Symmetrization Explained

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 x S \to A is called a if$$\backslash alpha(s,t)\; =\; \backslash alpha(t,s)\; \backslash quad\; \backslash text\; s,\; t\; \backslash in\; S.$$It is called an if instead $$\backslash alpha(s,t)\; =\; -\; \backslash alpha(t,s)\; \backslash quad\; \backslash text\; s,\; t\; \backslash in\; S.$$

The of a map

\alpha : S x S \to A is the map (x,y) \mapsto \alpha(x,y) + \alpha(y,x). Similarly, the or of a map \alpha : S x 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 (

S _{2} = 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,^[1] 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.

References

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

Notes and References

1. Hazewinkel (1990), [{{Google books|plainurl=y|id=kwMdtnhtUMMC|page=344|text=symmetrized}} p. 344]