Exterior square of the sl3 representation of highest weight (2,1) (${\bigwedge }^2{\Gamma }_{2,1}\cong {\Gamma }_{5,0}\oplus {\Gamma }_{2,3}\oplus {\Gamma }_{3,1}\oplus {\Gamma }_{1,2}\oplus {\Gamma }_{0,1}$)

The formal character identity ${\chi }_{{\wedge }^{2}U}(x)=\frac{1}{2}({\chi }_U(x{)}^2-{\chi }_U(x^2))$ counts unordered pairs of distinct basis vectors. Substituting ${\chi }_{2,1}=h_2(x)h_1(x^{-1})-h_1(x)$ and the Weyl character formula gives the displayed multiplicity-free decomposition. Its dimensions sum to $21+42+24+15+3=105=\left(\genfrac{}{}{0ex}{}{15}{2}\right)$. The sl3 interlacing character formula provides an exact weight-enumeration check.