{{Refimprove|date=November 2010}} In mathematics, '''symmetrization''' is a process that converts any function in <math>n</math> variables to a symmetric function in <math>n</math> variables. Similarly, '''antisymmetrization''' converts any function in <math>n</math> variables into an antisymmetric function.
==Two variables==
Let <math>S</math> be a set and <math>A</math> be an additive abelian group. A map <math>\alpha : S \times S \to A</math> is called a '''{{visible anchor|symmetric map}}''' if <math display=block>\alpha(s,t) = \alpha(t,s) \quad \text{ for all } s, t \in S.</math> It is called an '''{{visible anchor|antisymmetric map}}''' if instead <math display=block>\alpha(s,t) = - \alpha(t,s) \quad \text{ for all } s, t \in S.</math>
The '''{{visible anchor|symmetrization}}''' of a map <math>\alpha : S \times S \to A</math> is the map <math>(x,y) \mapsto \alpha(x,y) + \alpha(y,x).</math> Similarly, the '''{{visible anchor|antisymmetrization}}''' or '''{{visible anchor|skew-symmetrization}}''' of a map <math>\alpha : S \times S \to A</math> is the map <math>(x,y) \mapsto \alpha(x,y) - \alpha(y,x).</math>
The sum of the symmetrization and the antisymmetrization of a map <math>\alpha</math> is <math>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>\Z / 2\Z,</math> a function is skew-symmetric if and only if it is symmetric (as <math>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>\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>n</math> variables, one can symmetrize by taking the sum over all <math>n!</math> permutations of the variables,<ref>Hazewinkel (1990), [{{Google books|plainurl=y|id=kwMdtnhtUMMC|page=344|text=symmetrized}} p. 344]</ref> or antisymmetrize by taking the sum over all <math>n!/2</math> even permutations and subtracting the sum over all <math>n!/2</math> odd permutations (except that when <math>n \leq 1,</math> the only permutation is even).
Here symmetrizing a symmetric function multiplies by <math>n!</math> – thus if <math>n!</math> is invertible, such as when working over a field of characteristic <math>0</math> or <math>p > n,</math> then these yield projections when divided by <math>n!.</math>
In terms of representation theory, these only yield the subrepresentations corresponding to the trivial and sign representation, but for <math>n > 2</math> there are others – see representation theory of the symmetric group and symmetric polynomials.
==Bootstrapping==
Given a function in <math>k</math> variables, one can obtain a symmetric function in <math>n</math> variables by taking the sum over <math>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==
* {{annotated link|Alternating multilinear map}} * {{annotated link|Antisymmetric tensor}}
== Notes ==
{{reflist}} {{reflist|group=note}}
== References ==
* {{cite book|last1=Hazewinkel|first1=Michiel|author-link1=Michiel Hazewinkel|title=Encyclopaedia of mathematics: an updated and annotated translation of the Soviet "Mathematical encyclopaedia"|url=https://www.springer.com/mathematics/book/978-1-55608-005-0?cm_mmc=Google-_-Book%20Search-_-Springer-_-0|volume=6|year=1990|publisher=Springer|isbn=978-1-55608-005-0}}
{{Tensors}}
Category:Symmetric functions