Example

Tag 015X, permanent. Cite it as [Bourbaki, Tag 015X].

Lie Groups and Lie Algebras, chapter VIII, § 9. The formula of Hermann Weyl

Take $\mathfrak{g}=\mathfrak{s}\mathfrak{l}(2, k),\mathfrak{h}=kH$ . Let $\alpha$ be the root of $(\mathfrak{g},\mathfrak{h})$ such that $\alpha (H) = 2$ . The $\mathfrak{g}$ -module $V(m)$ has highest weight $(m/2)\alpha$ . Hence

ch(V( $m$ )) $= (e^{(m/2)\alpha+^1_2\alpha}-e^{-(m/2)\alpha-^1_2\alpha})/(e^{^1_2\alpha}-e^{-^1_2\alpha})$

=e^{-(m/2)\alpha}.(e^{(m+1)\alpha}-1)/(e^{\alpha}-1)

=e^{-(m/2)\alpha}(e^{m\alpha}+e^{(m-1)\alpha}+\cdots + 1)

=e^{(m/2)\alpha}+e^{(m-2)\alpha /2}+\cdots +e^{-(m/2)\alpha}

which also follows easily from §1, no. 2, Prop. 2.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

§1, no. 2, Prop. 2