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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Franz Gähler.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains the declarations for matrix groups over the rationals
##
#############################################################################
##
#C IsCyclotomicMatrixGroup( <G> )
##
## <#GAPDoc Label="IsCyclotomicMatrixGroup">
## <ManSection>
## <Filt Name="IsCyclotomicMatrixGroup" Arg='G' Type='Category'/>
##
## <Description>
## tests whether all matrices in <A>G</A> have cyclotomic entries.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonym( "IsCyclotomicMatrixGroup", IsCyclotomicCollCollColl and IsMatrixGroup );
#############################################################################
##
#P IsRationalMatrixGroup( <G> )
##
## <#GAPDoc Label="IsRationalMatrixGroup">
## <ManSection>
## <Prop Name="IsRationalMatrixGroup" Arg='G'/>
##
## <Description>
## tests whether all matrices in <A>G</A> have rational entries.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsRationalMatrixGroup", IsCyclotomicMatrixGroup );
InstallTrueMethod( IsCyclotomicMatrixGroup, IsRationalMatrixGroup );
#############################################################################
##
#P IsIntegerMatrixGroup( <G> )
##
## <#GAPDoc Label="IsIntegerMatrixGroup">
## <ManSection>
## <Prop Name="IsIntegerMatrixGroup" Arg='G'/>
##
## <Description>
## tests whether all matrices in <A>G</A> have integer entries.
## <Example><![CDATA[
## gap> A:=[[0,1,0],[0,0,1],[1,0,0]];;
## gap> B:=[[0,0,1],[0,1,0],[-1,0,0]];;
## gap> C:=[[E(4),0,0],[0,E(4)^(-1),0],[0,0,1]];;
## gap> IsIntegerMatrixGroup(Group(A, B));
## true
## gap> IsIntegerMatrixGroup(Group(A, C));
## false
## ]]></Example>
## <!-- Not <C>IsIntegralMatrixGroup</C> to avoid confusion with matrix groups of-->
## <!-- integral cyclotomic numbers. -->
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsIntegerMatrixGroup", IsCyclotomicMatrixGroup );
InstallTrueMethod( IsRationalMatrixGroup, IsIntegerMatrixGroup );
#############################################################################
##
#P IsNaturalGLnZ( <G> )
##
## <#GAPDoc Label="IsNaturalGLnZ">
## <ManSection>
## <Prop Name="IsNaturalGLnZ" Arg='G'/>
##
## <Description>
## tests whether <A>G</A> is <M>GL_n(&ZZ;)</M> in its natural representation
## by <M>n \times n</M> integer matrices.
## (The dimension <M>n</M> will be read off the generating matrices.)
## <Example><![CDATA[
## gap> IsNaturalGLnZ( GL( 2, Integers ) );
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonym( "IsNaturalGLnZ", IsNaturalGL and IsIntegerMatrixGroup );
#############################################################################
##
#P IsNaturalSLnZ( <G> )
##
## <#GAPDoc Label="IsNaturalSLnZ">
## <ManSection>
## <Prop Name="IsNaturalSLnZ" Arg='G'/>
##
## <Description>
## tests whether <A>G</A> is <M>SL_n(&ZZ;)</M> in its natural representation
## by <M>n \times n</M> integer matrices.
## (The dimension <M>n</M> will be read off the generating matrices.)
## <Example><![CDATA[
## gap> IsNaturalSLnZ( SL( 2, Integers ) );
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonym( "IsNaturalSLnZ", IsNaturalSL and IsIntegerMatrixGroup );
#############################################################################
##
#A ZClassRepsQClass( G ) . . . . . . . . . . . Z-class reps in Q-class of G
##
## <#GAPDoc Label="ZClassRepsQClass">
## <ManSection>
## <Attr Name="ZClassRepsQClass" Arg='G'/>
##
## <Description>
## The conjugacy class in <M>GL_n(&QQ;)</M> of the finite integer matrix
## group <A>G</A> splits into finitely many conjugacy classes in
## <M>GL_n(&ZZ;)</M>.
## <C>ZClassRepsQClass( <A>G</A> )</C> returns representative groups for these.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "ZClassRepsQClass", IsCyclotomicMatrixGroup );
#############################################################################
##
#A NormalizerInGLnZ( G ) . . . . . . . . . . . . . . . . . NormalizerInGLnZ
##
## <#GAPDoc Label="NormalizerInGLnZ">
## <ManSection>
## <Attr Name="NormalizerInGLnZ" Arg='G'/>
##
## <Description>
## is an attribute used to store the normalizer of <A>G</A> in
## <M>GL_n(&ZZ;)</M>, where <A>G</A> is an integer matrix group of dimension
## <A>n</A>. This attribute
## is used by <C>Normalizer( GL( n, Integers ), G )</C>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "NormalizerInGLnZ", IsCyclotomicMatrixGroup );
#############################################################################
##
#A CentralizerInGLnZ( G ) . . . . . . . . . . . . . . . . .CentralizerInGLnZ
##
## <#GAPDoc Label="CentralizerInGLnZ">
## <ManSection>
## <Attr Name="CentralizerInGLnZ" Arg='G'/>
##
## <Description>
## is an attribute used to store the centralizer of <A>G</A> in
## <M>GL_n(&ZZ;)</M>, where <A>G</A> is an integer matrix group of dimension
## <A>n</A>. This attribute
## is used by <C>Centralizer( GL( n, Integers ), G )</C>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "CentralizerInGLnZ", IsCyclotomicMatrixGroup );
#############################################################################
##
## RightAction or LeftAction
##
## <#GAPDoc Label="[1]{grpramat}">
## 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
## <Package>CrystGap</Package>, 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.
## <P/>
## For every such function, this fact is explicitly mentioned.
## The naming scheme is as follows: If <C>SomeThing</C> is such a function,
## there will be functions <C>SomeThingOnRight</C> and <C>SomeThingOnLeft</C>,
## assuming action from the right and from the left, respectively.
## In addition, there is a generic function <C>SomeThing</C>, which returns
## either the result of <C>SomeThingOnRight</C> or <C>SomeThingOnLeft</C>,
## depending on the global variable <Ref Var="CrystGroupDefaultAction"/>.
## <#/GAPDoc>
##
#############################################################################
##
#V CrystGroupDefaultAction
##
## <#GAPDoc Label="CrystGroupDefaultAction">
## <ManSection>
## <Var Name="CrystGroupDefaultAction"/>
##
## <Description>
## can have either of the two values <C>RightAction</C> and <C>LeftAction</C>.
## The initial value is <C>RightAction</C>. For functions which have
## variants OnRight and OnLeft, this variable determines which
## variant is returned by the generic form. The value of
## <Ref Var="CrystGroupDefaultAction"/> can be changed with the
## function <Ref Func="SetCrystGroupDefaultAction"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalName( "CrystGroupDefaultAction" );
BindGlobal( "LeftAction", Immutable( "LeftAction" ) );
BindGlobal( "RightAction", Immutable( "RightAction" ) );
#############################################################################
##
#F SetCrystGroupDefaultAction( <action> ) . . . . .RightAction or LeftAction
##
## <#GAPDoc Label="SetCrystGroupDefaultAction">
## <ManSection>
## <Func Name="SetCrystGroupDefaultAction" Arg='action'/>
##
## <Description>
## allows one to set the value of the global variable
## <Ref Var="CrystGroupDefaultAction"/>.
## Only the arguments <C>RightAction</C> and <C>LeftAction</C> are allowed.
## Initially, the value of <Ref Var="CrystGroupDefaultAction"/> is
## <C>RightAction</C>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "SetCrystGroupDefaultAction" );
#############################################################################
##
#P IsBravaisGroup( <G> ) . . . . . . . . . . . . . . . . . . .IsBravaisGroup
##
## <#GAPDoc Label="IsBravaisGroup">
## <ManSection>
## <Prop Name="IsBravaisGroup" Arg='G'/>
##
## <Description>
## test whether <A>G</A> coincides with its Bravais group
## (see <Ref Attr="BravaisGroup"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsBravaisGroup", IsCyclotomicMatrixGroup );
#############################################################################
##
#A BravaisGroup( <G> ) . . . . . . . . Bravais group of integer matrix group
##
## <#GAPDoc Label="BravaisGroup">
## <ManSection>
## <Attr Name="BravaisGroup" Arg='G'/>
##
## <Description>
## returns the Bravais group of a finite integer matrix group <A>G</A>.
## If <M>C</M> is the cone of positive definite quadratic forms <M>Q</M>
## invariant under <M>g \mapsto g Q g^{tr}</M> for all <M>g \in <A>G</A></M>,
## then the Bravais group of <A>G</A> is the maximal subgroup of
## <M>GL_n(&ZZ;)</M> leaving the forms in that same cone invariant.
## Alternatively, the Bravais group of <A>G</A>
## can also be defined with respect to the action <M>g \mapsto g^{tr} Q g</M>
## on positive definite quadratic forms <M>Q</M>. This latter definition
## is appropriate for groups <A>G</A> 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.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "BravaisGroup", IsCyclotomicMatrixGroup );
#############################################################################
##
#A BravaisSubgroups( <G> ) . . . . . . . .Bravais subgroups of Bravais group
##
## <#GAPDoc Label="BravaisSubgroups">
## <ManSection>
## <Attr Name="BravaisSubgroups" Arg='G'/>
##
## <Description>
## returns the subgroups of the Bravais group of <A>G</A>, which are
## themselves Bravais groups.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "BravaisSubgroups", IsCyclotomicMatrixGroup );
#############################################################################
##
#A BravaisSupergroups( <G> ) . . . . . .Bravais supergroups of Bravais group
##
## <#GAPDoc Label="BravaisSupergroups">
## <ManSection>
## <Attr Name="BravaisSupergroups" Arg='G'/>
##
## <Description>
## returns the subgroups of <M>GL_n(&ZZ;)</M> that contain the Bravais group
## of <A>G</A> and are Bravais groups themselves.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "BravaisSupergroups", IsCyclotomicMatrixGroup );
#############################################################################
##
#A NormalizerInGLnZBravaisGroup( <G> ) . norm. of Bravais group of G in GLnZ
##
## <#GAPDoc Label="NormalizerInGLnZBravaisGroup">
## <ManSection>
## <Attr Name="NormalizerInGLnZBravaisGroup" Arg='G'/>
##
## <Description>
## returns the normalizer of the Bravais group of <A>G</A> in the
## appropriate <M>GL_n(&ZZ;)</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "NormalizerInGLnZBravaisGroup", IsCyclotomicMatrixGroup );
#############################################################################
##
#A InvariantLattice( G )
##
## <#GAPDoc Label="InvariantLattice">
## <ManSection>
## <Attr Name="InvariantLattice" Arg='G'/>
##
## <Description>
## returns a matrix <M>B</M>, whose rows form a basis of a
## <M>&ZZ;</M>-lattice that is invariant under the rational matrix group
## <A>G</A> acting from the right.
## It returns <K>fail</K> if the group is not unimodular. The columns of the
## inverse of <M>B</M> span a <M>&ZZ;</M>-lattice invariant under <A>G</A>
## acting from the left.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "InvariantLattice", IsCyclotomicMatrixGroup );
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