Remark 3

Tag 00Y5, permanent. Cite it as [Bourbaki, Tag 00Y5].

Lie Groups and Lie Algebras, chapter VIII, § 1. The Lie algebra sl(2, k) and its representations

Let $\Phi$ be the bilinear form on $V(m)$ such that

$\Phi (e^{(m)}_n, e^{(m)}_{n'}) = 0$ if $n+n'\not=m$

\Phi (e^{(m)}_n, e^{(m)}_{m-n}) = (-1)^n(m)

n

If $x=au+bv$ and $y=cu+dv$ , then $\Phi (x^m, y^m) = (ad-bc)^m$ . It is now easy to check that $\Phi$ is invariant, and that $\Phi$ is symmetric for $m$ even, and alternating for $m$ odd.

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

Exercise 5, § 1
Proposition 12, § 7

Cites

Nothing.