Symmetrization

In mathematics, symmetrization is a process that converts any function in ${\displaystyle n}$ variables to a symmetric function in ${\displaystyle n}$ variables. Similarly, antisymmetrization converts any function in ${\displaystyle n}$ variables into an antisymmetric function.

Two variables

Let ${\displaystyle S}$  be a set and ${\displaystyle A}$  be an additive abelian group. A map ${\displaystyle \alpha :S\times S\to A}$  is called a symmetric map if

$${\displaystyle \alpha (s,t)=\alpha (t,s)\quad {\text{ for all }}s,t\in S.}$$

 

It is called an antisymmetric map if instead

$${\displaystyle \alpha (s,t)=-\alpha (t,s)\quad {\text{ for all }}s,t\in S.}$$

 

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

The sum of the symmetrization and the antisymmetrization of a map ${\displaystyle \alpha }$  is ${\displaystyle 2\alpha .}$  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 ${\displaystyle \mathbb {Z} /2\mathbb {Z} ,}$  a function is skew-symmetric if and only if it is symmetric (as ${\displaystyle 1=-1}$ ).

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 (${\displaystyle \mathrm {S} _{2}=\mathrm {C} _{2}}$ ), this corresponds to the discrete Fourier transform of order two.

n variables

More generally, given a function in ${\displaystyle n}$  variables, one can symmetrize by taking the sum over all ${\displaystyle n!}$  permutations of the variables,^[1] or antisymmetrize by taking the sum over all ${\displaystyle n!/2}$  even permutations and subtracting the sum over all ${\displaystyle n!/2}$  odd permutations (except that when ${\displaystyle n\leq 1,}$  the only permutation is even).

Here symmetrizing a symmetric function multiplies by ${\displaystyle n!}$  – thus if ${\displaystyle n!}$  is invertible, such as when working over a field of characteristic ${\displaystyle 0}$  or ${\displaystyle p>n,}$  then these yield projections when divided by ${\displaystyle n!.}$ 

In terms of representation theory, these only yield the subrepresentations corresponding to the trivial and sign representation, but for ${\displaystyle n>2}$  there are others – see representation theory of the symmetric group and symmetric polynomials.

Bootstrapping

Given a function in ${\displaystyle k}$  variables, one can obtain a symmetric function in ${\displaystyle n}$  variables by taking the sum over ${\displaystyle k}$ -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

Notes

1. Hazewinkel (1990), p. 344

References

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