Remark

Being given the basis $(e_i)_{1 \leq i \leq n}$ allows us to identify the algebra $S(M)$ with the polynomial algebra $A[X_1, \ldots, X_n]$ (§ 6, no. 6); formula (75) shows that the inner product by $e^{*\alpha}$ is just the differential operator $D^\alpha = D_1^{\alpha_1}D_2^{\alpha_2}\ldots D_n^{\alpha_n}$, where $D_i = \partial/\partial X_i$ for $1 \leq i \leq n$ (§ 10, no. 11, Example).

Consider finally the exterior algebra $\Lambda(M)$, which has as basis the set of elements $e_J$, where $J$ runs through the set of subsets of the interval $\{1, n\}$ of $\mathbf{N}$ (§ 7, no. 8, Theorem 1); similarly $\Lambda(M^*)$ has as basis the elements $e_J^*$. It follows from formula (68) of no. 9 that

(76)

where $\rho_{K, J-K}$ is the number defined by formula (19) of § 7, no. 8. There are analogous formulae with the roles of $M$ and $M^*$ interchanged.