Symmetrization

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 $$\alpha : S \times S \to A$$ is called a  if $$\alpha(s,t) = \alpha(t,s) \quad \text{ for all } s, t \in S.$$ It is called an  if instead $$\alpha(s,t) = - \alpha(t,s) \quad \text{ for all } s, t \in S.$$

The  of a map $$\alpha : S \times S \to A$$ is the map $$(x,y) \mapsto \alpha(x,y) + \alpha(y,x).$$ Similarly, the ' or ' of a map $$\alpha : S \times S \to A$$ is the map $$(x,y) \mapsto \alpha(x,y) - \alpha(y,x).$$

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:
 * exchanging variables gives a representation of the symmetric group on the space of functions in two variables,
 * the symmetric and antisymmetric functions are the subrepresentations corresponding to the trivial representation and the sign representation, and
 * symmetrization and antisymmetrization map a function into these subrepresentations – if one divides by 2, these yield projection maps.

As the symmetric group of order two equals the cyclic group of order two ($$\mathrm{S}_2 = \mathrm{C}_2$$), 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.