Exercise 9, § 9

We use the notations of Chap. VI, Plate I, in the case of an algebra $\mathfrak{g}$ of type $A_2$. Let $n, p$ be integers $\geq 0$.

$a)\mathfrak{P}(n\alpha_1+p\alpha_2) = 1 +$ inf($n, p$).

b) Let $\lambda =n\varpi_1+p\varpi_2$. Then dim $E(\lambda ) =\frac{1}{2}(n+ 1)(p+ 1)(n+p+ 2)$. The multiplicity of the weight 0 of $E(\lambda )$ is 0 if $\lambda$ is not radical, i.e. if $n\not\equiv p$ (mod. 3); if $\lambda$ is radical, it is 1 + inf($n, p$).