Symmetrization

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

Two variables

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

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

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

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 ([math]\displaystyle{ \mathrm{S}_2 = \mathrm{C}_2 }[/math]), this corresponds to the discrete Fourier transform of order two.

n variables

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

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

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

Bootstrapping

Given a function in [math]\displaystyle{ k }[/math] variables, one can obtain a symmetric function in [math]\displaystyle{ n }[/math] variables by taking the sum over [math]\displaystyle{ k }[/math]-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". Encyclopaedia of Mathematics. 6. Springer. ISBN 978-1-55608-005-0. https://www.springer.com/mathematics/book/978-1-55608-005-0?cm_mmc=Google-_-Book%20Search-_-Springer-_-0.