Tensor product of Lie algebra representations ($V\otimes W$)

The tensor product of Lie algebra representations has action $x(v\otimes w)=xv\otimes w+v\otimes xw$. The Lie algebra representation identity follows because operators on the two different tensor factors commute. This action descends to exterior powers and symmetric powers.

For the complex special linear Lie algebra ${\mathfrak{sl}}_3$, tensoring an irreducible highest-weight representation by the defining representation gives the displayed sum, omitting terms with negative Dynkin labels. It follows by multiplying its Weyl character formula by $x_1+x_2+x_3$. In particular ${\Gamma }_{2,1}\otimes {\Gamma }_{1,0}={\Gamma }_{3,1}\oplus {\Gamma }_{1,2}\oplus {\Gamma }_{2,0}$, with dimensions $24+15+6=45$.