# symmetric power

Let $X$ be a set and let

$${X}^{m}:=\underset{m-\text{times}}{\underset{\u23df}{X\times \mathrm{\cdots}\times X}}.$$ |

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

###### Definition.

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

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

###### 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 symmetric polynomial. 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 $R=\u2102.$ In this case, when we put on the natural complex manifold structure^{}
onto ${\u2102}_{sym}^{m},$ the map $\tau $ is a biholomorphism of ${\u2102}_{sym}^{m}$ and
${\u2102}^{m}.$

## References

- 1 Hassler Whitney. . Addison-Wesley, Philippines, 1972.

Title | symmetric power |
---|---|

Canonical name | SymmetricPower |

Date of creation | 2013-03-22 17:42:05 |

Last modified on | 2013-03-22 17:42:05 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 5 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 32A12 |

Classification | msc 05E05 |

Related topic | Multifunction |