QuantumGroupsPath = "~/projects/QuantumGroups/trunk/package/";
AppendTo[$Path, QuantumGroupsPath];
<<QuantumGroups`

LevelFromRoot[A1,16]

AlcoveWeights[A1,16]

AlcoveDimensions[A1,6]

N[%]

N[2+2Sqrt[2]]

RootFromLevel[A5,2]

AlcoveWeights[A5,RootFromLevel[A5,2]]

AlcoveDimensions[A5,2][[13]]

N[%]

N[(2+2Sqrt[2])/(2+Sqrt[2])]

SteinbergDecomposeRepresentation[A5,16][Irrep[A5][{1,0,0,0,0}]⊗Irrep[A5][{0,0,0,0,1}]]

SteinbergDecomposeRepresentation[A5,16][Irrep[A5][{1,0,0,0,1}]⊗Irrep[A5][{1,0,0,0,0}]]

SteinbergDecomposeRepresentation[A5,16][Irrep[A5][{0,1,0,0,1}]⊗Irrep[A5][{0,0,0,0,1}]]

SteinbergDecomposeRepresentation[A5,16][Irrep[A5][{0,1,0,1,0}]⊗Irrep[A5][{1,0,0,0,0}]]

SteinbergDecomposeRepresentation[A5,16][Irrep[A5][{0,0,1,1,0}]⊗Irrep[A5][{0,0,0,0,1}]]

SteinbergDecomposeRepresentation[A5,16][Irrep[A5][{0,0,2,0,0}]⊗Irrep[A5][{1,0,0,0,0}]]

Clear[p]

p[k_]:=p[k]=DecompositionMap[A1,Irrep[A1][{3}]⊗Irrep[A1][{3}],FundamentalBasis].DirectSumInclusionMap[(Subscript[A, 1]),DirectSum[Irrep[(Subscript[A, 1])][{6}],Irrep[(Subscript[A, 1])][{4}],Irrep[(Subscript[A, 1])][{2}],Irrep[(Subscript[A, 1])][{0}]],k,FundamentalBasis].DirectSumProjectionMap[(Subscript[A, 1]),DirectSum[Irrep[(Subscript[A, 1])][{6}],Irrep[(Subscript[A, 1])][{4}],Irrep[(Subscript[A, 1])][{2}],Irrep[(Subscript[A, 1])][{0}]],k,FundamentalBasis].InverseDecompositionMap[A1,Irrep[A1][{3}]⊗Irrep[A1][{3}],FundamentalBasis]

cap=Together[qInteger[4][q]DirectSumProjectionMap[(Subscript[A, 1]),DirectSum[Irrep[(Subscript[A, 1])][{6}],Irrep[(Subscript[A, 1])][{4}],Irrep[(Subscript[A, 1])][{2}],Irrep[(Subscript[A, 1])][{0}]],4,FundamentalBasis].InverseDecompositionMap[A1,Irrep[A1][{3}]⊗Irrep[A1][{3}],FundamentalBasis]]

cup=DecompositionMap[A1,Irrep[A1][{3}]⊗Irrep[A1][{3}],FundamentalBasis].DirectSumInclusionMap[(Subscript[A, 1]),DirectSum[Irrep[(Subscript[A, 1])][{6}],Irrep[(Subscript[A, 1])][{4}],Irrep[(Subscript[A, 1])][{2}],Irrep[(Subscript[A, 1])][{0}]],4,FundamentalBasis]

cap.cup

id=IdentityMap[A1,Irrep[A1][{3}],FundamentalBasis]

V=Irrep[(Subscript[A, 1])][{3}]

(id⊗cap)[[{2,4}]]

((id⊗cap).(InverseAssociator[A1,V,V,V,FundamentalBasis].(p[1]⊗id)))[[{2,4}]]

Clear[rotR]

rotR[k_]:=rotR[k]=((id⊗cap).(InverseAssociator[A1,V,V,V,FundamentalBasis].(p[k]⊗id)))/.{U_⊗Irrep[A1][{0}]:>U,Irrep[A1][{0}]⊗U_:>U}

((((id⊗rotR[1]).InverseAssociator[A1,V,V⊗V,V,FundamentalBasis]).(InverseAssociator[A1,V,V,V,FundamentalBasis]⊗id)).(cup⊗id⊗id))[[{2,4}]]

Clear[rot]

rot[k_]:=rot[k]=((((id⊗rotR[k]).InverseAssociator[A1,V,V⊗V,V,FundamentalBasis]).(InverseAssociator[A1,V,V,V,FundamentalBasis]⊗id)).(cup⊗id⊗id))/.{U_⊗Irrep[A1][{0}]:>U,Irrep[A1][{0}]⊗U_:>U}

rot[1];

rot[k_]:=rot[k]=((((id⊗id⊗cap).Inverse[Associator[A1,Irrep[(Subscript[A, 1])][{3}]⊗Irrep[(Subscript[A, 1])][{3}],Irrep[(Subscript[A, 1])][{3}],Irrep[(Subscript[A, 1])][{3}],FundamentalBasis]]).((Associator[A1,Irrep[(Subscript[A, 1])][{3}],Irrep[(Subscript[A, 1])][{3}],Irrep[(Subscript[A, 1])][{3}],FundamentalBasis].(id⊗p[k]))⊗id)).((Inverse[Associator[A1,Irrep[(Subscript[A, 1])][{3}],Irrep[(Subscript[A, 1])][{3}],Irrep[(Subscript[A, 1])][{3}],FundamentalBasis]].(cup⊗id))⊗id))/.{U_⊗Irrep[A1][{0}]:>U,Irrep[A1][{0}]⊗U_:>U}

candidates=(Plus@@#&/@Map[rot,Subsets[Range[3]],{2}])+rot[4];

P=candidates[[2]];

RepresentationTensor/:Entries[RepresentationTensor[_,_,_,_,_,matrices_]]:=Flatten[matrices[[All,2,3]]]

entries=Together[Entries[P.P-x P]];

sols=Solve[entries[[1;;6]]==0,{q,x}]

Simplify[entries/.sols]

Clear[ProjectionValues]

ProjectionValues[P_]:=ProjectionValues[P]=q/.Solve[Entries[P.P-x P]==0,{q,x}]

ProjectionValues[candidates[[3]]]

(q/.Solve[Entries[#.#-x #]==0,{q,x}])&/@candidates