 Mixed SU(3) tensor representation

A mixed tensor has $k$ defining indices and $l$ dual indices. The SU(3) action applies $U$ to every upper index and $U^{-1}$ to every lower index. Young symmetrizers select permutation types, while invariant Kronecker deltas take traces and invariant alternating tensors convert an antisymmetric pair of defining indices into a dual index, or conversely. Repeated fundamental and dual tensor rules decompose it with multiplicities: $(p,q)\otimes (1,0)=(p+1,q)\oplus (p-1,q+1)\oplus (p,q-1)$ and the conjugate rule, omitting negative labels.