Example

Tag 0164, permanent. Cite it as [Bourbaki, Tag 0164].

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

We return to the Example of no. 1. Let $\lambda = (n/2)\alpha , \mu= (p/2)\alpha$ , $\nu = (q/2)\alpha$ with $n\geq p$ . We have

m(\lambda , \mu, \nu ) =\mathfrak{P}(\frac{n}{2}\alpha +\frac{\alpha}{2}+\frac{p}{2}\alpha +\frac{\alpha}{2}-\frac{q}{2}\alpha -\alpha )

-\mathfrak{P}(\frac{n}{2}\alpha +\frac{\alpha}{2}-\frac{p}{2}\alpha -\frac{\alpha}{2}-\frac{q}{2}\alpha -\alpha )

-\mathfrak{P}(-\frac{n}{2}\alpha -\frac{\alpha}{2}+\frac{p}{2}\alpha +\frac{\alpha}{2}-\frac{q}{2}\alpha -\alpha )

+\mathfrak{P}(-\frac{n}{2}\alpha -\frac{\alpha}{2}-\frac{p}{2}\alpha -\frac{\alpha}{2}-\frac{q}{2}\alpha -\alpha )

=\mathfrak{P}(\frac{n + p-q}{2}\alpha )-\mathfrak{P}(\frac{n-p-q- 2}{2}\alpha )

This is zero if $n+p+q$ is not divisible by 2, or if $q\geq n+p$ . If

q=n+p-2r

with $r$ an integer $\geq 0$ , we have

m(\lambda , \mu, \nu ) =\mathfrak{P}(r\alpha )-\mathfrak{P}((r-p-1)\alpha )

hence $m(\lambda , \mu, \nu ) = 1$ if $r\leq p$ and $m(\lambda , \mu, \nu ) = 0$ if $r > p$ . Finally, the $\mathfrak{g}$ -module $V(n)\otimes V(p)$ is isomorphic to

V(n+p)\oplus V(n+p-2)\oplus V(n+p-4)\oplus \cdots \oplus V(n-p)

(Clebsch-Gordan formula).

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

Nothing.