QuantumGroupsPath = "c:\\scott\\projects\\QuantumGroups\\trunk\\package\\";
AppendTo[$Path, QuantumGroupsPath];
<<QuantumGroups`

TwistFactor[Γ_,V:Irrep[Γ_][λ_]]:=Simplify[((QuantumTrace[BraidingMap[Γ,(V^(⊗2)),FundamentalBasis]])/(qDimension[Γ][V]))]
TwistFactor[Γ_,V:Irrep[Γ_][λ_]]:=(q^(KillingForm[Γ][λ,λ+2ρ[Γ]]))

Eigenvalues[MatrixData[NormalisedBraidingMap[A1,((Irrep[A1][{1}])^(⊗2)),FundamentalBasis][{0}]]]/.q->Exp[-((2π I)/10)]

((Log[#])/(2π I))&/@%

Eigenvalues[MatrixData[NormalisedBraidingMap[A3,((Irrep[A3][{1,0,0}])^(⊗2)),FundamentalBasis][{1,0,-1}]]]/.q->Exp[-2((2π I)/28)]

((Log[#])/(2π I))&/@%

Union[Eigenvalues[MatrixData[NormalisedBraidingMap[D4,((Irrep[D4][{1,0,0,0}])^(⊗2)),FundamentalBasis][{0,0,0,0}]]]/.q->Exp[-2((2π I)/36)]]

((Log[#])/(2π I))&/@%

MatrixData[NormalisedBraidingMap[D4,((Irrep[D4][{1,0,0,0}])^(⊗2)),FundamentalBasis][{0,0,0,0}]]

Union[Eigenvalues[MatrixData[NormalisedBraidingMap[D4,((Irrep[D4][{1,0,0,0}])^(⊗2)),FundamentalBasis][{0,0,0,0}]]]]

Log[%/.q->Exp[-2((2π I)/36)]]/(2π I)

TwistFactor[D4,Irrep[D4][{1,0,0,0}]]

TwistFactor[D4,#]&/@List@@DecomposeRepresentation[D4][((Irrep[D4][{1,0,0,0}])^(⊗2))]

MatrixData[NormalisedBraidingMap[D5,((Irrep[D5][{1,0,0,0,0}])^(⊗2)),FundamentalBasis][{0,0,0,0,0}]]

Union[Eigenvalues[MatrixData[NormalisedBraidingMap[D5,((Irrep[D5][{1,0,0,0,0}])^(⊗2)),FundamentalBasis][{0,0,0,0,0}]]]]

Log[%/.q->Exp[-2((2π I)/44)]]/(2π I)

TwistFactor[D5,Irrep[D5][{1,0,0,0,0}]]

TwistFactor[D5,#]&/@List@@DecomposeRepresentation[D5][((Irrep[D5][{1,0,0,0,0}])^(⊗2))]

MatrixData[NormalisedBraidingMap[D6,((Irrep[D6][{1,0,0,0,0,0}])^(⊗2)),FundamentalBasis][{0,0,0,0,0,0}]]

TwistFactor[D6,Irrep[D6][{1,0,0,0,0,0}]]

TwistFactor[D6,#]&/@List@@DecomposeRepresentation[D6][((Irrep[D6][{1,0,0,0,0,0}])^(⊗2))]

{(q^(-22)),-(q^(-12)),(q^(-10))}

DecomposeRepresentation[D6][((Irrep[D6][{1,0,0,0,0,0}])^(⊗2))]

guess[(Subscript[D, n_])]:={(q^(-4n+2)),-(q^(-2n)),(q^(-2n+2))}

{λ1,λ2,λ3}=guess[(Subscript[D, n])]

Simplify[-(z/(a[n]))==(λ2+λ3)]

Simplify[-((a[n])^(-2))==λ1]

-λ2 λ3==λ1

Simplify[a[n]==PowerExpand[(1/(Sqrt[λ2 λ3]))]]

Simplify[z==PowerExpand[-((λ2+λ3)/(Sqrt[λ2 λ3]))]]

z:=I (q-(q^(-1)))
a[n_]:=-I (q^(2n-1))

{-((a[n])^(-2))}