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misc-gap's Issues

Another method to implement LinearRepresentationFromCharacter.

I noticed the function LinearRepresentationFromCharacter defined as follows:

misc-gap/minlin.g

Lines 90 to 112 in 8a583e8

LinearRepresentationFromCharacter := function( chi )
local grp, irrs, mults, reps, gens, imgs, i, mul, phi, j, img, k;
irrs := ConstituentsOfCharacter(chi);
mults := MatScalarProducts( irrs, [chi] )[1];
grp := UnderlyingGroup(chi);
reps := IrreducibleRepresentationsDixon(grp, irrs);
gens := GeneratorsOfGroup( grp );
imgs := List( gens, x -> fail );
for i in [1..Size(mults)] do
mul := mults[i];
phi := Irr(grp)[i];
for j in [1..Size(gens)] do
img := Image(reps[i], gens[j]);
for k in [1..mul] do
if imgs[j] = fail
then imgs[j] := img;
else imgs[j] := DirectSumMat( imgs[j], img );
fi;
od;
od;
od;
return GroupHomomorphismByImages( grp, gens, imgs );
end;

However, it seems that this can be done in a more concise way, as follows:

gap> gensSmallGroup16:=[
>   [[-E(4), 0, 0, 0],
>   [0, E(4), 0, 0],
>   [0, 0, 0, -E(4)],
>   [0, 0, -E(4), 0]],
>   [[0, 1, 0, 0],
>   [-1, 0, 0, 0],
>   [0, 0, 0, E(4)],
>   [0, 0, E(4), 0]]
>   ];;
gap> SmallGroup16:=Group(gensSmallGroup16);
<matrix group with 2 generators>
gap> chi := MinimalFaithfulLinearCharacterUsingChars( Irr( SmallGroup16 ) );
Character( CharacterTable( <matrix group of size 16 with 2 generators> ), [ 3, -1, -E(4), E(4), -1, 1, 1, -3, -E(4), E(4) ] )
gap> irrs := ConstituentsOfCharacter(chi);
[ Character( CharacterTable( <matrix group of size 16 with 2 generators> ), [ 1, 1, -E(4), E(4), -1, 1, -1, -1, -E(4), E(4) ] ), 
  Character( CharacterTable( <matrix group of size 16 with 2 generators> ), [ 2, -2, 0, 0, 0, 0, 2, -2, 0, 0 ] ) ]
gap> grp := UnderlyingGroup(chi);
<matrix group of size 16 with 2 generators>
gap> reps := IrreducibleRepresentationsDixon(grp, irrs);
[ CompositionMapping( [ [ [ 0, -1, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, -1, 0, 0, 0, 0, 0 ], 
          [ 0, 0, 0, 0, 0, 0, 0, -1 ], [ 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, -1, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 0 ] ], 
      [ [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ -1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, -1, 0, 0, 0, 0, 0, 0 ], 
          [ 0, 0, 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 0, 0, -1, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, -1, 0, 0, 0 ] ] ] -> [ [ [ -E(4) ] ], [ [ E(4) ] ] ],
     <mapping: Group([ [ [ -E(4), 0, 0, 0 ], [ 0, E(4), 0, 0 ], [ 0, 0, 0, -E(4) ], [ 0, 0, -E(4), 0 ] ], 
      [ [ 0, 1, 0, 0 ], [ -1, 0, 0, 0 ], [ 0, 0, 0, E(4) ], [ 0, 0, E(4), 0 ] ] ]) -> Group([ [ [ 0, -1, 0, 0, 0, 0, 0, 0 ], 
          [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, -1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, -1 ], [ 0, 0, 0, 0, 0, 0, 1, 0 ], 
          [ 0, 0, 0, 0, 0, -1, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 0 ] ], [ [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], 
          [ -1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, -1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 0, 0, -1, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0 ], 
          [ 0, 0, 0, 0, -1, 0, 0, 0 ] ] ]) > ), CompositionMapping( [ [ [ 0, -1, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], 
          [ 0, 0, -1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, -1 ], [ 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, -1, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 0 ] ], 
      [ [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ -1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, -1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1 ], 
          [ 0, 0, 0, 0, 0, 0, -1, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, -1, 0, 0, 0 ] ] ] -> [ [ [ 0, -E(4) ], [ -E(4), 0 ] ], [ [ -E(4), 0 ], [ 0, E(4) ] ] ],
     <mapping: Group([ [ [ -E(4), 0, 0, 0 ], [ 0, E(4), 0, 0 ], [ 0, 0, 0, -E(4) ], [ 0, 0, -E(4), 0 ] ], 
      [ [ 0, 1, 0, 0 ], [ -1, 0, 0, 0 ], [ 0, 0, 0, E(4) ], [ 0, 0, E(4), 0 ] ] ]) -> Group([ [ [ 0, -1, 0, 0, 0, 0, 0, 0 ], 
          [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, -1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, -1 ], [ 0, 0, 0, 0, 0, 0, 1, 0 ], 
          [ 0, 0, 0, 0, 0, -1, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 0 ] ], [ [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], 
          [ -1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, -1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 0, 0, -1, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0 ], 
          [ 0, 0, 0, 0, -1, 0, 0, 0 ] ] ]) > ) ]
gap> gens := GeneratorsOfGroup( grp );
[ [ [ -E(4), 0, 0, 0 ], [ 0, E(4), 0, 0 ], [ 0, 0, 0, -E(4) ], [ 0, 0, -E(4), 0 ] ], [ [ 0, 1, 0, 0 ], [ -1, 0, 0, 0 ], [ 0, 0, 0, E(4) ], [ 0, 0, E(4), 0 ] ] ]

gap> imgrep:=List(gens,x->List(reps, y-> x^y ));
[ [ [ [ -E(4) ] ], [ [ 0, -E(4) ], [ -E(4), 0 ] ] ], [ [ [ E(4) ] ], [ [ -E(4), 0 ], [ 0, E(4) ] ] ] ]
gap> imggen:=List(imgrep, x-> DirectSumMat(x) );
[ [ [ -E(4), 0, 0 ], [ 0, 0, -E(4) ], [ 0, -E(4), 0 ] ], [ [ E(4), 0, 0 ], [ 0, -E(4), 0 ], [ 0, 0, E(4) ] ] ]
gap> IsomorphismGroups(SmallGroup16,Group(imggen));
#I  Forcing finiteness test
CompositionMapping( [ [ [ 0, -1, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, -1, 0, 0, 0, 0, 0 ], 
      [ 0, 0, 0, 0, 0, 0, 0, -1 ], [ 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, -1, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 0 ] ], 
  [ [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ -1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, -1, 0, 0, 0, 0, 0, 0 ], 
      [ 0, 0, 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 0, 0, -1, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, -1, 0, 0, 0 ] ] ] -> 
[ [ [ -E(4), 0, 0 ], [ 0, 0, -E(4) ], [ 0, -E(4), 0 ] ], [ [ E(4), 0, 0 ], [ 0, -E(4), 0 ], [ 0, 0, E(4) ] ] ], <mapping: Group(
[ [ [ -E(4), 0, 0, 0 ], [ 0, E(4), 0, 0 ], [ 0, 0, 0, -E(4) ], [ 0, 0, -E(4), 0 ] ], [ [ 0, 1, 0, 0 ], [ -1, 0, 0, 0 ], [ 0, 0, 0, E(4) ], [ 0, 0, E(4), 0 ] ] ]) -> Group(
[ [ [ 0, -1, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, -1, 0, 0, 0, 0, 0 ], 
      [ 0, 0, 0, 0, 0, 0, 0, -1 ], [ 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, -1, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 0 ] ], 
  [ [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ -1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, -1, 0, 0, 0, 0, 0, 0 ], 
      [ 0, 0, 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 0, 0, -1, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, -1, 0, 0, 0 ] ] ]) > )

But the above result, aka, imggen is different from the results given by the following method:

gap> lpchi:=LinearRepresentationFromCharacter( chi );;
gap> IsGroupHomomorphism(lpchi);
true
gap> imgchi:=MappingGeneratorsImages(lpchi)[2];
[ [ [ -E(4), 0, 0 ], [ 0, 0, -1 ], [ 0, 1, 0 ] ], [ [ E(4), 0, 0 ], [ 0, -E(4), 0 ], [ 0, 0, E(4) ] ] ]

As you can see, imggen and imgchi are different. Any hints for these two different methods?

Regards,
Zhao

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