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

For an irreducible representation $V(\lambda )$, writeFormThe tensor product of highest-weight vectors is killed by all positive-root spaces and has weight $\lambda$. It therefore generates a highest-weight constituent isomorphic to $V(\lambda )$. By Complete reducibility of semisimple Lie algebra representations, this constituent is a subrepresentation of $T$. This proves the generation by fundamental representations statement.