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/ _______________________________________________ Forum mailing list Forum@gap-system.org https://mail.gap-system.org/mailman/listinfo/forum
