run "date";
needsPackage("LieAlgebraRepresentations");
sl6 = simpleLieAlgebra("A",5);
rhoStd = standardRepresentation(sl6);
rhoW2Std = exteriorPower(2,rhoStd);
rhoW2Stdstar = dual rhoW2Std;
-- Step 2: Get V02002 in W8W2Stdstar
V02002 = irreducibleLieAlgebraModule({0,2,0,0,2},sl6);
LAB = rhoStd#"Basis";
time L02002 = GTrepresentationMatrices(V02002);
rhoV02002 = lieAlgebraRepresentation(V02002,LAB,L02002);
-- Need a hwv for V02002 in W8W2Stdstar
-- Relatively easy because the weight {0,2,0,0,2} subspace in W8W2Stdstar is one-dimensional
wts = representationWeights(rhoW2Stdstar);
s8 = subsets(apply(#wts, i -> i),8);
t = first select(s8, s -> sum apply(s, j -> wts_j)=={0,2,0,0,2})
hwv = transpose matrix {apply(s8, s -> if s==t then 1/1 else 0/1)};
VInWedgekW(rhoV02002,8,rhoW2Stdstar,hwv,"SaveAsFunction"=>"V02002inW8W2Stdstar");
-- Step 3: Get V02000 in Sym2 V02002
V02000 = irreducibleLieAlgebraModule({0,2,0,0,0},sl6);
L02000 = GTrepresentationMatrices(V02000);
rhoV02000 = lieAlgebraRepresentation(V02000,LAB,L02000);
hwv2 = weightMuHighestWeightVectorsInSymdW({0,2,0,0,0},2,rhoV02002);
VInSymdW(rhoV02000,2,rhoV02002,hwv2_0,"SaveAsFunction"=>"V02000inS2V02002");
-- Step 4: Get the invariant in Sym3 V02000
hwv3 = weightMuHighestWeightVectorsInSymdW({0,0,0,0,0},3,rhoV02000);
rhoV00000=trivialRepresentation(sl6);
rhoV00000 = lieAlgebraRepresentation(rhoV00000#"Module",rhoV02000#"Basis",rhoV00000#"RepresentationMatrices");
V00000inS3V02000=VInSymdW(rhoV00000,3,rhoV02000,hwv3_0,"SaveAsFunction"=>"V00000inS3V02000");
run "date";