[GAP Forum] Repsn: constructing representations with real coefficients

[Dima Pasechnik]

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

While it's not (yet) implemented in GAP, it's always possible to realise such 
a representation over real cyclotomics, i.e. intersection of Q(zeta_n) and R,
see https://arxiv.org/abs/2107.03452

Basically, one needs to modify Dixon's algorithm to work with representations  induced from some
real 2-dimenional representations of certain subgroups of G.

Alternatively, one can always transform your representations realised over Q(zeta_n) to
ones over real cyclotomics; in the case of even dimension this needs solving
"norm equation over cyclotomics", something that does not seem to be implemented in GAP,
but this is implemented in e.g. Pari and Sagemath.

Best,
Dima
 

> 
> Best,
> 
> Denis Rosset
> 
> University of Geneva
> 
> 
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