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#############################################################################
##
#A fpgrp.gi Cryst library Bettina Eick
#A Franz G"ahler
#A Werner Nickel
##
#Y Copyright 1997-1999 by Bettina Eick, Franz G"ahler and Werner Nickel
##
#############################################################################
##
#M IsomorphismFpGroup( <P> ) . . . . . . . IsomorphismFpGroup for PointGroup
##
InstallMethod( IsomorphismFpGroup,
"for PointGroup", true, [ IsPointGroup ], 0,
function ( P )
local mono, N, F, gens, gensP, gensS, gensF, iso;
# compute an isomorphic permutation group
mono := NiceMonomorphism( P );
N := NiceObject( P );
# distinguish between solvable and non-solvable case
if IsSolvableGroup( N ) then
F := Image( IsomorphismFpGroupByPcgs( Pcgs( N ), "f" ) );
gens := AsList( Pcgs( N ) );
else
gens := GeneratorsOfGroup( N );
F := Image( IsomorphismFpGroupByGenerators( N, gens ) );
fi;
gensP := List( gens, x -> PreImagesRepresentativeNC( mono, x ) );
gensS := List( gens, x -> ImagesRepresentative( NiceToCryst( P ), x ) );
gensF := GeneratorsOfGroup( F );
iso := GroupHomomorphismByImagesNC( P, F, gensP, gensF );
SetMappingGeneratorsImages( iso, [ gensP, gensF ] );
SetIsBijective( iso, true );
SetKernelOfMultiplicativeGeneralMapping( iso, TrivialSubgroup( P ) );
iso!.preimagesInAffineCrystGroup := Immutable( gensS );
return iso;
end );
#############################################################################
##
#M IsomorphismFpGroup( <S> ) . . . . . . . for AffineCrystGroupOnLeftOrRight
##
InstallMethod( IsomorphismFpGroup,
"for AffineCrystGroup", true, [ IsAffineCrystGroupOnLeftOrRight ], 0,
function( S )
local P, hom, T, iso, F, gensP, relsP, matsP, d, n, t, R,
gensR, gensT, matsT, i, j, l, k, rels, relsR, rel, tail,
vec, word, gens, ims;
P := PointGroup( S );
hom := PointHomomorphism( S );
T := TranslationBasis( S );
iso := IsomorphismFpGroup( P );
F := Image( iso );
gensP := GeneratorsOfGroup( FreeGroupOfFpGroup( F ) );
relsP := RelatorsOfFpGroup( F );
matsP := iso!.preimagesInAffineCrystGroup;
d := DimensionOfMatrixGroup( S ) - 1;
n := Length( gensP );
t := Length( T );
R := FreeGroup( n + t );
gensR := GeneratorsOfGroup( R ){[1..n]};
gensT := GeneratorsOfGroup( R ){[n+1..n+t]};
matsT := List( gensT, x -> IdentityMat( d+1 ) );
for i in [1..Length( matsT )] do
if IsAffineCrystGroupOnRight( S ) then
matsT[i][d+1]{[1..d]} := T[i];
else
matsT[i]{[1..d]}[d+1] := T[i];
fi;
od;
rels := List( relsP, rel -> MappedWord( rel, gensP, gensR ) );
relsR := [];
# compute tails
for rel in rels do
tail := MappedWord( rel, gensR, matsP );
word := rel;
if t > 0 then
if IsAffineCrystGroupOnRight( S ) then
vec := SolutionMat( T, - tail[d+1]{[1..d]} );
else
vec := SolutionMat( T, - tail{[1..d]}[d+1] );
fi;
for i in [1..t] do
word := word * gensT[i]^vec[i];
od;
fi;
Add( relsR, word );
od;
# compute operation
for i in [1..n] do
for j in [1..t] do
rel := Comm( gensT[j], gensR[i] );
tail := Comm( matsT[j], matsP[i] );
if IsAffineCrystGroupOnRight( S ) then
vec := SolutionMat( T, - tail[d+1]{[1..d]} );
else
vec := SolutionMat( T, - tail{[1..d]}[d+1] );
fi;
word := rel;
for k in [1..t] do
word := word * gensT[k]^vec[k];
od;
Add( relsR, word );
od;
od;
# compute presentation of T
for i in [1..t-1] do
for j in [i+1..t] do
Add( relsR, Comm( gensT[j], gensT[i] ) );
od;
od;
# construct isomorphism
R := R / relsR;
gens := Concatenation( matsP, matsT );
ims := GeneratorsOfGroup( R );
iso := GroupHomomorphismByImagesNC( S, R, gens, ims );
SetMappingGeneratorsImages( iso, [ gens, ims ] );
SetIsFromAffineCrystGroupToFpGroup( iso, true );
SetIsBijective( iso, true );
SetKernelOfMultiplicativeGeneralMapping( iso, TrivialSubgroup( S ) );
return iso;
end );
#############################################################################
##
#M ImagesRepresentative( <iso>, <elm> ) for IsFromAffineCrystGroupToFpGroup
##
InstallMethod( ImagesRepresentative, FamSourceEqFamElm,
[IsGroupGeneralMappingByImages and IsFromAffineCrystGroupToFpGroup,
IsMultiplicativeElementWithInverse ], 0,
function( iso, elm )
local d, S, T, elmP, isoP, word, genP, len, genS, genF, elm2, v, i;
d := Length( elm ) - 1;
S := Source( iso );
T := TranslationBasis( S );
elmP := elm{[1..d]}{[1..d]};
isoP := IsomorphismFpGroup( PointGroup( S ) );
word := ImagesRepresentative( isoP, elmP );
genP := MappingGeneratorsImages( isoP )[2];
len := Length( genP );
genS := MappingGeneratorsImages( iso )[1];
genF := MappingGeneratorsImages( iso )[2];
elm2 := MappedWord( word, genP, genS{[1..len]} );
word := MappedWord( word, genP, genF{[1..len]} );
if Length( T ) > 0 then
if IsAffineCrystGroupOnRight( S ) then
v := SolutionMat( T, elm[d+1]{[1..d]} - elm2[d+1]{[1..d]} );
for i in [1..Length(v)] do
word := word * genF[len+i]^v[i];
od;
else
v := SolutionMat( T, elm{[1..d]}[d+1] - elm2{[1..d]}[d+1] );
for i in [1..Length(v)] do
word := genF[len+i]^v[i] * word;
od;
fi;
fi;
return word;
end );
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