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| | Emilia Shryock is my name but you can call me something you like. For a whilst I've been in South Dakota and my parents live close by. Doing ceramics is what her family members and her appreciate. Hiring is his profession.<br><br>my page; std home test ([http://withlk.com/board_RNsI08/10421 similar web site]) |
| In [[mathematics]], '''symmetrization''' is a process that converts any function in ''n'' variables to a [[symmetric function]] in ''n'' variables.
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| Conversely, '''anti-symmetrization''' converts any function in ''n'' variables into an [[antisymmetric]] function.
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| ==2 variables==
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| Let <math>S</math> be a set and <math>A</math> an [[Abelian group]]. Given a map <math>\alpha: S \times S \to A</math>, <math>\alpha</math> is termed a symmetric map if <math>\alpha(s,t) = \alpha(t,s)</math> for all <math>s,t \in S</math>.
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| The '''symmetrization''' of a map <math>\alpha \colon S \times S \to A</math> is the map <math>(x,y) \mapsto \alpha(x,y) + \alpha(y,x)</math>.
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| Conversely, the '''anti-symmetrization''' or '''skew-symmetrization''' of a map <math>\alpha \colon S \times S \to A</math> is the map <math>(x,y) \mapsto \alpha(x,y) - \alpha(y,x)</math>.
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| The sum of the symmetrization and the anti-symmetrization is <math>2\alpha.</math>
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| Thus, [[Localization of a ring#Terminology|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.
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| 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.
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| ===Bilinear forms===
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| 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.
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| 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 <math>\mathbf{Z}/2,</math> a function is skew-symmetric if and only if it is symmetric (as <math>1=-1</math>).
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| This leads to the notion of [[e-quadratic form|ε-quadratic form]]s and ε-symmetric forms.
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| ===Representation theory===
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| In terms of [[representation theory]]:
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| * exchanging variables gives a representation of the [[symmetric group]] on the space of functions in two variables,
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| * the symmetric and anti-symmetric functions are the [[subrepresentation]]s corresponding to the [[trivial representation]] and the [[sign representation]], and
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| * symmetrization and anti-symmetrization map a function into these subrepresentations – if one divides by 2, these yield [[projection map]]s.
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| As the symmetric group of order two equals the [[cyclic group]] of order two (<math>S_2=C_2</math>), this corresponds to the [[discrete Fourier transform]] of order two.
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| ==''n'' variables==
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| More generally, given a function in ''n'' variables, one can symmetrize by taking the sum over all <math>n!</math> permutations of the variables,<ref>Hazewinkel (1990), {{Google books quote|id=kwMdtnhtUMMC|page=344|text=symmetrized|p. 344}}</ref> or anti-symmetrize by taking the sum over all <math>n!/2</math> [[even permutation]]s and subtracting the sum over all <math>n!/2</math> odd permutations.
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| 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 <math>p > n,</math> then these yield projections.
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| In terms of representation theory, these only yield the subrepresentations corresponding to the trivial and sign representation, but for <math> n > 2</math> there are others – see [[representation theory of the symmetric group]] and [[symmetric polynomials]].
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| ==Bootstrapping==
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| 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 (statistics)|bootstrapping]], and the associated statistics are called [[U-statistics]].
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| == Notes ==
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| <references/>
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| == References ==
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| * {{cite book |last1=Hazewinkel |first1=Michiel |authorlink1=Michiel Hazewinkel |last2= |first2= |authorlink2= |title=Encyclopaedia of mathematics: an updated and annotated translation of the Soviet "Mathematical encyclopaedia" |url=http://www.springer.com/mathematics/book/978-1-55608-005-0?cm_mmc=Google-_-Book%20Search-_-Springer-_-0 |edition= |series=Encyclopaedia of Mathematics |volume=6 |year=1990 |publisher=Springer |location= |isbn=978-1-55608-005-0 |id= }}
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| [[Category:Symmetric functions]]
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Emilia Shryock is my name but you can call me something you like. For a whilst I've been in South Dakota and my parents live close by. Doing ceramics is what her family members and her appreciate. Hiring is his profession.
my page; std home test (similar web site)