[GAP Forum] Repsn: constructing representations with real coefficients

[Denis Rosset]

Dear forum members,

I'm constructing representations of finite groups using the Repsn 
package. It does not necessarily return representations whose images 
have real coefficients, when such constructions exist.

For example:

gap> G:=Group((1,2,3),(3,1));
Group([ (1,2,3), (1,3) ])
gap> tbl := CharacterTable(G);;
gap> chars := Irr(tbl);
[ Character( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 1, -1, 1 ] ), 
Character( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 2, 0, -1 ] ),
   Character( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 1, 1, 1 ] ) ]
gap> IrreducibleAffordingRepresentation(chars[2]);
[ (1,2,3), (1,3) ] -> [ [ [ E(3)^2, 0 ], [ 0, E(3) ] ], [ [ 0, E(3)^2 ], 
[ E(3), 0 ] ] ]

However, IrreducibleRepresentationsDixon returns a representation with 
real coefficients in that case:

gap> IrreducibleRepresentationsDixon(G);
[ [ (1,2,3), (1,3) ] -> [ [ [ 1 ] ], [ [ -1 ] ] ], [ (1,2,3), (1,3) ] -> 
[ [ [ -1, 1 ], [ -1, 0 ] ], [ [ 0, 1 ], [ 1, 0 ] ] ],
   [ (1,2,3), (1,3) ] -> [ [ [ 1 ] ], [ [ 1 ] ] ] ]

What is possible in GAP towards the construction of real-type 
(Frobenius-Schur indicator=1) representations with images having real 
coefficients?

Best,

Denis Rosset

University of Geneva