sl3 highest-weight tensor rule (${\Gamma }_{a,b}\otimes {\Gamma }_{1,0}\cong {\Gamma }_{a+1,b}\oplus {\Gamma }_{a-1,b+1}\oplus {\Gamma }_{a,b-1}$)

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 $24+15+6=45$.