# Symmetrization

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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

Let be a set and an Abelian group. Given a map , is termed a symmetric map if for all .

The **symmetrization** of a map is the map .

Conversely, the **anti-symmetrization** or **skew-symmetrization** of a map is the map .

The sum of the symmetrization and the anti-symmetrization is 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 a function is skew-symmetric if and only if it is symmetric (as ).

This leads to the notion of ε-quadratic forms and ε-symmetric forms.

### Representation theory

In terms of representation theory:

- exchanging variables gives a representation of the symmetric group on the space of functions in two variables,
- the symmetric and anti-symmetric functions are the subrepresentations corresponding to the trivial representation and the sign representation, and
- symmetrization and anti-symmetrization map a function into these subrepresentations – if one divides by 2, these yield projection maps.

As the symmetric group of order two equals the cyclic group of order two (), 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 permutations of the variables,^{[1]} or anti-symmetrize by taking the sum over all even permutations and subtracting the sum over all 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 then these yield projections.

In terms of representation theory, these only yield the subrepresentations corresponding to the trivial and sign representation, but for 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

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

## References

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