[GAP Forum] tensor product of lie algebra representations

[R.N. Tsai]

Dear gap-forum,
   
  I have question regarding tensor products of representations of semisimple lie algebra.
   
  It's easier to setup the question with an example :
   
  # first define the Lie algebra A2
L   := SimpleLieAlgebra("A",2,Rationals);
   
  # define V1 as module with highest weight [1,0]; (V1 is 3 dimensional)
w1  := [1,0];
V1  := HighestWeightModule(L,w1);
   
  # define V2 as module with highest weight [0,1]; (V2 is 3 dimensional)
w2  := [0,1];
V2  := HighestWeightModule(L,w2);
   
  # define V as tensor product of V1 and V2 (V is 9 dimensional)
V  := TensorProductOfAlgebraModules([V1,V2]);
   
  # GAP knows how V decomposes into irreducible modules :
w  := DecomposeTensorProduct(L,w1,w2);
   
  # this prints the decomposition, V = [1,1] (8 dimensional) + [0,0] (1 dimension)
for i in [1..Length(w[1])] do
 Print(" weight=",w[1][i]," multiplicity=",w[2][i]," dimension=", DimensionOfHighestWeightModule(L,w[1][i]),"\n");
od;
   
  # define basis for V
B := Basis(V);
   
  My question is the following :
   
  How would I get a basis for the irreducible constituents of V in terms of B.

  So for the example above, a basis for the 8 dimensional subspace and another for the 1 dimensional one.
   
  I hope I asked the question correctly and thanks for your help.
   
  Regards,
   
  R.N.

       
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