Adjoint representation of SU(3) ($8=R(1,1)$)

The complex Adjoint representation of SU(3) is the action by conjugation on traceless $3\times 3$ matrices. With $H_1=\mathrm{diag}(1,-1,0)$ and $H_2=\mathrm{diag}(0,1,-1)$, its highest-weight vector is $E_{13}$ of Dynkin labels $(1,1)$. The six off-diagonal matrix units have weights $(2,-1),(-2,1),(-1,2),(1,-2),(1,1),(-1,-1)$, and the two-dimensional Cartan subalgebra has weight zero. In coordinates $(h_1,(h_1+2h_2)/\sqrt{3})$, the weight diagram is a regular hexagon together with a central point of weight multiplicity two. This representation describes both the gluon colour multiplet and the approximate flavour baryon octet, under different physical SU(3) symmetries.

For traceless matrices $T,S$ transforming by conjugation under SU(3), the trace $\mathrm{tr}(TS)$ supplies a singlet, the commutator $[T,S]$ an antisymmetric octet, and the traceless anticommutator a symmetric octet. Symmetrizing the two upper and two lower indices and removing traces supplies the 27. Contracting an antisymmetric lower pair with the invariant alternating tensor and symmetrizing the remaining three upper indices supplies the 10, and the conjugate construction supplies the conjugate 10. These nonzero equivariant maps exhaust the dimensions: the symmetric part has $36=1+8+27$, the antisymmetric part $28=8+10+10$.