for these.
IsBravaisGroup( G ) P
test whether G coincides with its Bravais group (see BravaisGroup).
BravaisGroup( G ) A
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 ® g*Q*gtr for all g Î G, then the Bravais group of G is the maximal subgroup of GLn(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 ® gtr*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 ) A
returns the subgroups of the Bravais group of G, which are themselves Bravais groups.
BravaisSupergroups( G ) A
returns the subgroups of GLn(Z) that contain the Bravais group of G and are Bravais groups themselves.
NormalizerInGLnZBravaisGroup( G ) A
returns the normalizer of the Bravais group of G in the appropriate GLn(Z).
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GAP 4 manual