# Symmetrization

{{ safesubst:#invoke:Unsubst||$N=Refimprove |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} In mathematics, symmetrization is a process that converts any function in n variables to a symmetric function in n variables. Conversely, anti-symmetrization converts any function in n variables into an antisymmetric function.

## 2 variables

Conversely, the anti-symmetrization or skew-symmetrization of a map ${\displaystyle \alpha \colon S\times S\to A}$ is the map ${\displaystyle (x,y)\mapsto \alpha (x,y)-\alpha (y,x)}$.

The sum of the symmetrization and the anti-symmetrization is ${\displaystyle 2\alpha .}$ Thus, away from 2, meaning if 2 is invertible, such as for the real numbers, one can divide by 2 and express every function as a sum of a symmetric function and an anti-symmetric function.

The symmetrization of a symmetric map is simply its double, while the symmetrization of an alternating map is zero; similarly, the anti-symmetrization of a symmetric map is zero, while the anti-symmetrization of an anti-symmetric map is its double.

### Bilinear forms

The symmetrization and anti-symmetrization of a bilinear map are bilinear; thus away from 2, every bilinear form is a sum of a symmetric form and a skew-symmetric form, and there is no difference between a symmetric form and a quadratic form.

At 2, not every form can be decomposed into a symmetric form and a skew-symmetric form – for instance, over the integers, the associated symmetric form (over the rationals) may take half-integer values, while over ${\displaystyle \mathbf {Z} /2,}$ a function is skew-symmetric if and only if it is symmetric (as ${\displaystyle 1=-1}$).

### Representation theory

In terms of representation theory:

As the symmetric group of order two equals the cyclic group of order two (${\displaystyle S_{2}=C_{2}}$), this corresponds to the discrete Fourier transform of order two.

## n variables

More generally, given a function in n variables, one can symmetrize by taking the sum over all ${\displaystyle n!}$ permutations of the variables,[1] or anti-symmetrize by taking the sum over all ${\displaystyle n!/2}$ even permutations and subtracting the sum over all ${\displaystyle n!/2}$ odd permutations.

Here symmetrizing (respectively anti-symmetrizing) a symmetric function multiplies by n! – thus if n! is invertible, such as if one is working over the rationals or over a field of characteristic ${\displaystyle p>n,}$ then these yield projections.

In terms of representation theory, these only yield the subrepresentations corresponding to the trivial and sign representation, but for ${\displaystyle n>2}$ there are others – see representation theory of the symmetric group and symmetric polynomials.

## Bootstrapping

Given a function in k variables, one can obtain a symmetric function in n variables by taking the sum over k element subsets of the variables. In statistics, this is referred to as bootstrapping, and the associated statistics are called U-statistics.

## Notes

1. Hazewinkel (1990), [[[:Template:Google books]] p. 344]

## References

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