In mathematics, symmetrization is a process that converts any function in n variables to a symmetric function in n variables. Similarly, anti-symmetrization converts any function in n variables into an antisymmetric function.
Two variables
Let S be a set and A an abelian group. A map \( \alpha\colon S \times S \to A α {\displaystyle \alpha }\) \( \alpha \) is called symmetric 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 \( (x,y)\mapsto \alpha (x,y)+\alpha (y,x). \)
Similarly, 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 of a map α is 2α. 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 anti-symmetrization of a symmetric map is zero, while the anti-symmetrization of an anti-symmetric map is its double.
Bilinear forms
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, therefore 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 \( {\displaystyle \mathbf {Z} /2\mathbf {Z} ,} \) 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:
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.
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,[1] or anti-symmetrize by taking the sum over all n!/2 even permutations and subtracting the sum over all n!/2 odd permutations (except that when n ≤ 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 \( {\displaystyle 0} \) or \( {\displaystyle p>n} \), 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.
Notes
Hazewinkel (1990), p. 344
References
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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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