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 formula_1 be a set and formula_2 an abelian group. A map formula_3 formula_4 is called symmetric if formula_5 for all formula_6.
The symmetrization of a map formula_7 is the map formula_8.
Similarly, the anti-symmetrization or skew-symmetrization of a map formula_7 is the map formula_10.
The sum of the symmetrization and the anti-symmetrization of a map "α" is 2"α".
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, 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 formula_11 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:
As the symmetric group of order two equals the cyclic group of order two (formula_12), 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 formula_13 permutations of the variables, or anti-symmetrize by taking the sum over all formula_14 even permutations and subtracting the sum over all formula_14 odd permutations (except that when , the only permutation is even).
Here symmetrizing a symmetric function multiplies by formula_13 – thus if formula_13 is invertible, such as when working over a field of characteristic formula_18 or formula_19, then these yield projections when divided by formula_13.
In terms of representation theory, these only yield the subrepresentations corresponding to the trivial and sign representation, but for formula_21 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.