Symmetrization

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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 {{visible anchor|symmetric map}} if

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

It is called an {{visible anchor|antisymmetric map}} if instead

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

The {{visible anchor|symmetrization}} of a map \alpha : S \times S \to A is the map (x,y) \mapsto \alpha(x,y) + \alpha(y,x).

Similarly, the {{visible anchor|antisymmetrization}} or {{visible anchor|skew-symmetrization}} 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.

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.

=Bilinear forms=

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 \Z / 2\Z, a function is skew-symmetric if and only if it is symmetric (as 1 = - 1).

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

=Representation theory=

In terms of representation theory:

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

''n'' variables

More generally, given a function in n variables, one can symmetrize by taking the sum over all n! permutations of the variables,Hazewinkel (1990), [{{Google books|plainurl=y|id=kwMdtnhtUMMC|page=344|text=symmetrized}} p. 344] or antisymmetrize by taking the sum over all n!/2 even permutations and subtracting the sum over all n!/2 odd permutations (except that when n \leq 1, the only permutation is even).

Here symmetrizing a symmetric function multiplies by n! – thus if n! is invertible, such as when working over a field of characteristic 0 or p > n, then these yield projections when divided by n!.

In terms of representation theory, these only yield the subrepresentations corresponding to the trivial and sign representation, but for n > 2 there are others – see representation theory of the symmetric group and symmetric polynomials.

Bootstrapping

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.

See also

  • {{annotated link|Alternating multilinear map}}
  • {{annotated link|Antisymmetric tensor}}

Notes

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References

  • {{cite book|last1=Hazewinkel|first1=Michiel|author-link1=Michiel Hazewinkel|title=Encyclopaedia of mathematics: an updated and annotated translation of the Soviet "Mathematical encyclopaedia"|url=https://www.springer.com/mathematics/book/978-1-55608-005-0?cm_mmc=Google-_-Book%20Search-_-Springer-_-0|volume=6|year=1990|publisher=Springer|isbn=978-1-55608-005-0}}

{{Tensors}}

Category:Symmetric functions