PlanetMath: symmetric power

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symmetric power

(Definition)

Let be a set and let

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

Denote an element of

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 $x_i = x'_{\sigma{i}}$ .

Definition 1 The $m^{\text{th}}$ symmetric power of

is the set $X^m_{sym} := X^m / \sim.$ That is, the set of equivalence classes of

under the relation $\sim.$

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

Proposition 1 $f \colon X^m \to Y$ 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 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 th elementary symmetric polynomial. By the above lemma, we have a function $\tau \colon X^m_{sym} \to X^m$ , where $\tau' = \tau \circ \pi .$

Proposition 2 $\tau$ is one to one. If 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 ${\mathbb{C}}^m .$

Bibliography

1: Hassler Whitney. Complex Analytic Varieties. Addison-Wesley, Philippines, 1972.

"symmetric power" is owned by jirka.

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See Also: multifunction

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Cross-references: structure, complex manifold, algebraically closed, elementary symmetric polynomial, map, integral domain, function, symmetric, onto, projection, relation, equivalence classes, permutation, equivalence relation
There are 2 references to this entry.

This is version 2 of symmetric power, born on 2007-12-18, modified 2007-12-18.
Object id is 10143, canonical name is SymmetricPower.
Accessed 300 times total.

Classification:

AMS MSC:	05E05 (Combinatorics :: Algebraic combinatorics :: Symmetric functions)
	32A12 (Several complex variables and analytic spaces :: Holomorphic functions of several complex variables :: Multifunctions)

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