Symmetrization

Two variables

Let S be a set and A an abelian group. Given a map α : S × S → A , α is termed a symmetric map if α ( s , t ) = α ( t , s ) for all s , t ∈ S .

The symmetrization of a map α : S × S → A is the map ( x , y ) ↦ α ( x , y ) + α ( y , x ) .

Conversely, the anti-symmetrization or skew-symmetrization of a map α : S × S → A is the map ( x , y ) ↦ α ( x , y ) − α ( y , x ) .

The sum of the symmetrization and the anti-symmetrization of a map α is 2α. 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 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:

As the symmetric group of order two equals the cyclic group of order two ( S 2 = 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 anti-symmetrize by taking the sum over all n ! / 2 even permutations and subtracting the sum over all n ! / 2 odd permutations (except that when n ≤ 1, the only permutation is even).

Here symmetrizing (respectively anti-symmetrizing) a symmetric function multiplies by n ! – thus if n ! is invertible, such as when working 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.