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Inmathematics,symmetrization is a process that converts anyfunction in${\displaystyle n}$ variables to asymmetric function in${\displaystyle n}$ variables.Similarly,antisymmetrization converts any function in${\displaystyle n}$ variables into anantisymmetric function.

Let${\displaystyle S}$ be aset and${\displaystyle A}$ be anadditiveabelian group. A map${\displaystyle \alpha :S\times S\to A}$ is called asymmetric map if$${\displaystyle \alpha (s,t)=\alpha (t,s)\text{ for all }s,t\in S.}$$It is called anantisymmetric map if instead$${\displaystyle \alpha (s,t)=-\alpha (t,s)\text{ for all }s,t\in S.}$$

Thesymmetrization of a map${\displaystyle \alpha :S\times S\to A}$ is the map${\displaystyle (x,y)\mapsto \alpha (x,y)+\alpha (y,x).}$Similarly, theantisymmetrization orskew-symmetrization of a map${\displaystyle \alpha :S\times S\to A}$ is the map${\displaystyle (x,y)\mapsto \alpha (x,y)-\alpha (y,x).}$

The sum of the symmetrization and the antisymmetrization of a map${\displaystyle \alpha }$ is${\displaystyle 2\alpha .}$Thus,away from 2, meaning if 2 isinvertible, such as for thereal 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 analternating map is zero; similarly, the antisymmetrization of a symmetric map is zero, while the antisymmetrization of an anti-symmetric map is its double.

The symmetrization and antisymmetrization of abilinear 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 theintegers, the associated symmetric form (over therationals) may take half-integer values, while over${\displaystyle \mathbb{Z}/2\mathbb{Z},}$ a function is skew-symmetric if and only if it is symmetric (as${\displaystyle 1=-1}$).

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

In terms ofrepresentation theory:

• exchanging variables gives a representation of thesymmetric group on the space of functions in two variables,
• the symmetric and antisymmetric functions are thesubrepresentations corresponding to thetrivial representation and thesign representation, and
• symmetrization and antisymmetrization map a function into these subrepresentations – if one divides by 2, these yieldprojection maps.

As the symmetric group of order two equals thecyclic group of order two (${\displaystyle {\mathrm{S}}_2={\mathrm{C}}_2}$), this corresponds to thediscrete Fourier transform of order two.

More generally, given a function in${\displaystyle n}$ variables, one can symmetrize by taking the sum over all${\displaystyle n!}$ permutations of the variables,^[1] orantisymmetrize 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${\displaystyle n\le 1,}$ the only permutation is even).

Here symmetrizing a symmetric function multiplies by${\displaystyle n!}$ – thus if${\displaystyle n!}$ is invertible, such as when working over afield ofcharacteristic${\displaystyle 0}$ 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 – seerepresentation theory of the symmetric group andsymmetric polynomials.

Given a function in${\displaystyle k}$ variables, one can obtain a symmetric function in${\displaystyle n}$ variables by taking the sum over${\displaystyle k}$-elementsubsets of the variables. In statistics, this is referred to asbootstrapping, and the associated statistics are calledU-statistics.

• Alternating multilinear map – Multilinear map that is 0 whenever arguments are linearly dependent
• Antisymmetric tensor – Tensor equal to the negative of any of its transpositions

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

• Symmetric functions

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