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

Two variables

Let S be a set and A be an additive abelian group. A map is**** called**** a **** if**** It**** is**** called**** an**** **** if**** instead The **** of**** a map is**** the map Similarly, the **' or **' of a map is the map The sum of the symmetrization and the antisymmetrization of a map \alpha is 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 its double, while the symmetrization of an alternating map is zero; similarly, the antisymmetrization of a symmetric map is zero, while the antisymmetrization of an anti-symmetric map is its double.

Bilinear forms

The symmetrization and antisymmetrization 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 \Z / 2\Z, a function is skew-symmetric if and only if it is symmetric (as 1 = - 1). 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 n! permutations of the variables, or antisymmetrize by taking the sum over all n!/2 even permutations and subtracting the sum over all n!/2 odd permutations (except that when n \leq 1, the only permutation is even). Here symmetrizing a symmetric function multiplies by n! – thus if n! is invertible, such as when working over a field of characteristic 0 or p > n, then these yield projections when divided by n!. In terms of representation theory, these only yield the subrepresentations corresponding to the trivial and sign representation, but for 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.