Tensor square of the A2 representation of highest weight (2,0) ($R(2,0{)}^{\otimes 2}=R(4,0)\oplus R(2,1)\oplus R(0,2)$)

The tensor product of Lie algebra representations decomposes as $R(2,0)\otimes R(2,0)=R(4,0)\oplus R(2,1)\oplus R(0,2)$, of dimensions $15,15,6$. The exterior square is $R(2,1)$ and the symmetric square is $R(4,0)\oplus R(0,2)$. Highest-weight vectors and the dimension formula determine the decomposition.

The tensor square has fifteen distinct weights: three outer corners of multiplicity one, six near-corner edge weights of multiplicity two, three edge midpoints of multiplicity three, and three interior weights of multiplicity four. In the irreducible components only $R(2,1)$ is weight-degenerate: $(1,0),(-1,1),(0,-1)$ each have multiplicity two. All weights of $R(4,0)$ and $R(0,2)$ have multiplicity one.