Three times three for sl3

Given: AC, $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_3$ with its standard positive system and fundamental weights, and $V=L({\omega }_1)$ of dimension $3$ with weights ${\omega }_1,{\omega }_2-{\omega }_1,-{\omega }_2$, each of multiplicity one.

On the diagonal Cartan, the standard basis vectors of $V={\mathbb{C}}^3$ have weights ${\epsilon }_1={\omega }_1$, ${\epsilon }_2={\omega }_2-{\omega }_1$ and ${\epsilon }_3=-{\omega }_2$, since ${\omega }_1={\epsilon }_1$, ${\omega }_2={\epsilon }_1+{\epsilon }_2$ and ${\epsilon }_1+{\epsilon }_2+{\epsilon }_3=0$. The positive roots are ${\epsilon }_1-{\epsilon }_2$, ${\epsilon }_2-{\epsilon }_3$, ${\epsilon }_1-{\epsilon }_3$; their coroot pairings with ${\omega }_1$ are $1,0,1$, so ${\omega }_1$ is minuscule. Root reflections exchange the corresponding coordinates, giving the displayed three-element orbit. The matrix units $E_{ij}$ show that $V$ is simple: applying them to a nonzero vector produces every basis vector; $e_1$ is killed by upper-triangular root vectors and has highest weight ${\omega }_1$. The minuscule tensor rule applies (Root systems of the classical complex Lie algebras, Fundamental weights, Minuscule weights, Highest-weight classification, Minuscule weights have exactly the Weyl orbit as their weights, Tensor product with a minuscule representation).

The trivial module $L(0)$ is the one-dimensional module of highest weight $0$, and a dominant integral weight has all simple-coroot pairings $\ge 0$, whence $0\mathrm{\ne }{\omega }_1$, $0\mathrm{\ne }2{\omega }_1$, $0\mathrm{\ne }{\omega }_2$. Since $⟨{\omega }_i,{\alpha }_j^{\vee }⟩={\delta }_{ij}$, the weight ${\omega }_1-{\omega }_2$ has pairings $⟨{\omega }_1-{\omega }_2,{\alpha }_1^{\vee }⟩=1$ and $⟨{\omega }_1-{\omega }_2,{\alpha }_2^{\vee }⟩=-1$, so it is not dominant (Integral, dominant, and strictly dominant weights, Fundamental weights).

The flip $\tau (v\otimes w)=w\otimes v$ commutes with the diagonal Lie action. The projections $(I+\tau )\mathrm{/}2$ and $(I-\tau )\mathrm{/}2$ split $V\otimes V$ into symmetric and alternating subspaces, canonically isomorphic to the quotient powers of Symmetric and exterior powers over an arbitrary field via these projections. Their bases are $e_i\otimes e_i$ and $e_i\otimes e_j+e_j\otimes e_i$ for $i<j$, respectively $e_i\otimes e_j-e_j\otimes e_i$ for $i<j$, giving dimensions six and three. The nonzero vectors $e_1\otimes e_1$ and $e_1\otimes e_2-e_2\otimes e_1$ are killed by every upper-triangular root vector and have weights $2{\omega }_1$ and ${\omega }_2$. Complete reducibility therefore supplies a copy of each corresponding simple module in its respective subspace (Direct-sum, dual, Hom, and tensor representations, Weyl's complete reducibility theorem, Highest-weight classification).