Symmetrization

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. Let be a set and be an additive abelian group. A map is called a if It is called an if instead The of a map is the map Similarly, the or 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. 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. 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.