Exercise 17, § 7

Let $\lambda , \mu\in P_{++}$ and $\alpha \in B$ be such that $\lambda (H_{\alpha})\geq 1$ and $\mu(H_{\alpha})\geq 1$. Let $F = E(\lambda )\otimes E(\mu)$.

a) Show that dim $F^{\lambda+\mu}= 1$ and dim $F^{\lambda+\mu-\alpha}= 2$.

b) Show that $X_{\alpha}: F^{\lambda+\mu-\alpha}\rightarrow F^{\lambda+\mu}$ is surjective, and that the non-zero elements of its kernel are primitive (remark that, if $\beta \in B$ is distinct from $\alpha$, $\lambda +\mu-\alpha +\beta$ is not a weight of F).

c) Deduce that $E(\lambda )\otimes E(\mu)$ contains a unique submodule isomorphic to

d) Show that $\mathbf{S}^2(E(\lambda ))$ (resp. $\bigwedge^2E(\lambda )$) contains a unique submodule isomorphic to $E(2\lambda )$ (resp. to $E(2\lambda -\alpha )$).