Symmetrization

In mathematics, symmetrization is a process that converts any function in

variables to a symmetric function in

Similarly, antisymmetrization converts any function in

variables into an antisymmetric function.

be an additive abelian group.

is called a symmetric map if

It is called an antisymmetric map if instead

Similarly, the antisymmetrization or skew-symmetrization of a map

The sum of the symmetrization and the antisymmetrization of a map

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.

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

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.

variables, one can symmetrize by taking the sum over all

permutations of the variables,[1] or antisymmetrize by taking the sum over all

even permutations and subtracting the sum over all

is invertible, such as when working over a field of characteristic

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

variables, one can obtain a symmetric function in

variables by taking the sum over

In statistics, this is referred to as bootstrapping, and the associated statistics are called U-statistics.