Exercise 10, § 9

Tag 016E, permanent. Cite it as [Bourbaki, Tag 016E].

Lie Groups and Lie Algebras, VIII, 257

We use the notations of Chap. VI, Plate II, in the case of an algebra of type $B_2$ . Let $n, p$ be integers $\geq 0$ .

a) We have

\mathfrak{P}(n\alpha_1+p(\alpha_1+ 2\alpha_2)) = 1 +\frac{1}{2}p(p+ 3)

\mathfrak{P}(n\alpha_2+p(\alpha_1+\alpha_2)) = [p^2/4] +p+ 1

\mathfrak{P}(n(\alpha_1+\alpha_2) +p(\alpha_1+ 2\alpha_2)) = [n^2/4] +n+ 1 +np+\frac{1}{2}p(p+ 3)

b) Let $\lambda =n(\alpha_1+\alpha_2) +p(\alpha_1+ 2\alpha_2)$ . The multiplicity of the weight 0 in $E(\lambda )$ is

[n/2] + 1 +np+p

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