Dear Imran,
You asked about the trace form on representations of semisimple
Lie algebras. The following example computes the matrix of this
form for an irreducible representation of the Lie algebra of
type G2. I hope this helps you.
gap> L:= SimpleLieAlgebra("G",2,Rationals);;
gap> V:= HighestWeightModule( L, [0,2] );
<77-dimensional left-module over <Lie algebra of dimension 14 over Rationals>>
gap> m:= List( Basis(L), x -> MatrixOfAction( Basis(V), x ) );;
gap> tracemat:= List( m, x -> List( m, y -> TraceMat(x*y) ) );
[ [ 0, 0, 0, 0, 0, 0, 330, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 110, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 330, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 330, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 110, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 110, 0, 0 ],
[ 330, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 110, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 330, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 330, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 110, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 110, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 660, -330 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -330, 220 ] ]
Best wishes,
Willem de Graaf
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