[GAP Forum] Repsn: constructing representations with real coefficients

[Dima Pasechnik]

Dear Denis,
On Sat, Mar 20, 2021 at 07:33:46PM +0100, Denis Rosset wrote:
> 
> 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?

While Repsn won't in general compute unitary representations, RepnDecomp can do this for you:
https://gap-packages.github.io/RepnDecomp/doc/chap3.html

This would be a first step to making a representation real (if possible).
If I recall correctly, from this point on it's just linear algebra,
finding an invariant quadratic form and diagonalising, but details escape me now.

HTH,
Dima
http://users.ox.ac.uk/~coml0531/