Remark

— Let n and k be two positive integers. We denote by $\Delta_{k,n}$ the set of elements of length k in $N^n$ and further equip $A$, , with the order relation, written $a \leq \beta$, induced by the lexicographic order on $N^n$ (Set Theory, III, p. 157), and define an action of the group $\mathfrak{S}_n$ on $N^n$ by $(\sigma a)(i) = a(\sigma^{-1}(i))$ for $\sigma \in \mathfrak{S}_n, a \in N^n$ and $l \leq i \leq n$. Moreover, we denote by $D_k$ the set of elements $a = (\alpha(1), \ldots, \alpha(k))$ of $\mathbf{N}^k$ such that

Suppose that $k \leq n$ and let us identify $\mathbf{N}^k$ with a subset of $N^n$ by the mapping $(a(1), \ldots, a(k)) \mapsto (a(1), \ldots, a(k), 0, \ldots, 0)$. Then $D_k$ consists of the elements $a$ of $\Delta_{k,n}$ such that $\sigma a \leq a$ for all $\sigma \in \mathfrak{S}_n$. It follows that every orbit of the group $\mathfrak{S}_n$ in $A$, , contains a unique element of $D_k$. For every $a \in D_k$ let $O(a)$ be the orbit of $a$ in $\Delta_{k,n}$ under the operation of $\mathfrak{S}_n$; put

It follows from Lemma 1 of IV, p. 47, that the family $(\mathbf{M}(\alpha))_{\alpha \in D_k}$ is a basis of the A-module $S_k^{(n)}$.

For each $\alpha \in D_k$ put

since we have $\sum_{i=1}^k i \cdot (a(i) - a(i+1)) = \sum_{i=1}^k a(i) = k$, the symmetric polynomial $S(\alpha)$ is homogeneous of degree $k$. It follows directly from Th. 1 (IV, p. 62) that the family $(S(\alpha))_{\alpha \in D_k}$ is a basis of the A-module $S_k^{(n)}$.

Let $\alpha, \beta$ be in $D_k$, and let $c_{\alpha\beta}$ be the coefficient of the monomial $X^\beta$ in the polynomial $S(\alpha)$ defined by (12) in the case $A = \mathbf{Z}$ and $k = n$. This is then a positive integer, independent of the ring A and of the integer n. By formula (9) (IV, p. 64) we then have

It may be shown (cf. IV, p. 101, Exercise 13) that the matrix $C = (c_{\alpha\beta})_{\alpha,\beta \in D_k}$ is such that $c_{\alpha\alpha} = 1$ and $c_{\alpha\beta} = 0$ for $\alpha < \beta$; generalizing the terminology introduced in II, p. 351, we shall say that $C$ belongs to the lower total triangular group. The same is true of the matrix D inverse to C. The values of the matrices C and D when $2 \leq k \leq 5$, are given in the table (IV, p. 103-105).

Suppose now that $n < k$ and let us identify $N^n$ with a subset of $N^k$ by the mapping $(a(1), \ldots, a(n)) \mapsto (a(1), \ldots, a(n), 0, \ldots, 0)$. For every $a$ in $D_k \cap N^n$ we again denote by $O(\alpha)$ the orbit of $a$ in $\Delta_{k,n}$ under the operation of $\mathfrak{S}_n$ and define $\mathbf{M}(\alpha)$ by (11). Further we define $S(\alpha)$ by (12), then the families $(\mathbf{M}(a))_{a \in D_k \cap N^n}$ and $(S(a))_{a \in D_k \cap N^n}$ are bases of the A-module $S_k^{(n)}$. By formula (9) of IV, p. 64, we have a formula analogous to (13) where $D_k$ is replaced by $D_k \cap N^n$, with the same integers $c_{\alpha\beta}$.