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Concept# Symmetrization

Summary

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 \times S \to A is called a if
\alpha(s,t) = \alpha(t,s) \quad \text{ for all } s, t \in S.
It is called an if instead
\alpha(s,t) = - \alpha(t,s) \quad \text{ for all } s, t \in S.
The of a map \alpha : S \times S \to A is the map (x,y) \mapsto \alpha(x,y) + \alpha(y,x).
Similarly, the or of a map \alpha : S \times 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.

Official source

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