{Rank[RepresentationCategory[A2,14]],Rank[RepresentationCategory[A3,14]]}

Tally[RootReduce[FullSimplify[Dimensions[RepresentationCategory[A2,14]][[1]]]]]

Tally[RootReduce[FullSimplify[Dimensions[RepresentationCategory[A3,14]][[1]]]]]

AlcoveRoots[A2,14]

TwistFactor[A2,Irrep[A2][#]]&/@AlcoveRoots[A2,14]/.(q^(n_)):>(q^(Mod[n,14]))

AlcoveRoots[A3,14]

TwistFactor[A3,Irrep[A3][#]]&/@AlcoveRoots[A3,14]/.(q^(n_)):>(q^(Mod[-n,14]))

TensorProductTable[Γ_,l_]:=Outer[DecomposeRepresentation[Γ,l][Irrep[Γ][#1]⊗Irrep[Γ][#2]]&,AlcoveWeights[Γ,l],AlcoveWeights[Γ,l],1]

SymmetryGroupTable[Γ_,l_]:=Outer[DecomposeRepresentation[Γ,l][#1⊗#2]&,SymmetryGroup[Γ,l],SymmetryGroup[Γ,l],1]

RootTensorProductTable[Γ_,l_]:=Outer[DecomposeRepresentation[Γ,l][Irrep[Γ][#1]⊗Irrep[Γ][#2]]&,AlcoveRoots[Γ,l],AlcoveRoots[Γ,l],1]

RootTensorProductTable[A2,14]/.DirectSum->List//TableForm

AlcoveRoots[A3,14]

RootTensorProductTable[A3,14]/.DirectSum->List//TableForm

Here's the example of (Subscript[A, 3])/3 modularising to some (Subscript[B, 2]) thing:

RootTensorProductTable[A2,18]/.DirectSum->List//TableForm