Definition 1

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

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

If there exists $n_0 \in \mathbf{Z}$ such that $M_n = 0$ for $n \leq n_0$ , the formal series $\sum_{n \geq n_0} \mathrm{rk}_K(M_n) T^n$ , which is an element of $\mathbf{Q}((T))$ , is called the Poincaré series of M and denoted by $P_M(T)$ .

Let $M'$ be another graded K-module of type $\mathbf{Z}$ , and $(M'_n)_{n \in \mathbf{Z}}$ its grading. Assume that $M'_n$ is zero for n less than a certain number. Then

P_{M \oplus M'}(T) = P_M(T) + P_{M'}(T)

and, if $M \otimes_K M'$ is provided with the total grading (Algebra, Chap. II, § 11, no. 5),

P_{M \otimes M'}(T) = P_M(T) P_{M'}(T).

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

Cited by

Nothing.

Cites

Algebra, Chap. II, § 11, no. 5