How getting the unitarized irreducible representations with GAP?

Question

The function IrreducibleRepresentations on GAP gives non-necessarily unitary representations, for example:

gap> R:=IrreducibleRepresentations(SymmetricGroup(5));;
gap> m:=(2,3)^R[5];;
gap> Display(m);
[ [   1,   0,   0,   0,   0 ],
  [   0,   1,   0,   0,   0 ],
  [   0,   0,   0,   0,   1 ],
  [  -1,  -1,   0,  -1,   0 ],
  [   0,   0,   1,   0,   0 ] ]

A unitary representation of the transposition (2,3) is supposed to be self-adjoint, but the matrix above is not.

Question: How getting the unitarized irreducible representations with GAP?

Answer 1

The algorithm by Dixon that is used by IrreducibleRepresentationDixon allows for unitary representations. The calculations however depend on square roots being available (otherwise the calculation can fail), and can involve rather awkward cyclotomics.

In case anybody cannot wait, I put the relevant code under:

https://www.dropbox.com/s/2ag7yd3mom1ulta/unitarydixon.g?dl=0

Read in the file with Reread and call IrreducibleRepresentationDixon with the unitary option:

gap> R:=IrreducibleRepresentationsDixon(SymmetricGroup(5):unitary);;
gap> m:=(2,3)^R[5];;
gap> m*GaloisCyc(TransposedMat(m),-1);  # unitary, jay!
[ [ 1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ]
gap> 10*m[2][3]; # ugly !
-E(60)^7-E(60)^11+E(60)^19+E(60)^23+E(60)^31-E(60)^43+E(60)^47-E(60)^59