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. Conversely, anti-symmetrization converts any function in n variables into an antisymmetric function.

2 variables[edit]

Let S be a set and A an Abelian group. Given a map \alpha: S \times S \to A, \alpha is termed a symmetric map if \alpha(s,t) = \alpha(t,s) for all s,t \in S.

The symmetrization of a map \alpha \colon S \times S \to A is the map (x,y) \mapsto \alpha(x,y) + \alpha(y,x).

Conversely, the anti-symmetrization or skew-symmetrization of a map \alpha \colon S \times S \to A is the map (x,y) \mapsto \alpha(x,y) - \alpha(y,x).

The sum of the symmetrization and the anti-symmetrization 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 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.

Bilinear forms[edit]

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 \mathbf{Z}/2, 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[edit]

In terms of representation theory:

As the symmetric group of order two equals the cyclic group of order two (S_2=C_2), this corresponds to the discrete Fourier transform of order two.

n variables[edit]

More generally, given a function in n variables, one can symmetrize by taking the sum over all n! permutations of the variables,[1] or anti-symmetrize by taking the sum over all n!/2 even permutations and subtracting the sum over all n!/2 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 p > n, then these yield projections.

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[edit]

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.

Notes[edit]

  1. ^ Hazewinkel (1990), p. 344

References[edit]