[GAP Forum] Unitary representations

[Laurent Bartholdi]

Dear all,
I'm looking for the 2-dimensional unitary representation of SL(2,5). I
thought GAP would give it to me, but
gap> rep :=
First(IrreducibleRepresentations(SL(2,5)),r->Length(Image(r).1)=2);
CompositionMapping( [
(2,5,4,3)(6,11,16,21)(7,15,19,23)(8,12,20,24)(9,13,17,25)(10,14,18,22),
  (2,16,9)(3,21,15)(4,6,17)(5,11,23)(7,22,10)(8,12,13)(14,18,19)(20,24,25)
] ->
[ [ [ -E(5)+E(5)^4, E(5)^2+E(5)^3 ], [ E(5)^2+E(5)^3, E(5)-E(5)^4 ] ],
  [ [ E(5)+E(5)^2, -E(5)^2-E(5)^3 ], [ 1, E(5)^3+E(5)^4 ] ] ], <action
isomorphism> )

does *not* give unitary matrices.

Now, classically, I can define s =
Sum(SL(2,5),g->g^rep*TransposedMat(ComplexConjugate(g^rep))), which is the
Gram matrix of a positive-definite invariant sesquilinear form; but I don't
know how to factor s as t*TransposedMat(ComplexConjugate(t)) so as to
conjugate rep by t.

Any ideas?

Thanks in advance, Laurent