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Symmetrization

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Source updated February 21, 2024

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.

01Two variables

Let $S$ be a set and $A$ be an additive abelian group. A map $\alpha :S\times S\to A$ is called a symmetric map if $\alpha (s,t)=\alpha (t,s)\quad {\text{ for all }}s,t\in S.$ It is called an antisymmetric map if instead $\alpha (s,t)=-\alpha (t,s)\quad {\text{ for all }}s,t\in S.$

The symmetrization of a map $\alpha :S\times S\to A$ is the map $(x,y)\mapsto \alpha (x,y)+\alpha (y,x).$ Similarly, the antisymmetrization or skew-symmetrization 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 $\mathbb {Z} /2\mathbb {Z} ,$ 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.

02n 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.

03Bootstrapping

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.

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Sources and credits

This article is adapted from the Wikipedia article “Symmetrization”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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