Symmetrization


In mathematics, symmetrization is a process that converts any function in n variables to a symmetric function in n variables.
Similarly, anti-symmetrization converts any function in n variables into an antisymmetric function.

Two variables

Let be a set and an abelian group. A map is called symmetric if for all.
The symmetrization of a map is the map.
Similarly, the anti-symmetrization or skew-symmetrization of a map is the map.
The sum of the symmetrization and the anti-symmetrization of a map α is 2α.
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 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, therefore 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 may take half-integer values, while over a function is skew-symmetric if and only if it is symmetric.
This leads to the notion of ε-quadratic forms and ε-symmetric forms.

Representation theory

In terms of representation theory:
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, or anti-symmetrize by taking the sum over all even permutations and subtracting the sum over all odd permutations.
Here symmetrizing a symmetric function multiplies by – thus if is invertible, such as when working over a field of characteristic or, then these yield projections when divided by.
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.