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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