Symmetrization

Let ${\displaystyle S}$ be a set and ${\displaystyle A}$ be an additive abelian group. A map ${\displaystyle \alpha :S\times S\to A}$ is called a symmetric map if

$${\displaystyle \alpha (s,t)=\alpha (t,s)\quad {\text{ for all }}s,t\in S.}$$

It is called an antisymmetric map if instead

$${\displaystyle \alpha (s,t)=-\alpha (t,s)\quad {\text{ for all }}s,t\in S.}$$

The symmetrization of a map ${\displaystyle \alpha :S\times S\to A}$ is the map ${\displaystyle (x,y)\mapsto \alpha (x,y)+\alpha (y,x).}$ Similarly, the antisymmetrization or skew-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 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.

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] or antisymmetrize 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 ${\displaystyle n\leq 1,}$ 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 a field of characteristic ${\displaystyle 0}$ 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 – see representation theory of the symmetric group and symmetric polynomials.

Given a function in ${\displaystyle k}$ variables, one can obtain a symmetric function in ${\displaystyle n}$ variables by taking the sum over ${\displaystyle k}$-element subsets of the variables. In statistics, this is referred to as bootstrapping, and the associated statistics are called U-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". Encyclopaedia of Mathematics. Vol. 6. Springer. ISBN 978-1-55608-005-0.