Symmetrization

 ( Symmetric Functions ) 

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.

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Two variables Let $S$ be a set and $A$ be an Additive group abelian group. A map $\backslash alpha\; :\; S\; \backslash times\; S\; \backslash to\; A$ is called a if $$\backslash alpha(s,t)\; =\; \backslash alpha(t,s)\; \backslash quad\; \backslash text\{\; for\; all\; \}\; s,\; t\; \backslash in\; S.$$ It is called an if instead $$\backslash alpha(s,t)\; =\; -\; \backslash alpha(t,s)\; \backslash quad\; \backslash text\{\; for\; all\; \}\; s,\; t\; \backslash in\; S.$$

The of a map $\backslash alpha\; :\; S\; \backslash times\; S\; \backslash to\; A$ is the map $(x,y)\; \backslash mapsto\; \backslash alpha(x,y)\; +\; \backslash alpha(y,x).$ Similarly, the ' or ' of a map $\backslash alpha\; :\; S\; \backslash times\; S\; \backslash to\; A$ is the map $(x,y)\; \backslash mapsto\; \backslash alpha(x,y)\; -\; \backslash alpha(y,x).$

The sum of the symmetrization and the antisymmetrization of a map $\backslash alpha$ is $2\; \backslash alpha.$ Thus, away from 2, meaning if 2 is invertible, such as for the , 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.

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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 , the associated symmetric form (over the Rational number) may take half-integer values, while over $\backslash Z\; /\; 2\backslash Z,$ a function is skew-symmetric if and only if it is symmetric (as $1\; =\; -\; 1$).

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

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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 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 .

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

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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), or antisymmetrize by taking the sum over all $n!/2$ and subtracting the sum over all $n!/2$ odd permutations (except that when $n\; \backslash leq\; 1,$ 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.

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Bootstrapping Given a function in $k$ variables, one can obtain a symmetric function in $n$ variables by taking the sum over $k$-element of the variables. In statistics, this is referred to as bootstrapping, and the associated statistics are called U-statistics.

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See also

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Notes

• (1990). 9781556080050, Springer. . ISBN 9781556080050

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