Symmetric map

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In mathematics, symmetrization is a process that converts any function in n {\displaystyle n} {\displaystyle n} variables to a symmetric function in n {\displaystyle n} {\displaystyle n} variables. Similarly, antisymmetrization converts any function in n {\displaystyle n} {\displaystyle n} variables into an antisymmetric function.

Two variables

Let S {\displaystyle S} {\displaystyle S} be a set and A {\displaystyle A} {\displaystyle A} be an additive abelian group. A map α : S × S → A {\displaystyle \alpha :S\times S\to A} {\displaystyle \alpha :S\times S\to A} is called a symmetric map if α ( s , t ) = α ( t , s )  for all  s , t ∈ S . {\displaystyle \alpha (s,t)=\alpha (t,s)\quad {\text{ for all }}s,t\in S.} {\displaystyle \alpha (s,t)=\alpha (t,s)\quad {\text{ for all }}s,t\in S.} It is called an antisymmetric map if instead α ( s , t ) = − α ( t , s )  for all  s , t ∈ S . {\displaystyle \alpha (s,t)=-\alpha (t,s)\quad {\text{ for all }}s,t\in S.} {\displaystyle \alpha (s,t)=-\alpha (t,s)\quad {\text{ for all }}s,t\in S.}

The symmetrization of a map α : S × S → A {\displaystyle \alpha :S\times S\to A} {\displaystyle \alpha :S\times S\to A} is the map ( x , y ) ↦ α ( x , y ) + α ( y , x ) . {\displaystyle (x,y)\mapsto \alpha (x,y)+\alpha (y,x).} {\displaystyle (x,y)\mapsto \alpha (x,y)+\alpha (y,x).} Similarly, the antisymmetrization or skew-symmetrization of a map α : S × S → A {\displaystyle \alpha :S\times S\to A} {\displaystyle \alpha :S\times S\to A} is the map ( x , y ) ↦ α ( x , y ) − α ( y , x ) . {\displaystyle (x,y)\mapsto \alpha (x,y)-\alpha (y,x).} {\displaystyle (x,y)\mapsto \alpha (x,y)-\alpha (y,x).}

The sum of the symmetrization and the antisymmetrization of a map α {\displaystyle \alpha } {\displaystyle \alpha } is 2 α . {\displaystyle 2\alpha .} {\displaystyle 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 , {\displaystyle \mathbb {Z} /2\mathbb {Z} ,} {\displaystyle \mathbb {Z} /2\mathbb {Z} ,} a function is skew-symmetric if and only if it is symmetric (as 1 = − 1 {\displaystyle 1=-1} {\displaystyle 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 ( S 2 = C 2 {\displaystyle \mathrm {S} _{2}=\mathrm {C} _{2}} {\displaystyle \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 {\displaystyle n} {\displaystyle n} variables, one can symmetrize by taking the sum over all n ! {\displaystyle n!} {\displaystyle n!} permutations of the variables,[1] or antisymmetrize by taking the sum over all n ! / 2 {\displaystyle n!/2} {\displaystyle n!/2} even permutations and subtracting the sum over all n ! / 2 {\displaystyle n!/2} {\displaystyle n!/2} odd permutations (except that when n ≤ 1 , {\displaystyle n\leq 1,} {\displaystyle n\leq 1,} the only permutation is even).

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

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

Bootstrapping

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

See also

Notes

  1. Hazewinkel (1990), p. 344

References