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
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