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

The Classification of finite-dimensional sl2 representations says that for every $n\ge 0$ there is one irreducible module $V(n)$ of dimension $n+1$, with weightsof multiplicity one, and every finite-dimensional ${\mathfrak{sl}}_2(\mathbb{C})$-module is a direct sum of these irreducibles.

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The weight lattice isFor each root $\alpha$, restrict $V$ to the subalgebraIf $V_{\lambda }\ne 0$, the ${\mathfrak{sl}}_2$ classification shows that the $h_{\alpha }$-eigenvalue $\lambda (h_{\alpha })=⟨\lambda ,{\alpha }^{\vee }⟩$ is an integer. Hence $\lambda \in X$.

The same classification makes every $\alpha$-string symmetric underand preserves weight multiplicity. Since the Weyl group is generated by these simple reflections,

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Take $\mathfrak{g}={\mathfrak{sl}}_3(\mathbb{C})$ and its adjoint representation $V=\mathfrak{g}$. This representation is irreducible because $\mathfrak{g}$ is simple. Its zero-weight space is the Cartan subalgebra $\mathfrak{t}$, so

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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.

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