Remark

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

Lie Groups and Lie Algebras, chapter V, § 5. Invariants in the symmetric algebra

Let $S = K[X_1, \ldots, X_n]$ be a graded polynomial algebra over K, $d_i$ the degree of $X_i$ and $F(X_1, \ldots, X_n)$ a homogeneous element of degree m of S. Then

\sum_{i=1}^n d_i X_i \frac{\partial F}{\partial X_i} = mF.

Indeed, it is immediate that the K-linear map D from S to S that transforms every homogeneous element z of degree p into pz is a derivation of S. Thus

mF(X_1, \ldots, X_n) = D(F(X_1, \ldots, X_n)) = \sum_{i=1}^n D(X_i) \frac{\partial F}{\partial X_i} = \sum_{i=1}^n d_i X_i \frac{\partial F}{\partial X_i}.

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

Cited by

Nothing.

Cites

Nothing.