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 if
$\alpha(s,t) = \alpha(t,s) \quad \text s, t \in S.$
It is called an if instead
$\alpha(s,t) = - \alpha(t,s) \quad \text s, t \in S.$

The of a map $\alpha : S \times S \to A$ is the map $(x,y) \mapsto \alpha(x,y) + \alpha(y,x).$
Similarly, the or 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 $\Z / 2\Z,$ 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 ($\mathrm_2 = \mathrm_2$), 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,Hazewinkel (1990), p. 344/ref> 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

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Notes

References

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{{Tensors

Symmetric functions