#############################################################################
##
#A cryst.gi Cryst library Bettina Eick
#A Franz G"ahler
#A Werner Nickel
##
#Y Copyright 1997-2012 by Bettina Eick, Franz G"ahler and Werner Nickel
##
## Methods for affine crystallographic groups
##
#############################################################################
##
## Utility functions
##
#############################################################################
#############################################################################
##
#M IsAffineMatrixOnRight( <mat> ) . . . . . . . affine matrix action OnRight
##
InstallGlobalFunction( IsAffineMatrixOnRight, function( mat )
local d, v;
if not IsMatrix( mat ) or not IsCyclotomicCollColl( mat ) then
return false;
fi;
d := Length( mat );
if not DimensionsMat( mat ) = [d,d] then
return false;
fi;
v := 0 * [1..d]; v[d] := 1;
return mat{[1..d]}[d] = v;
end );
#############################################################################
##
#M IsAffineMatrixOnLeft( <mat> ) . . . . . . . . affine matrix action OnLeft
##
InstallGlobalFunction( IsAffineMatrixOnLeft, function( mat )
local d, v;
if not IsMatrix( mat ) or not IsCyclotomicCollColl( mat ) then
return false;
fi;
d := Length( mat );
if not DimensionsMat( mat ) = [d,d] then
return false;
fi;
v := 0 * [1..d]; v[d] := 1;
return mat[d] = v;
end );
#############################################################################
##
## Methods and functions for CrystGroups and PointGroups
##
#############################################################################
#############################################################################
##
#M IsAffineCrystGroupOnLeftOrRight( <S> ) . . . . . AffineCrystGroup acting
#M . . . . . . . . . . . . . . . . . . . . . . . . either OnLeft or OnRight
##
InstallTrueMethod(IsAffineCrystGroupOnLeftOrRight,IsAffineCrystGroupOnRight);
InstallTrueMethod(IsAffineCrystGroupOnLeftOrRight,IsAffineCrystGroupOnLeft);
#############################################################################
##
#M TransposedMatrixGroup( <S> ) . . . . . . . .transpose of AffineCrystGroup
##
InstallMethod( TransposedMatrixGroup,
true, [ IsAffineCrystGroupOnLeftOrRight ], 0,
function( S )
local gen, grp;
gen := List( GeneratorsOfGroup( S ), TransposedMat );
grp := Group( gen, One( S ) );
if IsAffineCrystGroupOnRight( S ) then
SetIsAffineCrystGroupOnLeft( grp, true );
else
SetIsAffineCrystGroupOnRight( grp, true );
fi;
if HasTranslationBasis( S ) then
AddTranslationBasis( grp, TranslationBasis( S ) );
fi;
SetTransposedMatrixGroup( grp, S );
UseIsomorphismRelation( S, grp );
return grp;
end );
#############################################################################
##
#M InternalBasis( S ) . . . . . . . . . . . . . . . . . . . . internal basis
##
InstallMethod( InternalBasis,
true, [ IsAffineCrystGroupOnLeftOrRight ], 0,
function( S )
local d, T, basis, comp, i, j, k, mat;
d := DimensionOfMatrixGroup( S ) - 1;
T := TranslationBasis( S );
if Length( T ) = d then
basis := T;
elif Length( T ) = 0 then
basis := IdentityMat( d );
else
comp := NullMat( d - Length(T), d );
i:=1; j:=1; k:=1;
while i <= Length( T ) do
while T[i][j] = 0 do
comp[k][j] := 1;
k := k+1; j:=j+1;
od;
i := i+1; j := j+1;
od;
while j <= d do
comp[k][j] := 1;
k := k+1; j:=j+1;
od;
basis := Concatenation( T, comp );
fi;
SetIsStandardAffineCrystGroup( S, basis = IdentityMat( d ) );
if not IsStandardAffineCrystGroup( S ) then
mat := IdentityMat( d+1 );
mat{[1..d]}{[1..d]} := basis;
if IsAffineCrystGroupOnRight( S ) then
S!.lconj := mat;
S!.rconj := mat^-1;
else
mat := TransposedMat( mat );
S!.lconj := mat^-1;
S!.rconj := mat;
fi;
fi;
return basis;
end );
#############################################################################
##
#F TranslationBasisFun( S ) . . . . . determine basis of translation lattice
##
TranslationBasisFun := function ( S )
local d, P, Sgens, Pgens, trans, g, m, F, Fgens, rel, new;
if IsAffineCrystGroupOnLeft( S ) then
Error( "use only for an AffineCrystGroupOnRight" );
fi;
d := DimensionOfMatrixGroup( S ) - 1;
P := PointGroup( S );
Pgens := [];
Sgens := [];
trans := [];
# first the obvious translations
for g in GeneratorsOfGroup( S ) do
m := g{[1..d]}{[1..d]};
if IsOne( m ) then
Add( trans, g[d+1]{[1..d]} );
else
Add( Sgens, g );
Add( Pgens, m );
fi;
od;
# then the hidden translations
if not IsTrivial( P ) then
F := Image( IsomorphismFpGroupByGenerators( P, Pgens ) );
Fgens := GeneratorsOfGroup( FreeGroupOfFpGroup( F ) );
for rel in RelatorsOfFpGroup( F ) do
new := MappedWord( rel, Fgens, Sgens );
Add( trans, new[d+1]{[1..d]} );
od;
fi;
# make translations invariant under point group
trans := Set( Union( Orbits( P, trans ) ) );
return ReducedLatticeBasis( trans );
end;
#############################################################################
##
#M AddTranslationBasis( S, basis ) . . . . .add basis of translation lattice
##
InstallGlobalFunction( AddTranslationBasis, function ( S, basis )
local T;
if not IsAffineCrystGroupOnLeftOrRight( S ) then
Error("S must be an AffineCrystGroup");
fi;
T := ReducedLatticeBasis( basis );
if HasTranslationBasis( S ) then
if T <> TranslationBasis( S ) then
Error("adding incompatible translation basis attempted");
fi;
else
SetTranslationBasis( S, T );
if not IsStandardAffineCrystGroup( S ) then
InternalBasis( S ); # computes S!.lconj, S!.rconj
fi;
fi;
end );
#############################################################################
##
#M TranslationBasis( S ) . . . . . . . . . . . .basis of translation lattice
##
InstallMethod( TranslationBasis, true, [ IsAffineCrystGroupOnLeftOrRight ],0,
function( S )
local T;
if IsAffineCrystGroupOnRight( S ) then
T := TranslationBasisFun( S );
else
T := TranslationBasis( TransposedMatrixGroup( S ) );
fi;
AddTranslationBasis( S, T );
return T;
end );
#############################################################################
##
#M CheckTranslationBasis( S ) . . . . . . check basis of translation lattice
##
InstallGlobalFunction( CheckTranslationBasis, function( S )
local T;
if IsAffineCrystGroupOnRight( S ) then
T := TranslationBasisFun( S );
else
T := TranslationBasisFun( TransposedMatrixGroup( S ) );
fi;
if HasTranslationBasis( S ) then
if T <> TranslationBasis( S ) then
Print( "#W Warning: translations are INCORRECT - you better\n",
"#W start again with a fresh group!\n" );
fi;
else
AddTranslationBasis( S, T );
fi;
end );
#############################################################################
##
#M \^( S, conj ) . . . . . . . . . . . . . . . . . . . . . . . change basis
##
InstallOtherMethod( \^,
IsCollsElms, [ IsAffineCrystGroupOnRight, IsMatrix ], 0,
function ( S, conj )
local d, c, C, Ci, gens, i, R, W, r, w, t;
d := DimensionOfMatrixGroup( S ) - 1;
if not IsAffineMatrixOnRight( conj ) then
Error( "conj must represent an affine transformation" );
fi;
# get the conjugators;
C := conj;
Ci := conj^-1;
c := C {[1..d]}{[1..d]};
t := C [d+1]{[1..d]}; # Translation
# conjugate the generators of S
gens := ShallowCopy( GeneratorsOfGroup( S ) );
for i in [1..Length(gens)] do
gens[i] := Ci * gens[i] * C;
od;
R := AffineCrystGroupOnRight( gens, One( S ) );
# add translations if known
if HasTranslationBasis( S ) then
AddTranslationBasis( R, TranslationBasis( S ) * c );
fi;
# add Wyckoff positions if known
if HasWyckoffPositions( S ) then
W := [];
for w in WyckoffPositions( S ) do
r := rec( basis := w!.basis,
translation := w!.translation*c + t,
class := w!.class,
spaceGroup := R );
if r.basis <> [] then r.basis := r.basis * c; fi;
ReduceAffineSubspaceLattice( r );
Add( W, WyckoffPositionObject( r ) );
od;
SetWyckoffPositions( R, W );
fi;
return R;
end );
InstallOtherMethod( \^,
IsCollsElms, [ IsAffineCrystGroupOnLeft, IsMatrix ], 0,
function ( S, conj )
local d, c, C, Ci, gens, i, R, W, r, w, t;
d := DimensionOfMatrixGroup( S ) - 1;
if not IsAffineMatrixOnLeft( conj ) then
Error( "conj must represent an affine transformation" );
fi;
# get the conjugators;
C := conj;
Ci := conj^-1;
c := TransposedMat( C {[1..d]}{[1..d]} );
t := C {[1..d]}[d+1]; # Translation
# conjugate the generators of S
gens := ShallowCopy( GeneratorsOfGroup( S ) );
for i in [1..Length(gens)] do
gens[i] := C * gens[i] * Ci;
od;
R := AffineCrystGroupOnLeft( gens, One( S ) );
# add translations if known
if HasTranslationBasis( S ) then
AddTranslationBasis( R, TranslationBasis( S ) * c );
fi;
# add Wyckoff positions if known
if HasWyckoffPositions( S ) then
W := [];
for w in WyckoffPositions( S ) do
r := rec( basis := w!.basis,
translation := w!.translation*c + t,
class := w!.class,
spaceGroup := R );
if r.basis <> [] then r.basis := r.basis * c; fi;
ReduceAffineSubspaceLattice( r );
Add( W, WyckoffPositionObject( r ) );
od;
SetWyckoffPositions( R, W );
fi;
return R;
end );
#############################################################################
##
#M StandardAffineCrystGroup( S ) . . . . . . . . . . change basis to std rep
##
InstallGlobalFunction( StandardAffineCrystGroup, function( S )
local B, d, C;
if IsAffineCrystGroupOnRight( S ) then
B := InternalBasis( S );
elif IsAffineCrystGroupOnLeft( S ) then
B := TransposedMat( InternalBasis( S ) );
else
Error( "S must be an AffineCrystGroup" );
fi;
d := DimensionOfMatrixGroup( S ) - 1;
C := IdentityMat( d+1 );
C{[1..d]}{[1..d]} := B^-1;
return S^C;
end );
#############################################################################
##
#M Size( S ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Size
##
InstallMethod( Size,
"for AffineCrystGroup",
true, [ IsAffineCrystGroupOnLeftOrRight ], 0,
function( S )
if Length( TranslationBasis( S ) ) > 0 then
return infinity;
else
return Size( PointGroup( S ) );
fi;
end );
#############################################################################
##
#M IsFinite( S ) . . . . . . . . . . . . . . . . . . . . . . . . . .IsFinite
##
InstallMethod( IsFinite, "for AffineCrystGroup",
true, [ IsAffineCrystGroupOnLeftOrRight ], 0,
S -> Length( TranslationBasis( S ) ) = 0 );
#############################################################################
##
#M EnumeratorSorted( S ) . . . . . . . . . . EnumeratorSorted for CrystGroup
##
InstallMethod( EnumeratorSorted,
"for AffineCrystGroup",
true, [ IsAffineCrystGroupOnLeftOrRight ], 0,
function( S )
if not IsFinite( S ) then
Error("S is infinite");
else
TryNextMethod();
fi;
end );
#############################################################################
##
#M Enumerator( S ) . . . . . . . . . . . . . . . . Enumerator for CrystGroup
##
InstallMethod( Enumerator,
"for AffineCrystGroup",
true, [ IsAffineCrystGroupOnLeftOrRight ], 0,
function( S )
if not IsFinite( S ) then
Error("S is infinite");
else
TryNextMethod();
fi;
end );
#############################################################################
##
#M TransParts( S ) reduced transl. parts of GeneratorsSmallest of point grp
##
InstallMethod( TransParts, true, [ IsAffineCrystGroupOnLeftOrRight ], 0,
function( S )
local T, P, H, d, gens;
T := TranslationBasis( S );
P := PointGroup( S );
H := PointHomomorphism( S );
d := DimensionOfMatrixGroup( P );
gens := GeneratorsSmallest( P );
gens := List( gens, x -> PreImagesRepresentative( H, x ) );
if IsAffineCrystGroupOnRight( S ) then
gens := List( gens, x -> VectorModL( x[d+1]{[1..d]}, T ) );
else
gens := List( gens, x -> VectorModL( x{[1..d]}[d+1], T ) );
fi;
return gens;
end );
#############################################################################
##
#M \<( S1, S2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . \<
##
AffineCrystGroupLessFun := function( S1, S2 )
local T1, T2, P1, P2;
# first compare the translation lattices
T1 := TranslationBasis( S1 );
T2 := TranslationBasis( S2 );
if not T1 = T2 then
return T1 < T2;
fi;
# then the point groups
P1 := PointGroup( S1 );
P2 := PointGroup( S2 );
if not P1 = P2 then
return P1 < P2;
fi;
# finally the translation parts
return TransParts( S1 ) < TransParts( S2 );
end;
InstallMethod( \<, "two AffineCrystGroupOnRight", IsIdenticalObj,
[ IsAffineCrystGroupOnRight, IsAffineCrystGroupOnRight ], 0,
AffineCrystGroupLessFun
);
InstallMethod( \<, "two AffineCrystGroupOnLeft", IsIdenticalObj,
[ IsAffineCrystGroupOnLeft, IsAffineCrystGroupOnLeft ], 0,
AffineCrystGroupLessFun
);
#############################################################################
##
#M \in( m, S ) . . . . . . . . . . . . . . . .check membership in CrystGroup
##
InstallMethod( \in, "for CrystGroup",
IsElmsColls, [ IsMatrix, IsAffineCrystGroupOnLeftOrRight ], 0,
function( m, S )
local d, P, mm, t;
if not DimensionsMat( m ) = DimensionsMat( One(S) ) then
return false;
fi;
d := DimensionOfMatrixGroup( S ) - 1;
P := PointGroup( S );
mm := m{[1..d]}{[1..d]};
if not mm in P then
return false;
fi;
mm := PreImagesRepresentativeNC( PointHomomorphism( S ), mm );
if IsAffineCrystGroupOnRight( S ) then
if not IsAffineMatrixOnRight( m ) then
return false;
fi;
t := m[d+1]{[1..d]} - mm[d+1]{[1..d]};
else
if not IsAffineMatrixOnLeft( m ) then
return false;
fi;
t := m{[1..d]}[d+1] - mm{[1..d]}[d+1];
fi;
return 0*t = VectorModL( t, TranslationBasis( S ) );
end );
#############################################################################
##
#M IsSpaceGroup( S ) . . . . . . . . . . . . . . . . . . is S a space group?
##
InstallMethod( IsSpaceGroup, true, [ IsCyclotomicMatrixGroup ], 0,
function( S )
local d;
if IsAffineCrystGroupOnLeftOrRight( S ) then
d := DimensionOfMatrixGroup( S ) - 1;
return d = Length( TranslationBasis( S ) );
else
return false;
fi;
end );
#############################################################################
##
#M IsSymmorphicSpaceGroup( S ) . . . . . . . . . . . . . . .is S symmorphic?
##
InstallMethod( IsSymmorphicSpaceGroup,
"generic method", true, [ IsAffineCrystGroupOnLeftOrRight ], 0,
function ( S )
local d, gen, mat, vec, sol;
d := DimensionOfMatrixGroup( S ) - 1;
gen := GeneratorsOfGroup( StandardAffineCrystGroup(S) );
gen := Filtered( List( gen, g -> g - One(S) ),
m -> not IsZero( m{[1..d]}{[1..d]} ) );
if IsEmpty( gen ) then gen := [ One(S) ]; fi;
if IsAffineCrystGroupOnLeft(S) then
gen := List( gen, TransposedMat );
fi;
mat := List( [1..d], i -> Concatenation( List( gen, g -> g[i]{[1..d]} ) ) );
vec := List( Concatenation( List( gen, g -> g[d+1]{[1..d]} ) ), FractionModOne );
sol := SolveInhomEquationsModZ( mat, vec, true );
return not IsEmpty( sol[1] );
end );
#############################################################################
##
#M IsStandardAffineCrystGroup( S ) . . . . . . . . . is S in standard form?
##
InstallMethod( IsStandardAffineCrystGroup,
true, [ IsCyclotomicMatrixGroup ], 0,
function( S )
local d, T;
if IsAffineCrystGroupOnLeftOrRight( S ) then
d := DimensionOfMatrixGroup( S ) - 1;
return InternalBasis( S ) = IdentityMat( d );
else
return false;
fi;
end );
#############################################################################
##
#F PointGroupHomomorphism( S ) . . . . . . . . . . . .PointGroupHomomorphism
##
PointGroupHomomorphism := function( S )
local d, gen, im, I, Pgens, Sgens, i, P, nice, N, perms, lift, H;
d := DimensionOfMatrixGroup( S ) - 1;
gen := GeneratorsOfGroup( S );
im := List( gen, m -> m{[1..d]}{[1..d]} );
I := IdentityMat( d );
Pgens := [];
Sgens := [];
for i in [1..Length( im )] do
if im[i] <> I and not im[i] in Pgens then
Add( Pgens, im[i] );
Add( Sgens, MutableMatrix( gen[i] ) );
fi;
od;
P := GroupByGenerators( Pgens, I );
SetIsPointGroup( P, true );
SetAffineCrystGroupOfPointGroup( P, S );
if not IsFinite( P ) then
Error( "AffineCrystGroups must have a *finite* point group" );
fi;
nice := NiceMonomorphism( P );
N := NiceObject( P );
perms := List( Pgens, x -> ImagesRepresentative( nice, x ) );
lift := GroupGeneralMappingByImagesNC( N, S, perms, Sgens : noassert );
SetNiceToCryst( P, lift );
H := GroupHomomorphismByImagesNC( S, P, gen, im : noassert );
SetIsPointHomomorphism( H, true );
return [ P, H ];
end;
#############################################################################
##
#M PointGroup( S ) . . . . . . . . . . . . PointGroup of an AffineCrystGroup
##
InstallMethod( PointGroup, true, [ IsAffineCrystGroupOnLeftOrRight ], 0,
function( S )
local res;
res := PointGroupHomomorphism( S );
SetPointHomomorphism( S, res[2] );
return res[1];
end );
#############################################################################
##
#M PointHomomorphism( S ) . . . . . PointHomomorphism of an AffineCrystGroup
##
InstallMethod( PointHomomorphism, true, [IsAffineCrystGroupOnLeftOrRight], 0,
function( S )
local res;
res := PointGroupHomomorphism( S );
SetPointGroup( S, res[1] );
return res[2];
end );
#############################################################################
##
#M IsPointGroup( <P> ) . . . . . . . . . PointGroup of an AffineCrystGroup?
##
# PointGroups always know that they are PointGroups
InstallMethod( IsPointGroup,
"fallback method", true, [ IsCyclotomicMatrixGroup ], 0, P -> false );
#############################################################################
##
#M IsSubset( <G>, <U> ) . . . . . . . . . . . . . . . for AffineCrystGroups
##
InstallMethod( IsSubset, IsIdenticalObj,
[ IsAffineCrystGroupOnRight, IsAffineCrystGroupOnLeft ], 0, ReturnFalse);
InstallMethod( IsSubset, IsIdenticalObj,
[ IsAffineCrystGroupOnLeft, IsAffineCrystGroupOnRight ], 0, ReturnFalse);
#############################################################################
##
## Identification and construction of affine crystallographic groups
##
#############################################################################
#############################################################################
##
#M IsAffineCrystGroupOnRight( <S> ) . . . . AffineCrystGroup acting OnRight
##
# Subgroups of AffineCrystGroups are AffineCrystGroups
InstallSubsetMaintenance( IsAffineCrystGroupOnRight,
IsAffineCrystGroupOnRight, IsCollection );
# AffineCrystGroups always know that they are AffineCrystGroups
InstallMethod( IsAffineCrystGroupOnRight,
"fallback method", true, [ IsCyclotomicMatrixGroup ], 0, S -> false );
#############################################################################
##
#M IsAffineCrystGroupOnLeft( <S> ) . . . . . AffineCrystGroup acting OnLeft
##
# Subgroups of AffineCrystGroups are AffineCrystGroups
InstallSubsetMaintenance( IsAffineCrystGroupOnLeft,
IsAffineCrystGroupOnLeft, IsCollection );
# AffineCrystGroups always know that they are AffineCrystGroups
InstallMethod( IsAffineCrystGroupOnLeft,
"fallback method", true, [ IsCyclotomicMatrixGroup ], 0, S -> false );
#############################################################################
##
#M IsAffineCrystGroup( <S> ) . . . . . . . . . . . . AffineCrystGroup acting
#M . . . . . . . . . . . . . . . . . as specified by CrystGroupDefaultAction
##
InstallGlobalFunction( IsAffineCrystGroup, function( S )
if CrystGroupDefaultAction = RightAction then
return IsAffineCrystGroupOnRight( S );
elif CrystGroupDefaultAction = LeftAction then
return IsAffineCrystGroupOnLeft( S );
else
Error(" CrystGroupDefaultAction must be RightAction or LeftAction" );
fi;
end );
#############################################################################
##
#M AffineCrystGroupOnRight( <gens> ) . . . . . . . . . . . . . . . . . . . .
#M AffineCrystGroupOnRight( <genlist> ) . . . . . . . . . . . . . . . . . .
#M AffineCrystGroupOnRight( <genlist>, <identity> ) . . . . . . constructor
##
InstallGlobalFunction( AffineCrystGroupOnRight, function( arg )
local G;
G := CallFuncList( Group, arg );
return AsAffineCrystGroupOnRight( G );
end );
InstallGlobalFunction( AffineCrystGroupOnRightNC, function( arg )
local G;
G := CallFuncList( Group, arg );
SetIsAffineCrystGroupOnRight( G, true );
return G;
end );
#############################################################################
##
#M AffineCrystGroupOnLeft( <gens> ) . . . . . . . . . . . . . . . . . . . .
#M AffineCrystGroupOnLeft( <genlist> ) . . . . . . . . . . . . . . . . . . .
#M AffineCrystGroupOnLeft( <genlist>, <identity> ) . . . . . . . constructor
##
InstallGlobalFunction( AffineCrystGroupOnLeft, function( arg )
local G;
G := CallFuncList( Group, arg );
return AsAffineCrystGroupOnLeft( G );
end );
InstallGlobalFunction( AffineCrystGroupOnLeftNC, function( arg )
local G;
G := CallFuncList( Group, arg );
SetIsAffineCrystGroupOnLeft( G, true );
return G;
end );
#############################################################################
##
#M AffineCrystGroup( <gens> ) . . . . . . . . . . . . . . . . . . . . . . .
#M AffineCrystGroup( <genlist> ) . . . . . . . . . . . . . . . . . . . . . .
#M AffineCrystGroup( <genlist>, <identity> ) . . . . . . . . . . constructor
##
InstallGlobalFunction( AffineCrystGroup, function( arg )
local G;
G := CallFuncList( Group, arg );
if CrystGroupDefaultAction = RightAction then
return AsAffineCrystGroupOnRight( G );
else
return AsAffineCrystGroupOnLeft( G );
fi;
end );
InstallGlobalFunction( AffineCrystGroupNC, function( arg )
local G;
G := CallFuncList( Group, arg );
if CrystGroupDefaultAction = RightAction then
SetIsAffineCrystGroupOnRight( G, true );
return G;
else
SetIsAffineCrystGroupOnLeft( G, true );
return G;
fi;
end );
#############################################################################
##
#M AsAffineCrystGroupOnRight( S ) . . . . . . . . . . . convert matrix group
##
InstallGlobalFunction( AsAffineCrystGroupOnRight, function( S )
local ph;
if HasIsAffineCrystGroupOnRight( S ) then
if IsAffineCrystGroupOnRight( S ) then
return S;
else
S := Group( GeneratorsOfGroup( S ), One( S ) );
fi;
fi;
# an AffineCrystGroup cannot act both OnLeft and OnRight
if IsAffineCrystGroupOnLeft( S ) then
S := Group( GeneratorsOfGroup( S ), One( S ) );
fi;
# do a few basic checks
if ForAny( GeneratorsOfGroup( S ),
x -> not IsAffineMatrixOnRight( x ) ) then
Error("this group can not be made an AffineCrystGroupOnRight");
fi;
# check if PointGroup is finite
ph := PointGroupHomomorphism( S );
# if check did not fail, we can make S an AffineCrystGroupOnRight
SetIsAffineCrystGroupOnRight( S, true );
SetPointGroup( S, ph[1] );
SetPointHomomorphism( S, ph[2] );
return S;
end );
#############################################################################
##
#M AsAffineCrystGroupOnLeft( S ) . . . . . . . . . . . convert matrix group
##
InstallGlobalFunction( AsAffineCrystGroupOnLeft, function( S )
local ph;
if HasIsAffineCrystGroupOnLeft( S ) then
if IsAffineCrystGroupOnLeft( S ) then
return S;
else
S := Group( GeneratorsOfGroup( S ), One( S ) );
fi;
fi;
# an AffineCrystGroup cannot act both OnLeft and OnRight
if IsAffineCrystGroupOnRight( S ) then
S := Group( GeneratorsOfGroup( S ), One( S ) );
fi;
# do a few basic checks
if ForAny( GeneratorsOfGroup( S ),
x -> not IsAffineMatrixOnLeft( x ) ) then
Error("this group can not be made an AffineCrystGroupOnLeft");
fi;
# check if PointGroup is finite
ph := PointGroupHomomorphism( S );
# if check did not fail, we can make S an AffineCrystGroupOnLeft
SetIsAffineCrystGroupOnLeft( S, true );
SetPointGroup( S, ph[1] );
SetPointHomomorphism( S, ph[2] );
return S;
end );
#############################################################################
##
#F AsAffineCrystGroup( <S> ) . . . . . . . . . . . . . convert matrix group
##
InstallGlobalFunction( AsAffineCrystGroup, function( S )
if CrystGroupDefaultAction = RightAction then
return AsAffineCrystGroupOnRight( S );
else
return AsAffineCrystGroupOnLeft( S );
fi;
end );
#############################################################################
##
#M CanEasilyTestMembership( <grp> )
##
InstallTrueMethod( CanEasilyTestMembership, IsAffineCrystGroupOnLeftOrRight);
#############################################################################
##
#M CanComputeSize( <grp> )
##
InstallTrueMethod( CanComputeSize, IsAffineCrystGroupOnLeftOrRight );
#############################################################################
##
#M CanComputeSizeAnySubgroup( <grp> )
##
InstallTrueMethod(CanComputeSizeAnySubgroup,IsAffineCrystGroupOnLeftOrRight);
#############################################################################
##
#M CanComputeIndex( <G>, <H> )
##
InstallMethod( CanComputeIndex, IsIdenticalObj,
[IsAffineCrystGroupOnRight,IsAffineCrystGroupOnRight], 0, ReturnTrue );
InstallMethod( CanComputeIndex, IsIdenticalObj,
[IsAffineCrystGroupOnLeft,IsAffineCrystGroupOnLeft], 0, ReturnTrue );
#############################################################################
##
#M CanComputeIsSubset( <G>, <H> )
##
InstallMethod( CanComputeIsSubset, IsIdenticalObj,
[IsAffineCrystGroupOnRight,IsAffineCrystGroupOnRight], 0, ReturnTrue );
InstallMethod( CanComputeIsSubset, IsIdenticalObj,
[IsAffineCrystGroupOnLeft,IsAffineCrystGroupOnLeft], 0, ReturnTrue );
#############################################################################
##
#M HirschLength( <S> ) . . . . . . . . . . . . . . . . .Hirsch length of <S>
##
InstallMethod( HirschLength,
true, [ IsAffineCrystGroupOnLeftOrRight ], 0,
function( S )
return Length( TranslationBasis( S ) );
end );
