In
mathematics, symmetrization is a process that converts any
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
in
variables to a
symmetric function
In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f ...
in
variables.
Similarly, antisymmetrization converts any function in
variables into an
antisymmetric function.
Two variables
Let
be a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
and
be an
additive
Additive may refer to:
Mathematics
* Additive function, a function in number theory
* Additive map, a function that preserves the addition operation
* Additive set-functionn see Sigma additivity
* Additive category, a preadditive category with fi ...
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
. A map
is called a if
It is called an if instead
The of a map
is the map
Similarly, the or of a map
is the map
The sum of the symmetrization and the antisymmetrization of a map
is
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
In mathematics, more specifically in multilinear algebra, an alternating multilinear map is a multilinear map with all arguments belonging to the same vector space (for example, a bilinear form or a multilinear form) that is zero whenever any pai ...
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
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.
Definition
Vector spaces
Let V, ...
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
a function is skew-symmetric if and only if it is symmetric (as
).
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 antisymmetric functions are the
subrepresentation In representation theory, a subrepresentation of a representation (\pi, V) of a group ''G'' is a representation (\pi, _W, W) such that ''W'' is a vector subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also kn ...
s corresponding to the
trivial representation and the
sign representation
A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...
, and
* symmetrization and antisymmetrization 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 (
), this corresponds to the
discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comp ...
of order two.
''n'' variables
More generally, given a function in
variables, one can symmetrize by taking the sum over all
permutations of the variables,
[Hazewinkel (1990), p. 344/ref> or antisymmetrize by taking the sum over all even permutations and subtracting the sum over all odd permutations (except that when the only permutation is even).
Here symmetrizing a symmetric function multiplies by – thus if is invertible, such as when working over a ]field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of characteristic or then these yield projections when divided by
In terms of representation theory, these only yield the subrepresentations corresponding to the trivial and sign representation, but for there are others – see representation theory of the symmetric group and symmetric polynomials
In mathematics, a symmetric polynomial is a polynomial in variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, is a ''symmetric polynomial'' if for any permutation of the subscripts one h ...
.
Bootstrapping
Given a function in variables, one can obtain a symmetric function in variables by taking the sum over -element subsets of the variables. In statistics, this is referred to as bootstrapping
In general, bootstrapping usually refers to a self-starting process that is supposed to continue or grow without external input.
Etymology
Tall boots may have a tab, loop or handle at the top known as a bootstrap, allowing one to use fingers ...
, and the associated statistics are called U-statistics
In statistical theory, a U-statistic is a class of statistics that is especially important in estimation theory; the letter "U" stands for unbiased. In elementary statistics, U-statistics arise naturally in producing minimum-variance unbiased est ...
.
See also
*
*
*
Notes
References
*
{{Tensors
Symmetric functions