Symmetric function: Difference between revisions

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

{{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}}

In [[mathematics]], a [[Function (mathematics)|function]] of ''<math>n''</math> variables is '''symmetric''' if its value is the same no matter the order of its [[argumentArgument of a function|arguments]]. For example, ifa function <math>f=f\left(x_1,x_2\right)</math> of two arguments is a symmetric function, thenif and only if <math>f\left(x_1,x_2\right) = f\left(x_2,x_1\right)</math> for all <math>x_1</math> and <math>x_2</math> such that <math>\left(x_1,x_2\right)</math> and <math>\left(x_2,x_1\right)</math> are in the [[Domain of a function|___domain]] of ''<math>f''.</math> The most commonly encountered symmetric functions are [[polynomial function]]s, which are given by the [[symmetric polynomial]]s.

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. Antisymmetric functions are not the opposite of symmetric functions: it exists functions which are both symmetric and antisymmetric.<ref>{{cite journal | author = Petitjean, M. | title = Symmetry, antisymmetry and chirality: Use and misuse of terminology | journal=Symmetry | year=2021 | volume=13 |issue=4 | at=603 | no-pp=yes | doi=10.3390/sym13040603 | doi-access=free }}</ref>

== Symmetrization ==

{{main|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==

<ul>

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

::<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>

: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 function

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

::<math display=block>f(x_1x,x_2,...,x_ny) = f(x_2,x_1,...,x_n)x^2 =+ y^2 f(x_3,x_1,- r^2...,x_n,x_{n-1})</math> etc.

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

::<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

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

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

::<math> (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>

: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) = ay^2 + bx^2 - r^2.</math>

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

* Consider the function

</li>

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

</ul>

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

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

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

* 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==

*[[Symmetrization]]

*[[Elementary symmetric{{annotated link|Alternating polynomial]]}}

*[[Alternating {{annotated link|Elementary symmetric polynomial]]s}}

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

*[[Vandermonde polynomial]]

* {{annotated link|Exchangeable random variables}}

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

* [[Ring of symmetric functions]]

* [[Even{{annotated andlink|Ring oddof symmetric functions]]}}

*[[ {{annotated link|Symmetrization]]}}

*[[ {{annotated link|Vandermonde polynomial]]}}

==References==

{{Reflist}}

{{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:Combinatorics]]