Symmetrization

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In mathematics, symmetrization is a process that converts any function in n variables to a symmetric function in n variables. Conversely, anti-symmetrization converts any function in n variables into an antisymmetric function.

Contents 1 2 variables 1.1 Bilinear forms 1.2 Representation theory 2 n variables 3 Bootstrapping

[edit] 2 variables

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

The symmetrization of a map is the map .

Conversely, the anti-symmetrization or skew-symmetrization of a map is the map .

The sum of the symmetrization and the anti-symmetrization 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 simply 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.

[edit] 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 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.

[edit] 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 anti-symmetric functions are the subrepresentations corresponding to the trivial representation and the sign representation, and
• symmetrization and anti-symmetrization 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 (S₂ = C₂), this corresponds to the discrete Fourier transform of order two.

[edit] 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.

Here symmetrizing (respectively anti-symmetrizing) a symmetric (respectively anti-symmetric) function multiplies by n! – thus if n! is invertible, such as if one is working over the rationals or over a field of characteristic p > n, then these yield projections.

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.

[edit] 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.