Symmetrization

Summary

In mathematics, symmetrization is a process that converts any function in variables to a symmetric function in variables. Similarly, antisymmetrization converts any function in variables into an antisymmetric function.

Two variablesEdit

Let   be a set and   be an additive abelian group. A map   is called a symmetric map if

 
It is called an antisymmetric map if instead
 

The symmetrization of a map   is the map   Similarly, the antisymmetrization or skew-symmetrization of a map   is the map  

The sum of the symmetrization and the antisymmetrization of a map   is   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 formsEdit

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   a function is skew-symmetric if and only if it is symmetric (as  ).

This leads to the notion of ε-quadratic forms and ε-symmetric forms.

Representation theoryEdit

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 ( ), this corresponds to the discrete Fourier transform of order two.

n variablesEdit

More generally, given a function in   variables, one can symmetrize by taking the sum over all   permutations of the variables,[1] or antisymmetrize by taking the sum over all   even permutations and subtracting the sum over all   odd permutations (except that when   the only permutation is even).

Here symmetrizing a symmetric function multiplies by   – thus if   is invertible, such as when working over a field of characteristic   or   then these yield projections when divided by  

In terms of representation theory, these only yield the subrepresentations corresponding to the trivial and sign representation, but for   there are others – see representation theory of the symmetric group and symmetric polynomials.

BootstrappingEdit

Given a function in   variables, one can obtain a symmetric function in   variables by taking the sum over  -element subsets of the variables. In statistics, this is referred to as bootstrapping, and the associated statistics are called U-statistics.

See alsoEdit

NotesEdit

  1. ^ Hazewinkel (1990), p. 344

ReferencesEdit

  • 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.