Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 102 / 4 / e / Solution

Let $V=V({\omega }_1)$, so $V^{\ast }=V({\omega }_2)$. First,The highest-weight tensor-product rule givesandThe dimensions $15+3+6+3=27$ check the decomposition. Therefore the Triple tensor decomposition for the defining sl3 representation isThus one may take