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 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.

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.

See also

• Alternating multilinear map – Multilinear map that is 0 whenever arguments are linearly dependent
• Antisymmetric tensor – Tensor equal to the negative of any of its transpositions

References

• Hazewinkel, Michiel (1990). Encyclopaedia of mathematics: an updated and annotated translation of the Soviet "Mathematical encyclopaedia". Vol. 6. Springer. ISBN 978-1-55608-005-0.