Exercise 6, § 13

The notations are those of no. 2.IV, type $B_l$. Show that, if $1\leq r\leq l-1$, the dimension of $E(\varpi_r)$ is $(^{2l+1}_r)$, and deduce another proof of the fact that $\bigwedge^r\sigma$ is a fundamental representation of highest weight $\varpi_r$.