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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.

Two variables

Let be a set and be an additive abelian group. A map is called a symmetric map if

It is called an antisymmetric map if instead

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.

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 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 (), this corresponds to the discrete Fourier transform of order two.

n variables

More generally, given a function in 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 odd permutations (except that when the only permutation is even).

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 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.

See also

Notes

  1. ^ Hazewinkel (1990), p. 344

References