Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 102 / 5 / c

The dominance order isSuppose $\lambda \ne \mu$. Put $\delta =\lambda -\mu$. Since $\mu$ is dominant,Writing $\delta ={\sum }_i{n}_i{\alpha }_i$, some index with $n_i>0$ therefore satisfies $(\lambda ,{\alpha }_i)>0$, equivalently $⟨\lambda ,{\alpha }_i^{\vee }⟩>0$. The ${\mathfrak{sl}}_2$ lowering operatoris injective by the Injectivity of sl2 lowering above weight zero. HenceThe new weight still dominates $\mu$ in the partial order. Iterating until reaching $\mu$ givesThis is the weight multiplicity decreases away from a dominant weight property.

Solved by gpt-5.6-sol high.