Proposition 1

*For every pair of positive integers k, n, let $S_k^{(n)}$ be the A-module consisting of all symmetric polynomials in $X_1, \ldots, X_n$, which are homogeneous of degree k. If the integer m satisfies $0 \leq k \leq m \leq n$, then the mapping $f \mapsto f (X_1, \ldots, X_m, 0, \ldots, 0)$ is an isomorphism of $S_k^{(n)}$ onto $S_k^{(m)}$.

By the commutativity of the diagram (10) it is enough to show that every isobaric polynomial of weight k in $S_1, \ldots, S_n$ depends only on $S_1, \ldots, S_m$ under the hypotheses $0 \leq k \leq m \leq n$. Now the weight of a monomial $S_1^{\alpha(1)} \cdots S_n^{\alpha(n)}$ is equal to the integer $\alpha(1) + 2\alpha(2) + \ldots + n\alpha(n)$; since the integers $\alpha(1), \ldots, \alpha(n)$ are positive, the relation

implies $\alpha(j) = 0$ for $k < j \leq n$, whence the assertion.