symmetric power

Let $X$ be a set and let

$$X^m:=\underset{m-\text{times}}{\underbrace{X\times \mathrm{\cdots }\times X}}.$$

Denote an element of $X^m$ by $x=(x_1,\mathrm{\dots },x_m).$Define an equivalence relationby $x\sim x^{\prime }$ if and only if there exists apermutation $\sigma$ of $(1,\mathrm{\dots },m),$ such that$x_i=x_{\sigma i}^{\prime }$.

Definition.

The $m^{\text{th}}$ symmetric power of $X$ isthe set $X_{sym}^m:=X^m/\sim .$ That is, the set of equivalence classes of $X^m$ under therelation $\sim .$

Let $\pi$ be the natural projection of $X^m$ onto $X_{sym}^m$.

Proposition.

$f:X^m\to Y$ is a symmetric function if and only if there exists a function$g\mathrm{:}{X}_{s\mathit{}y\mathit{}m}^m\mathrm{\to }Y$ such that $f\mathrm{=}g\mathrm{\circ }\pi \mathrm{.}$

From now on let $R$ be an integral domain. Let ${\tau }^{\prime }:X^m\to X^m$ be the map${\tau }^{\prime }(x):=({\tau }_1(x),\mathrm{\dots },{\tau }_m(x)),$ where ${\tau }_k$ is the $k^{\text{th}}$ elementary symmetricpolynomial. By the above lemma, we have a function $\tau :X_{sym}^m\to X^m$, where${\tau }^{\prime }=\tau \circ \pi .$

Proposition.

$\tau$ is one to one. If $R$ is algebraically closed, then $\tau$ is onto.

A very useful case is when $R=ℂ.$ In this case, when we put on the natural complex manifold structureonto ${ℂ}_{sym}^m,$ the map $\tau$ is a biholomorphism of ${ℂ}_{sym}^m$ and${ℂ}^m.$