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#############################################################################
##
#A cryst2.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
##
## More methods for affine crystallographic groups
##
#############################################################################
##
#M IsSolvableGroup( S ) . . . . . . . . . . . . . . . . . . .IsSolvableGroup
##
InstallMethod( IsSolvableGroup,
"for AffineCrystGroup, via PointGroup",
true, [ IsAffineCrystGroupOnLeftOrRight ], 0,
S -> IsSolvableGroup( PointGroup( S ) ) );
#############################################################################
##
#M IsCyclic( S ) . . . . . . . . . . . . . . . . . . . . . . . . . .IsCyclic
##
InstallMethod( IsCyclic,
"for AffineCrystGroup",
true, [ IsAffineCrystGroupOnLeftOrRight ], 0,
function( S )
local P, T;
P := PointGroup(S);
T := TranslationBasis(S);
if Length(T) = 0 then
return IsCyclic(P);
elif Length(T) > 1 then
return false;
elif IsTrivial(P) then
return true;
else
return IsCyclic(P) and T=CocVecs(S);
fi;
end );
#############################################################################
##
#M Index( G, H ) . . . . . . . . . . . . . . . . . . . . . . . . . . . Index
##
InstallMethod( IndexOp, "AffineCrystGroupOnRight", IsIdenticalObj,
[ IsAffineCrystGroupOnRight, IsAffineCrystGroupOnRight ], 0,
function( G, H )
if not IsSubgroup( G, H ) then
Error( "H must be a subgroup of G" );
fi;
return IndexNC( G, H );
end );
InstallMethod( IndexOp, "AffineCrystGroupOnLeft", IsIdenticalObj,
[ IsAffineCrystGroupOnLeft, IsAffineCrystGroupOnLeft ], 0,
function( G, H )
if not IsSubgroup( G, H ) then
Error( "H must be a subgroup of G" );
fi;
return IndexNC( G, H );
end );
InstallMethod( IndexNC, "AffineCrystGroupOnRight", IsIdenticalObj,
[ IsAffineCrystGroupOnRight, IsAffineCrystGroupOnRight ], 0,
function( G, H )
local TG, TH, IP, M;
TG := TranslationBasis( G );
TH := TranslationBasis( H );
if Length( TG ) > Length( TH ) then return infinity; fi;
IP := Index( PointGroup( G ), PointGroup( H ) );
if IsFinite( G ) then
return IP;
else
M := List( TH, x -> SolutionMat( TG, x ) );
return IP * DeterminantMat( M );
fi;
end );
InstallMethod( IndexNC, "AffineCrystGroupOnLeft", IsIdenticalObj,
[ IsAffineCrystGroupOnLeft, IsAffineCrystGroupOnLeft ], 0,
function( G, H )
local TG, TH, IP, M;
TG := TranslationBasis( G );
TH := TranslationBasis( H );
if Length( TG ) > Length( TH ) then return infinity; fi;
IP := Index( PointGroup( G ), PointGroup( H ) );
if IsFinite( G ) then
return IP;
else
M := List( TH, x -> SolutionMat( TG, x ) );
return IP * DeterminantMat( M );
fi;
end );
#############################################################################
##
#M ClosureGroup( G, elm ) . . . . . . . . .closure of a group and an element
##
InstallMethod( ClosureGroup,
"AffineCrystGroupOnRight method for group and element", IsCollsElms,
[ IsAffineCrystGroupOnRight, IsMultiplicativeElementWithInverse ], 0,
function( G, elm )
local gens, C;
if not IsAffineMatrixOnRight( elm ) then
Error( "elm must be an affine matrix acting OnRight" );
fi;
gens:= GeneratorsOfGroup( G );
# try to avoid adding an element to a group that already contains it
if elm in gens or elm^-1 in gens or elm = One( G ) then
return G;
fi;
# make the closure group
C := AffineCrystGroupOnRightNC( Concatenation( gens, [ elm ] ) );
# if <G> is infinite then so is <C>
if HasIsFinite( G ) and not IsFinite( G ) then
SetIsFinite( C, false );
SetSize( C, infinity );
fi;
return C;
end );
InstallMethod( ClosureGroup,
"AffineCrystGroupOnLeft method for group and element", IsCollsElms,
[ IsAffineCrystGroupOnLeft, IsMultiplicativeElementWithInverse ], 0,
function( G, elm )
local gens, C;
if not IsAffineMatrixOnLeft( elm ) then
Error( "elm must be an affine matrix acting OnLeft" );
fi;
gens:= GeneratorsOfGroup( G );
# try to avoid adding an element to a group that already contains it
if elm in gens or elm^-1 in gens or elm = One( G ) then
return G;
fi;
# make the closure group
C := AffineCrystGroupOnLeftNC( Concatenation( gens, [ elm ] ) );
# if <G> is infinite then so is <C>
if HasIsFinite( G ) and not IsFinite( G ) then
SetIsFinite( C, false );
SetSize( C, infinity );
fi;
return C;
end );
#############################################################################
##
#M ConjugateGroup( <G>, <g> ) . . . . . . . . . . . . . . . . ConjugateGroup
##
##
InstallMethod( ConjugateGroup,
"method for AffineCrystGroupOnRight and element", IsCollsElms,
[ IsAffineCrystGroupOnRight, IsMultiplicativeElementWithInverse ], 0,
function( G, g )
local gen, H, d, T;
if not IsAffineMatrixOnRight( g ) then
Error( "g must be an affine matrix action OnRight" );
fi;
# if <G> is trivial conjugating is trivial
if IsTrivial(G) then
return G;
fi;
# create the domain
gen := List( GeneratorsOfGroup( G ), x -> g^-1 * x * g );
H := AffineCrystGroupOnRightNC( gen );
if HasTranslationBasis( G ) then
d := DimensionOfMatrixGroup( G ) - 1;
T := TranslationBasis( G );
AddTranslationBasis( H, T*g{[1..d]}{[1..d]} );
fi;
# maintain useful information
UseIsomorphismRelation( G, H );
return H;
end );
InstallMethod( ConjugateGroup,
"method for AffineCrystGroupOnLeft and element", IsCollsElms,
[ IsAffineCrystGroupOnLeft, IsMultiplicativeElementWithInverse ], 0,
function( G, g )
local gen, H, d, T;
if not IsAffineMatrixOnLeft( g ) then
Error( "g must be an affine matrix action OnLeft" );
fi;
# if <G> is trivial conjugating is trivial
if IsTrivial(G) then
return G;
fi;
# create the domain
gen := List( GeneratorsOfGroup( G ), x -> g * x * g^-1 );
H := AffineCrystGroupOnLeftNC( gen );
if HasTranslationBasis( G ) then
d := DimensionOfMatrixGroup( G ) - 1;
T := TranslationBasis( G );
AddTranslationBasis( H, T*g{[1..d]}{[1..d]} );
fi;
# maintain useful information
UseIsomorphismRelation( G, H );
return H;
end );
#############################################################################
##
#M RightCosets( G, H ) . . . . . . . . . . . . . . . . . . . . . RightCosets
##
InstallMethod( RightCosetsNC, "AffineCrystGroupOnRight",
true, [ IsAffineCrystGroupOnRight, IsAffineCrystGroupOnRight ], 0,
function( G, H )
local orb, pnt, img, gen, rep;
# first some simple checks
if Length( TranslationBasis(G) ) <> Length( TranslationBasis(H) ) then
Error("sorry, there are infinitely many cosets");
fi;
orb := [ RightCoset( H, One( H ) ) ];
for pnt in orb do
rep := Representative( pnt );
for gen in GeneratorsOfGroup( G ) do
img := RightCoset( H, rep*gen );
if not img in orb then
Add( orb, img );
fi;
od;
od;
return orb;
end );
InstallMethod( RightCosetsNC, "AffineCrystGroupOnLeft",
true, [ IsAffineCrystGroupOnLeft, IsAffineCrystGroupOnLeft ], 0,
function( G, H )
local orb, pnt, img, gen, rep;
# first some simple checks
if Length( TranslationBasis(G) ) <> Length( TranslationBasis(H) ) then
Error("sorry, there are infinitely many cosets");
fi;
orb := [ RightCoset( H, One( H ) ) ];
for pnt in orb do
rep := Representative( pnt );
for gen in GeneratorsOfGroup( G ) do
img := RightCoset( H, rep*gen );
if not img in orb then
Add( orb, img );
fi;
od;
od;
return orb;
end );
#############################################################################
##
#M CanonicalRightCosetElement( S, rep ) . . . . . CanonicalRightCosetElement
##
InstallMethod( CanonicalRightCosetElement, "for AffineCrystGroupOnRight",
IsCollsElms, [ IsAffineCrystGroupOnRight, IsObject ], 0,
function( S, rep )
local P, d, m, T, mm, res;
P := PointGroup( S );
d := DimensionOfMatrixGroup( P );
m := rep{[1..d]}{[1..d]};
T := ReducedLatticeBasis( TranslationBasis( S )*m );
mm := CanonicalRightCosetElement( P, m );
res := PreImagesRepresentativeNC( PointHomomorphism( S ), mm*m^-1 ) * rep;
res := MutableCopyMat( res );
res[d+1]{[1..d]} := VectorModL( res[d+1]{[1..d]}, T );
return res;
end );
InstallMethod( CanonicalRightCosetElement, "for AffineCrystGroupOnLeft",
IsCollsElms, [ IsAffineCrystGroupOnLeft, IsObject ], 0,
function( S, rep )
local P, d, m, T, mm, res;
P := PointGroup( S );
d := DimensionOfMatrixGroup( P );
m := rep{[1..d]}{[1..d]};
T := ReducedLatticeBasis( TranslationBasis( S )*m );
mm := CanonicalRightCosetElement( P, m );
res := PreImagesRepresentativeNC( PointHomomorphism( S ), mm*m^-1 ) * rep;
res := MutableCopyMat( res );
res{[1..d]}[d+1] := VectorModL( res{[1..d]}[d+1], T );
return res;
end );
#############################################################################
##
#M Intersection2( G1, G2 ) . . . . . . . . . intersection of two CrystGroups
##
InstallMethod( Intersection2, "two AffineCrystGroupsOnRight", IsIdenticalObj,
[ IsAffineCrystGroupOnRight, IsAffineCrystGroupOnRight ], 0,
function( G1, G2 )
local d, P1, P2, P, T1, T2, T, L, gen, gen1, gen2, orb, set,
rep, stb, pnt, i, img, sch, new, g, g1, g2, t1, t2, s, t, R;
# get the intersections of the point groups and the translation groups
d := DimensionOfMatrixGroup( G1 ) - 1;
P1 := PointGroup(G1);
P2 := PointGroup(G2);
P := Intersection( P1, P2 );
T1 := TranslationBasis( G1 );
T2 := TranslationBasis( G2 );
T := IntersectionModule( T1, T2 );
L := UnionModule( T1, T2 );
gen := GeneratorsOfGroup( P );
gen1 := List( gen, x -> PreImagesRepresentativeNC(
PointHomomorphism( G1 ), x) );
gen2 := List( gen, x -> PreImagesRepresentativeNC(
PointHomomorphism( G2 ), x)^-1 );
orb := [ MutableMatrix( One( G1 ) ) ];
set := [ One( G1 ) ];
rep := [ One( P ) ];
stb := TrivialSubgroup( P );
# get the subgroup of P that can be lifted to the intersection
for pnt in orb do
for i in [1..Length( gen )] do
img := gen2[i]*pnt*gen1[i];
img[d+1]{[1..d]} := VectorModL( img[d+1]{[1..d]}, L );
if not img in set then
Add( orb, img );
AddSet( set, img );
Add( rep, rep[Position(orb,pnt)]*gen[i] );
else
sch := rep[Position(orb,pnt)]*gen[i]
/ rep[Position(orb,img)];
if not sch in stb then
stb := ClosureGroup( stb, sch );
fi;
fi;
od;
od;
# determine the lift of stb
new := [];
for g in GeneratorsOfGroup( stb ) do
g1 := PreImagesRepresentativeNC( PointHomomorphism( G1 ), g );
g1 := AffMatMutableTrans( g1 );
if Length(T1) > 0 then
g2 := PreImagesRepresentativeNC( PointHomomorphism( G2 ), g );
t1 := g1[d+1]{[1..d]};
t2 := g2[d+1]{[1..d]};
s := IntSolutionMat( Concatenation( T1, -T2 ), t2-t1 );
g1[d+1]{[1..d]} := t1+s{[1..Length(T1)]}*T1;
fi;
Add( new, g1 );
od;
# add the translations
for t in T do
g1 := IdentityMat( d+1 );
g1[d+1]{[1..d]} := t;
Add( new, g1 );
od;
R := AffineCrystGroupOnRightNC( new, One( G1 ) );
AddTranslationBasis( R, T );
return R;
end );
InstallMethod( Intersection2, "two AffineCrystGroupsOnLeft", IsIdenticalObj,
[ IsAffineCrystGroupOnLeft, IsAffineCrystGroupOnLeft ], 0,
function( G1, G2 )
local T1, T2, I;
T1 := TransposedMatrixGroup( G1 );
T2 := TransposedMatrixGroup( G2 );
I := Intersection2( T1, T2 );
return TransposedMatrixGroup( I );
end );
#############################################################################
##
#M NormalizerPointGroupInGLnZ( <P> ) . . . . . . .Normalizer of a PointGroup
##
InstallMethod( NormalizerPointGroupInGLnZ,
true, [ IsPointGroup ], 0,
function( P )
local S, T;
S := AffineCrystGroupOfPointGroup( P );
T := InternalBasis( S );
if T <> One(P) then
return NormalizerInGLnZ( P^(T^-1) )^T;
else
return NormalizerInGLnZ( P );
fi;
end );
#############################################################################
##
#M CentralizerPointGroupInGLnZ( G ) . . . . . . .Centralizer of a PointGroup
##
InstallMethod( CentralizerPointGroupInGLnZ, "via NormalizerPointGroupInGLnZ",
true, [ IsPointGroup ], 0,
function( G )
return Centralizer( NormalizerPointGroupInGLnZ( G ), G );
end );
#############################################################################
##
#F CentralizerElement
##
CentralizerElement := function( G, u, TT )
local d, I, U, L, orb, set, rep, stb, pnt, gen, img, sch, v;
d := DimensionOfMatrixGroup( G ) - 1;
I := IdentityMat( d );
U := List( TT, t -> t * (u{[1..d]}{[1..d]} - I) );
L := ReducedLatticeBasis( U );
orb := [ MutableMatrix( u ) ];
set := [ u ];
rep := [ MutableMatrix( One( G ) ) ];
stb := TrivialSubgroup( G );
for pnt in orb do
for gen in GeneratorsOfGroup( G ) do
img := pnt^gen;
# reduce image mod L
img[d+1]{[1..d]} := VectorModL( img[d+1]{[1..d]}, L );
if not img in set then
Add( orb, img );
AddSet( set, img );
Add( rep, rep[Position(orb,pnt)]*gen );
else
sch := rep[Position(orb,pnt)]*gen
/ rep[Position(orb,img)];
# check if a translation conjugate of sch is in stabilizer
v := u^sch - u;
v := v[d+1]{[1..d]};
if v <> 0 * v then
Assert(0, U <> [] );
v := IntSolutionMat( U, v );
Assert( 0, v <> fail );
sch[d+1]{[1..d]} := sch[d+1]{[1..d]} + v*TT;
fi;
stb := ClosureGroup( stb, sch );
fi;
od;
od;
return stb;
end;
#############################################################################
##
#F CentralizerAffineCrystGroup( G, obj ) . . centralizer of subgroup/element
##
CentralizerAffineCrystGroup := function ( G, obj )
local d, P, T, e, I, M, m, L, i, U, o, gen, Q, C, u;
d := DimensionOfMatrixGroup( G ) - 1;
P := PointGroup( G );
T := TranslationBasis( G );
e := Length( T );
I := IdentityMat( d );
# we first determine the subgroup of G that centralizes the
# point group and the translation group of obj or its span
if IsGroup( obj ) then
M := PointGroup( obj );
L := List( [1..e], x -> [] );
gen := GeneratorsOfGroup( M );
for i in [ 1..Length( gen ) ] do
L{[1..e]}{[1..d]+(i-1)*d} := T*(gen[i]-I);
od;
P := Centralizer( P, M );
if not IsEmpty( TranslationBasis( obj ) ) then
P := Stabilizer( P, TranslationBasis( obj ), OnRight );
fi;
U := Filtered( GeneratorsOfGroup(obj), x -> x{[1..d]}{[1..d]} <> I );
else
if not IsAffineMatrixOnRight( obj ) then
Error( "obj must be an affine matrix acting OnRight" );
fi;
M := obj{[1..d]}{[1..d]};
L := T*(M - I);
P := Centralizer( P, M );
o := Order( M );
m := obj^o;
P := Stabilizer( P, m[d+1]{[1..d]} );
if o > 1 then U := [ obj ]; else U := []; fi;
fi;
gen := List( GeneratorsOfGroup( P ),
x -> PreImagesRepresentativeNC( PointHomomorphism( G ), x ) );
# if G is finite
if e = 0 then
return SubgroupNC( G, gen );
fi;
# we keep only translation generators which centralize obj
Q := IdentityMat( e );
if L <> [] then
L := RowEchelonFormT( L, Q );
fi;
for i in [ Length( L )+1..e ] do
Add( gen, AugmentedMatrix( I, Q[i]*T ) );
od;
# C centralizes the point group and the translation group of obj
C := SubgroupNC( G, gen );
# now find the centralizer for each u in U
for u in U do
C := CentralizerElement( C, u, T );
T := TranslationBasis( C );
od;
return C;
end;
#############################################################################
##
#M Centralizer( G, obj ) . . . . . . . . . . centralizer of subgroup/element
##
InstallMethod( CentralizerOp, "AffineCrystGroupOnRight and element",
IsCollsElms, [ IsAffineCrystGroupOnRight, IsMatrix ], 0,
function( G, m )
if not IsAffineMatrixOnRight( m ) then
Error( "m must be an affine matrix acting OnRight" );
fi;
return CentralizerAffineCrystGroup( G, m );
end );
InstallMethod( CentralizerOp, "AffineCrystGroupOnLeft and element",
IsCollsElms, [ IsAffineCrystGroupOnLeft, IsMatrix ], 0,
function( G, m )
local T, C;
if not IsAffineMatrixOnLeft( m ) then
Error( "m must be an affine matrix acting OnLeft" );
fi;
T := TransposedMatrixGroup( G );
C := CentralizerAffineCrystGroup( T, TransposedMat( m ) );
return TransposedMatrixGroup( C );
end );
InstallMethod( CentralizerOp, "two AffineCrystGroupsOnRight", IsIdenticalObj,
[ IsAffineCrystGroupOnRight, IsAffineCrystGroupOnRight ], 0,
function( G1, G2 )
return CentralizerAffineCrystGroup( G1, G2 );
end );
InstallMethod( CentralizerOp, "two AffineCrystGroupsOnLeft", IsIdenticalObj,
[ IsAffineCrystGroupOnLeft, IsAffineCrystGroupOnLeft ], 0,
function( G1, G2 )
local G, U, C;
G := TransposedMatrixGroup( G1 );
U := TransposedMatrixGroup( G2 );
C := CentralizerAffineCrystGroup( G, U );
return TransposedMatrixGroup( C );
end );
#############################################################################
##
#M TranslationNormalizer( S ) . . . . . . . . . . . . translation normalizer
##
InstallMethod( TranslationNormalizer, "for SpaceGroup acting OnRight",
true, [ IsAffineCrystGroupOnRight and IsSpaceGroup ], 0,
function( S )
local P, T, d, N, M, I, Pgens, B, invB, g, i, Q, L, K, k, l, j, gen;
P := PointGroup( S );
T := TranslationBasis( S );
d := DimensionOfMatrixGroup( S ) - 1;
if Size( P ) = 1 then
N := GroupByGenerators( [], IdentityMat( d+1 ) );
N!.continuousTranslations := IdentityMat( d );
return N;
fi;
M := List( [1..d], i->[] ); i := 0;
I := IdentityMat( d );
Pgens := GeneratorsOfGroup( P );
if not IsStandardAffineCrystGroup( S ) then
B := InternalBasis( S );
invB := B^-1;
Pgens := List( Pgens, x -> B * x * invB );
fi;
for g in Pgens do
g := g - I;
M{[1..d]}{[1..d]+i*d} := g;
i := i+1;
od;
# first diagonalize M
Q := IdentityMat( Length(M) );
M := TransposedMat(M);
M := RowEchelonForm( M );
while Length(M) > 0 and not IsDiagonalMat(M) do
M := TransposedMat(M);
M := RowEchelonFormT(M,Q);
if not IsDiagonalMat(M) then
M := TransposedMat(M);
M := RowEchelonForm(M);
fi;
od;
# and then determine the solutions of x*M=0 mod Z
if Length(M)>0 then
L := List( [1..Length(M)], i -> [ 0 .. M[i][i]-1 ] / M[i][i] );
L := List( Cartesian( L ), l -> l * Q{[1..Length(M)]} );
else
L := NullMat( 1, Length(Q) );
fi;
# get the kernel
if Length(M) < Length(Q) then
K := Q{[Length(M)+1..Length(Q)]};
TriangulizeMat( K );
else
K := [];
fi;
# reduce to basis modulo kernel
Append( L, IdentityMat( d ) );
for k in K do
j := PositionProperty( k, x -> x=1 );
for l in L do
l := l-l[j]*k;
od;
od;
L := ReducedLatticeBasis( L );
# conjugate if not standard
if not IsStandardAffineCrystGroup( S ) then
L := L*T;
fi;
# get generators
gen := List( L, x -> IdentityMat( d+1 ) );
for i in [1..Length(L)] do
gen[i][d+1]{[1..d]} := L[i];
od;
N := GroupByGenerators( gen, IdentityMat( d+1 ) );
N!.continuousTranslations := K;
return N;
end );
InstallMethod( TranslationNormalizer, "for SpaceGroup acting OnLeft",
true, [ IsAffineCrystGroupOnLeft and IsSpaceGroup ], 0,
function( S )
local N1, gen, N;
N1 := TranslationNormalizer( TransposedMatrixGroup( S ) );
gen := List( GeneratorsOfGroup( N1 ), TransposedMat );
N := GroupByGenerators( gen, One( N1 ) );
N!.continuousTranslations := N1!.continuousTranslations;
return N;
end );
RedispatchOnCondition( TranslationNormalizer, true,
[IsAffineCrystGroupOnRight],
[IsAffineCrystGroupOnRight and IsSpaceGroup], 0);
RedispatchOnCondition( TranslationNormalizer, true,
[IsAffineCrystGroupOnLeft],
[IsAffineCrystGroupOnLeft and IsSpaceGroup], 0);
#############################################################################
##
#F AffineLift( pnt, d )
##
AffineLift := function( pnt, d )
local M, b, i, I, p, m, Q, j, s;
M := List( [1..d], i->[] ); b := []; i := 0;
I := IdentityMat( d );
for p in pnt do
m := p[1]{[1..d]}{[1..d]} - I;
M{[1..d]}{[1..d]+i*d} := m;
Append( b, p[2] - p[1][d+1]{[1..d]} );
i := i+1;
od;
Q := IdentityMat( d );
M := TransposedMat(M);
M := RowEchelonFormVector( M,b );
while Length(M) > 0 and not IsDiagonalMat(M) do
M := TransposedMat(M);
M := RowEchelonFormT(M,Q);
if not IsDiagonalMat(M) then
M := TransposedMat(M);
M := RowEchelonFormVector(M,b);
fi;
od;
## Check if we have any solutions modulo Z.
for j in [Length(M)+1..Length(b)] do
if not IsInt( b[j] ) then
return [];
fi;
od;
s := List( [1..Length(M)], i -> b[i]/M[i][i] );
for i in [Length(M)+1..d] do
Add( s, 0);
od;
return s*Q;
end;
#############################################################################
##
#M AffineNormalizer( S ) . . . . . . . . . . . . . . . . . affine normalizer
##
InstallMethod( AffineNormalizer, "for SpaceGroup acting OnRight", true,
[ IsAffineCrystGroupOnRight and IsSpaceGroup ], 0,
function( S )
local d, P, H, T, N, Pgens, Sgens, invT, gens, Pi, Si, hom, opr, orb,
g, m, set, rep, lst, pnt, img, t, sch, n, nn, normgens, TN, AN;
d := DimensionOfMatrixGroup( S ) - 1;
P := PointGroup( S );
H := PointHomomorphism( S );
T := TranslationBasis( S );
N := NormalizerPointGroupInGLnZ( P );
Pgens := GeneratorsOfGroup( P );
Sgens := List( Pgens, x -> PreImagesRepresentativeNC( H, x ) );
# we work in a standard representation
if not IsStandardAffineCrystGroup( S ) then
invT := T^-1;
gens := List( GeneratorsOfGroup( N ), x -> T * x * invT );
Pgens := List( Pgens, x -> T * x * invT );
Sgens := List( Sgens, x -> S!.lconj * x * S!.rconj );
else
gens := GeneratorsOfGroup( N );
fi;
Pi := Group( Pgens, One( P ) );
Si := Group( Sgens, One( S ) );
hom := GroupHomomorphismByImagesNC( Si, Pi, Sgens, Pgens );
# the operation we shall need in the stabilizer algorithm
opr := function( data, g )
local m, mm, res;
m := data[1]{[1..d]}{[1..d]};
mm := m^g;
if m = mm then
res := [ data[1], List( data[2]*g, FractionModOne ) ];
else
m := AffMatMutableTrans( PreImagesRepresentativeNC( hom, mm ) );
m[d+1]{[1..d]} := List( m[d+1]{[1..d]}, FractionModOne );
res := [ m, List( data[2]*g, FractionModOne ) ];
fi;
return res;
end;
orb := [];
for g in Sgens do
m := AffMatMutableTrans( g );
m[d+1]{[1..d]} := List( m[d+1]{[1..d]}, FractionModOne );
Add( orb, [ m, m[d+1]{[1..d]} ] );
od;
orb := [ orb ];
set := ShallowCopy( orb );
rep := [ One( N ) ];
lst := [];
for pnt in orb do
for g in gens do
img := List( pnt, x -> opr( x, g ) );
if not img in set then
Add( orb, img );
AddSet( set, img );
Add( rep, rep[Position(orb,pnt)]*g );
else
t := AffineLift( img, d );
if t<>[] then
sch := rep[Position(orb,pnt)]*g;
n := IdentityMat( d+1 );
n{[1..d]}{[1..d]} := sch;
n[d+1]{[1..d]} := t;
AddSet( lst, n );
fi;
fi;
od;
od;
if IsFinite( N ) then
nn := Subgroup( N, [] );
normgens := [];
for g in lst do
m := g{[1..d]}{[1..d]};
if not m in nn then
Add( normgens, g );
nn := ClosureGroup( nn, m );
fi;
od;
else
normgens := lst;
fi;
m := IdentityMat( d+1 );
m{[1..d]}{[1..d]} := T;
if not IsStandardAffineCrystGroup( S ) then
normgens := List( normgens, x -> x^m );
fi;
TN := TranslationNormalizer( S );
Append( normgens, GeneratorsOfGroup( TN ) );
AN := Group( normgens, One( S ) );
AN!.continuousTranslations := TN!.continuousTranslations;
# can AN be made an AffineCrystGroup?
if IsFinite( N ) and AN!.continuousTranslations = [] then
SetIsAffineCrystGroupOnRight( AN, true );
lst := List( GeneratorsOfGroup( TN ), x -> x[d+1]{[1..d]} );
lst := ReducedLatticeBasis( lst );
AddTranslationBasis( AN, lst );
fi;
return AN;
end );
InstallMethod( AffineNormalizer, "for SpaceGroup acting OnLeft", true,
[ IsAffineCrystGroupOnLeft and IsSpaceGroup ], 0,
function( S )
local A1, A, gen;
A1 := AffineNormalizer( TransposedMatrixGroup( S ) );
if IsAffineCrystGroupOnRight( A1 ) then
A := TransposedMatrixGroup( A1 );
else
gen := List( GeneratorsOfGroup( A1 ), TransposedMat );
A := Group( gen, One( A1 ) );
fi;
A!.continuousTranslations := A1!.continuousTranslations;
return A;
end );
RedispatchOnCondition( AffineNormalizer, true,
[IsAffineCrystGroupOnRight],
[IsAffineCrystGroupOnRight and IsSpaceGroup], 0);
RedispatchOnCondition( AffineNormalizer, true,
[IsAffineCrystGroupOnLeft],
[IsAffineCrystGroupOnLeft and IsSpaceGroup], 0);
#############################################################################
##
#M AffineInequivalentSubgroups( S, subs ) . . reps of affine ineq. subgroups
##
InstallGlobalFunction( AffineInequivalentSubgroups, function( S, subs )
local C, A, opr, reps, orb, grp, gen, img;
if subs = [] then
return subs;
fi;
C := ShallowCopy( subs );
A := AffineNormalizer( S );
if A!.continuousTranslations <> [] then
return fail;
fi;
reps := [];
while C <> [] do
if not IsSubgroup( S, C[1] ) then
Error( "subs must be a list of subgroups of S" );
fi;
orb := [ C[1] ];
for grp in orb do
for gen in GeneratorsOfGroup( A ) do
img := List( GeneratorsOfGroup( grp ), x -> x^gen );
if not ForAny( orb, g -> ForAll( img, x -> x in g ) ) then
Add( orb, ConjugateGroup( grp, gen ) );
fi;
od;
od;
Add( reps, orb[1] );
C := Filtered( C, x -> not x in orb );
od;
return reps;
end );
[ Dauer der Verarbeitung: 0.42 Sekunden
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