Proposition 5

*In the universal decomposition algebra $E_f$ the family of monomials $x_1^{\nu(1)} ... x_n^{\nu(n)}$ such that $0 \leq \nu(i) < i$ for $1 \leq i \leq n$ is a basis of the A-module $E_f$. In particular $E_f$ is a free A-module of rank $n!$.

Write $B = A[X_1, ..., X_n]$ and $C = A[X_1, ..., X_n]^{\text{sym}}$. By Th. 1 (IV, p. 62) we have $C = A[s_1, ..., s_n]$ and $s_1, ..., s_n$ are algebraically independent over A. The polynomials without constant term in $s_1, ..., s_n$ form an ideal $C^+$ in C, supplementary to A and generated by $s_1, ..., s_n$. Let c be the ideal of C generated by $s_1 + a_1, s_2 - a_2, ..., s_n + (-1)^{n+1} a_n$. There exists an A-algebra automorphism of C which maps $s_k$ to $s_k + (-1)^{k+1} a_k$ for $1 \leq k \leq n$, and hence maps $C^+$ onto c ; therefore we have $C = A \oplus c$. Moreover Th. 1, c) of IV, p. 62 shows that

where S is the set of all $\nu \in \mathbf{N}^n$ such that $0 \leq \nu(i) < i$ for $1 \leq i \leq n$. The ideal a of B is generated by c, whence $a = Bc = \bigoplus_{\nu \in S} c \cdot X^\nu$. Since $C = A \oplus c$, we find

whence Prop. 5, because $E_f = B/a$.