symmetric power

Let X be a set and let


Denote an element of Xm by x=(x1,,xm). Define an equivalence relationMathworldPlanetmath by xx if and only if there exists a permutationMathworldPlanetmath σ of (1,,m), such that xi=xσi.


The mth symmetric power of X is the set Xsymm:=Xm/. That is, the set of equivalence classesMathworldPlanetmath of Xm under the relationMathworldPlanetmathPlanetmath .

Let π be the natural projection of Xm onto Xsymm.


f:XmY is a symmetric function if and only if there exists a function g:XsymmY such that f=gπ.

From now on let R be an integral domain. Let τ:XmXm be the map τ(x):=(τ1(x),,τm(x)), where τk is the kth elementary symmetric polynomial. By the above lemma, we have a function τ:XsymmXm, where τ=τπ.


τ is one to one. If R is algebraically closedMathworldPlanetmath, then τ is onto.

A very useful case is when R=. In this case, when we put on the natural complex manifold structureMathworldPlanetmath onto symm, the map τ is a biholomorphism of symm and m.


  • 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