symmetric powerSymmetricPower2007-12-18 11:32:032007-12-18 12:08:30Definition% this is the default PlanetMath preamble. as your knowledge
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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
$x_i = x'_{\sigma{i}}$.
\begin{defn}
The $m^{\text{th}}$ 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 $\sim.$
\end{defn}
Let $\pi$ be the natural projection of $X^m$ onto $X^m_{sym}$.
\begin{prop}
$f \colon X^m \to Y$ is a symmetric function if and only if there exists a function
$g \colon X^m_{sym} \to Y$ such that $f = g \circ \pi.$
\end{prop}
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 $k^\text{th}$ elementary symmetric
polynomial. By the above lemma, we have a function $\tau \colon X^m_{sym} \to X^m$, where
$\tau' = \tau \circ \pi .$
\begin{prop}
$\tau$ is one to one. If $R$ is algebraically closed, then $\tau$ is onto.
\end{prop}
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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
${\mathbb{C}}^m .$
\begin{thebibliography}{9}
\bibitem{Whitney:varieties}
Hassler Whitney.
{\em \PMlinkescapetext{Complex Analytic Varieties}}.
Addison-Wesley, Philippines, 1972.
\end{thebibliography}