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In mathematics, symmetrization is a process that converts any function in n variables to a symmetric function in n variables. Conversely, anti-symmetrization converts any function in n variables into an antisymmetric function.
Let be a set and an Abelian group. Given a map , is termed a symmetric map if for all .
The symmetrization of a map is the map .
Conversely, the anti-symmetrization or skew-symmetrization of a map is the map .
The sum of the symmetrization and the anti-symmetrization 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 simply its double, while the symmetrization of an alternating map is zero; similarly, the anti-symmetrization of a symmetric map is zero, while the anti-symmetrization of an anti-symmetric map is its double.
The symmetrization and anti-symmetrization 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 anti-symmetric functions are the subrepresentations corresponding to the trivial representation and the sign representation, and
- symmetrization and anti-symmetrization map a function into these subrepresentations – if one divides by 2, these yield projection maps.
More generally, given a function in n variables, one can symmetrize by taking the sum over all permutations of the variables, or anti-symmetrize by taking the sum over all even permutations and subtracting the sum over all odd permutations.
Here symmetrizing (respectively anti-symmetrizing) a symmetric (respectively anti-symmetric) function multiplies by n! – thus if n! is invertible, such as if one is working over the rationals or over a field of characteristic then these yield projections.
In terms of representation theory, these only yield the subrepresentations corresponding to the trivial and sign representation, but for there are others – see representation theory of the symmetric group and symmetric polynomials.
Given a function in k variables, one can obtain a symmetric function in n variables by taking the sum over k element subsets of the variables. In statistics, this is referred to as bootstrapping, and the associated statistics are called U-statistics.
- Hazewinkel (1990),
- Hazewinkel, Michiel (1990). Encyclopaedia of mathematics: an updated and annotated translation of the Soviet "Mathematical encyclopaedia". Encyclopaedia of Mathematics 6. Springer. ISBN 978-1-55608-005-0.