Lemma 8

Tag 0AY9, permanent. Cite it as [Bourbaki, Tag 0AY9].

Lie Groups and Lie Algebras, chapter VI, § 4. Classification of root systems

Let $i_1, i_2, \ldots, i_p$ be vertices of X such that $\{i_1, i_2\}, \{i_2, i_3\}, \ldots, \{i_{p-1}, i_p\}$ are edges of order 3. Then $q(\sum_{r=1}^p r e_{i_r}) = \frac{1}{2} p(p+1)$ .

We have $(e_{i_r}|e_{i_r}) = 1, (e_{i_r}|e_{i_{r+1}}) = -\frac{1}{2}, (e_{i_r}|e_{i_s}) = 0$ if $s > r + 1$ . Thus

q(\sum_{r=1}^p r e_{i_r}) = \sum_{r=1}^p r^2 - 2 \sum_{r=1}^{p-1} \frac{1}{2} r(r+1) = p^2 - \sum_{r=1}^{p-1} r.

By Theory of Sets, Chap. III. § 5, no. 8. Cor. to Prop. 14, this is equal to

p^2 - \frac{1}{2} p(p-1) = \frac{1}{2} p(p+1).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Lemma 10, § 4
Lemma 9, § 4

Cites

Nothing.