Symmetric function: Difference between revisions

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{{Short description|Function that is invariant under all permutations of its variables}}

{{About|general properties of symmetric functions|the ring of symmetric functions in algebraic combinatorics|ring of symmetric functions}}

{{About|functions that are invariant under all permutations of their variables|the generalization of symmetric polynomials to infinitely many variables (in algebraic combinatorics)|ring of symmetric functions|symmetric functions on elements of a vector space |symmetric tensor}}

{{technical|date=March 2013}}

In [[mathematics]], a '''symmetric[[Function (mathematics)|function]] of ''<math>n''</math> variables is '''symmetric''' if its value is onethe whosesame valueno atmatter anythe ''n''-[[tuple]]order of its [[argumentArgument of a function|arguments]]. isFor theexample, samea asfunction its<math>f\left(x_1,x_2\right)</math> valueof attwo anyarguments [[permutation]]is ofa thatsymmetric ''n''-tuple. So,function if e.g.and only if <math>f\left(x_1,x_2\bold{x}right) = f\left(x_1,x_2,x_3x_1\right)</math>, the function can be symmetric onfor all its variables, or just<math>x_1</math> onand <math>(x_1,x_2)</math>, such that <math>\left(x_2x_1,x_3x_2\right)</math>, orand <math>\left(x_1x_2,x_3x_1\right)</math>. Whileare thisin notionthe can[[Domain applyof toa any typefunction|___domain]] of function<math>f.</math> whoseThe ''n''most argumentscommonly haveencountered thesymmetric samefunctions ___domain set, it is most often used forare [[polynomial function]]s, in which case these are the functions given by the [[symmetric polynomialspolynomial]]. There is very little systematic theory of symmetric non-polynomial functions of ''n'' variables, so this sense is little-used, except as a general definitions.

A related notion is [[alternating polynomial]]s, which change sign under an interchange of variables. Aside from polynomial functions, [[Symmetric tensor|tensors]] that act as functions of several vectors can be symmetric, and in fact the space of symmetric <math>k</math>-tensors on a [[vector space]] <math>V</math> is [[isomorphic]] to the space of [[homogeneous polynomials]] of degree <math>k</math> on <math>V.</math> Symmetric functions should not be confused with [[even and odd functions]], which have a different sort of symmetry.

== Symmetrization ==

{{main|Symmetrization}}

Given any function ''f'' in ''n'' variables with values in an abelian group, a symmetric function can be constructed by summing values of ''f'' over all permutations of the arguments. Similarly, an anti-symmetric function can be constructed by summing over [[even permutation]]s and subtracting the sum over [[odd permutation]]s. These operations are of course not invertible, and could well result in a function that is identically zero for nontrivial functions ''f''. The only general case where ''f'' can be recovered if both its symmetrization and anti-symmetrization are known is when ''n''&nbsp;=&nbsp;2 and the abelian group admits a division by 2 (inverse of doubling); then ''f'' is equal to half the sum of its symmetrization and its anti-symmetrization.

Given any function ''<math>f''</math> in ''<math>n''</math> variables with values in an [[abelian group]], a symmetric function can be constructed by summing values of ''<math>f''</math> over all permutations of the arguments. Similarly, an anti-symmetric function can be constructed by summing over [[even permutation]]s and subtracting the sum over [[odd permutation]]s. These operations are of course not invertible, and could well result in a function that is identically zero for nontrivial functions ''<math>f''.</math> The only general case where ''<math>f''</math> can be recovered if both its symmetrization and anti-symmetrizationantisymmetrization are known is when ''<math>n''&nbsp; =&nbsp; 2</math> and the abelian group admits a division by 2 (inverse of doubling); then ''<math>f''</math> is equal to half the sum of its symmetrization and its anti-symmetrizationantisymmetrization.

== Examples ==

== Examples ==

* Consider the real function

::<math>f(x_1,x_2,x_3)=(x-x_1)(x-x_2)(x-x_3)</math>

<ul>

:By definition, a symmetric function with n variables has the property that

<li>Consider the [[Real number|real]] function

::<math>f(x_1,x_2,...,x_n)=f(x_2,x_1,...,x_n)=f(x_3,x_1,...,x_n,x_{n-1})</math> etc.

::<math display=block>f(x_1,x_2,x_3) = (x-x_1)(x-x_2)(x-x_3).</math>

:By definition, a symmetric function with <math>n</math> variables has the property that

::<math display=block>f(x_1,x_2,...\ldots,x_n) = f(x_2,x_1,...\ldots,x_n) = f(x_3,x_1,...\ldots,x_n,x_{n-1}), \quad \text{ etc.}</math> etc.

:In general, the function remains the same for every [[permutation]] of its variables. This means that, in this case,

<math display=block>(x-x_1)(x-x_2)(x-x_3) = (x-x_2)(x-x_1)(x-x_3) = (x-x_3)(x-x_1)(x-x_2)</math>

:and so on, for all permutations of <math>x_1, x_2, x_3.</math>

</li>

* <li>Consider the real function

:In general, the function remains the same for every [[permutation]] of its variables. This means that, in this case,

::<math> (x-x_1)(x-x_2)(x-x_3)display=block>f(x-x_2)(x-x_1)(x-x_3,y) =( x^2 + y^2 -x_3)(x-x_1)(x-x_2) r^2.</math>

:If ''<math>x''</math> and ''<math>y''</math> are interchanged the function becomes

:and so on, for all permutations of <math>x_1,x_2,x_3</math>

::<math display=block>f(y,x) = y^2 + x^2 - r^2,</math>

:which yields exactly the same results as the original ''<math>f''(''x'','' y'').</math>

</li>

* <li>Consider now the function

::<math display=block>f(x,y) =x ax^2+yby^2-r^2.</math>

:If ''<math>x''</math> and ''<math>y''</math> are interchanged, the function becomes

::<math display=block>f(y,x) = ay^2 + bx^2 - r^2.</math>

:If ''x'' and ''y'' are interchanged the function becomes

:This function is obviously not the same as the original if {{nowrap|1=''<math>a'' ≠\neq ''b''}},</math> which makes it non-symmetric.

::<math>f(y,x)=y^2+x^2-r^2</math>

</li>

:which yields exactly the same results as the original ''f''(''x'',''y'').

</ul>

* Consider now the function

::<math>f(x,y)=ax^2+by^2-r^2</math>

:If ''x'' and ''y'' are interchanged, the function becomes

::<math>f(y,x)=ay^2+bx^2-r^2.</math>

:This function is obviously not the same as the original if {{nowrap|1=''a'' ≠ ''b''}}, which makes it non-symmetric.

== Applications ==

=== U-statistics ===

{{main|U-statistic}}

In [[statistics]], an ''<math>n''</math>-sample statistic (a function in ''<math>n''</math> variables) that is obtained by [[bootstrappingBootstrapping (statistics)|bootstrapping]] symmetrization of a ''<math>k''</math>-sample statistic, yielding a symmetric function in ''<math>n''</math> variables, is called a [[U-statistic]]. Examples include the [[sample mean]] and [[sample variance]].

==See also==

* [[Elementary symmetric polynomial]]

* {{annotated link|Alternating polynomial}}

* [[Quasisymmetric function]]

* [[Ring{{annotated oflink|Elementary symmetric functions]]polynomial}}

* {{annotated link|Even and odd functions}}

* {{annotated link|Exchangeable random variables}}

* [[{{annotated link|Quasisymmetric function]]}}

* {{annotated link|Ring of symmetric functions}}

* {{annotated link|Symmetrization}}

* {{annotated link|Vandermonde polynomial}}

==References==

{{reflist}}

{{reflist|group=note}}

* [[F. N. David]], [[M. G. Kendall]] & D. E. Barton (1966) ''Symmetric Function and Allied Tables'', [[Cambridge University Press]].

* Joseph P. S. Kung, [[Gian-Carlo Rota]], & [[Catherine Yan|Catherine H. Yan]] (2009) ''[[Combinatorics: The Rota Way]]'', §5.1 Symmetric functions, pp 222–5, Cambridge University Press, {{isbn|978-0-521-73794-4}} .

{{Tensors}}

[[Category:Symmetric functions| ]]

[[Category:Combinatorics]]

[[Category:Symmetric functions| ]]

[[Category:Properties of binary operations]]