Symmetrization

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.

Two variables

Let ${\displaystyle S}$ be a set and ${\displaystyle A}$ an abelian group. Given a map ${\displaystyle \alpha \colon S\times S\to A}$, ${\displaystyle \alpha }$ is termed a symmetric map if ${\displaystyle \alpha (s,t)=\alpha (t,s)}$ for all ${\displaystyle s,t\in S}$.

The symmetrization of a map ${\displaystyle \alpha \colon S\times S\to A}$ is the map ${\displaystyle (x,y)\mapsto \alpha (x,y)+\alpha (y,x)}$.

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

The sum of the symmetrization and the anti-symmetrization of a map α 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 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.

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 ${\displaystyle \mathbf {Z} /2\mathbf {Z} ,}$ 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:

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 n variables, one can symmetrize by taking the sum over all ${\displaystyle n!}$ permutations of the variables,[1] or anti-symmetrize 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 n ≤ 1, the only permutation is even).

Here symmetrizing (respectively anti-symmetrizing) a symmetric function multiplies by ${\displaystyle n!}$ – thus if ${\displaystyle n!}$ is invertible, such as when working 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 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.

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. 6. Springer. ISBN 978-1-55608-005-0.