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Let $X$ be a set and let \begin{equation*} X^m := \underbrace{X \times \cdots \times X}_{m-\text{times}} . \end{equation*}Denote an element of $X^m$ by $x = (x_1,\ldots,x_m).$ Define an equivalence relation by $x \sim x'$ if and only if there exists a permutation $\sigma$ of $(1,\ldots,m),$ such that $x_i = x'_{\sigma{i}}$ .
Definition 1 The $m^{{th}}$ symmetric power of $X$ is the set $X^m_{sym} := X^m / \sim.$ That is, the set of equivalence classes of $X^m$ under the relation $\sim.$
Let $\pi$ be the natural projection of $X^m$ onto $X^m_{sym}$ .
Proposition 1 $f \colon X^m \to Y$ is a symmetric function if and only if there exists a function $g \colon X^m_{sym} \to Y$ such that $f = g \circ \pi.$
From now on let $R$ be an integral domain. Let $\tau' \colon X^m \to X^m$ be the map $\tau'(x) := (\tau_1(x),\ldots,\tau_m(x)),$ where $\tau_k$ is the $k^{th}$ elementary symmetric polynomial. By the above lemma, we have a function $\tau \colon X^m_{sym} \to X^m$ , where $\tau' = \tau \circ \pi .$
A very useful case is when $R = \mathbb{C}.$ In this case, when we put on the natural complex manifold structure onto ${\mathbb{C}}^m_{sym},$ the map $\tau$ is a biholomorphism of ${\mathbb{C}}^m_{sym}$ and ${\mathbb{C}}^m .$
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- Hassler Whitney. Complex Analytic Varieties. Addison-Wesley, Philippines, 1972.
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