# symmetric power

Let $X$ be a set and let

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

Denote an element of $X^{m}$ by $x=(x_{1},\ldots,x_{m}).$ Define an equivalence relation by $x\sim x^{\prime}$ if and only if there exists a permutation $\sigma$ of $(1,\ldots,m),$ such that $x_{i}=x^{\prime}_{\sigma{i}}$.

###### Definition.

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 $\sim.$

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

###### Proposition.

$f\colon X^{m}\to Y$ 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 $\tau^{\prime}\colon X^{m}\to X^{m}$ be the map $\tau^{\prime}(x):=(\tau_{1}(x),\ldots,\tau_{m}(x)),$ 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 $\tau^{\prime}=\tau\circ\pi.$

###### Proposition.

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

A very useful case is when $R=\mathbb{C}.$ 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 ${\mathbb{C}}^{m}.$

## References

• 1 Hassler Whitney. . Addison-Wesley, Philippines, 1972.
Title symmetric power SymmetricPower 2013-03-22 17:42:05 2013-03-22 17:42:05 jirka (4157) jirka (4157) 5 jirka (4157) Definition msc 32A12 msc 05E05 Multifunction