In mathematics, symmetrization is a process that converts any function in variables to a symmetric function in variables. Similarly, antisymmetrization converts any function in variables into an antisymmetric function.
The symmetrization of a map is the map Similarly, the antisymmetrization or skew-symmetrization of a map is the map
The sum of the symmetrization and the antisymmetrization of a map 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 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.
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 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.
In terms of representation theory:
More generally, given a function in variables, one can symmetrize by taking the sum over all permutations of the variables, or antisymmetrize by taking the sum over all even permutations and subtracting the sum over all odd permutations (except that when the only permutation is even).
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.
Given a function in variables, one can obtain a symmetric function in variables by taking the sum over -element subsets of the variables. In statistics, this is referred to as bootstrapping, and the associated statistics are called U-statistics.