Symmetric function: Difference between revisions

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

In [[mathematics]], the term "symmetric function" can mean two different concepts.

{{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 [[Argument of a function|arguments]]. For example, a function <math>f\left(x_1,x_2\right)</math> of two arguments is a symmetric function if 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.

A '''symmetric function of ''n'' variables''' is one whose value at any ''n''-[[tuple]] of arguments is the same as its value at any permutation of that ''n''-tuple. While this notion can apply to any type of function whose ''n'' arguments live in the same set, it is most often used for [[polynomial function]]s, in which case these are the functions given by [[symmetric polynomials]]. There is very little systematic theory of symmetric non-polynomial functions of ''n'' variables, so this sense is little-used, except as a general definition.

In [[algebra]] and in particular in [[algebraic combinatorics]], the term "symmetric function" is often used instead to refer to elements of the [[ring of symmetric functions]], where that ring is a specific limit of the rings of symmetric polynomials in ''n'' indeterminates, as ''n'' goes to infinity. This ring serves as universal structure in which relations between symmetric polynomials can be expressed in a way independent of the number ''n'' of indeterminates (but its elements are neither polynomials nor functions). Among other things, this ring plays an important role in the [[representation theory of the symmetric group]]s.

For these specific uses, see the corresponding articles; the remainder of this article addresses general properties of symmetric functions in ''n'' variables.

== 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>

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

<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.

<ul>

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

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

::<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>

:and so on, for all permutations of <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>f(x-x_1,)(x-x_2,)(x-x_3) = (x-x_2)(x-x_1)(x-x_2x_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

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

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

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

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

</li>

:If<li>Consider ''x'' and ''y'' are interchangednow the function becomes

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

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

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

::<math display=block>f(x,y,x) =ax ay^2 +by 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 now the function

</li>

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

</ul>

: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==

* {{annotated link|Alternating polynomial}}

* [[Ring of symmetric functions]]

* {{annotated link|Elementary symmetric polynomial}}

* [[Quasisymmetric function]]

* {{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==

[[Category:Symmetric functions| ]]

{{reflist}}

[[ca:Funció simètrica]]

{{reflist|group=note}}

[[eo:Simetria funkcio]]

[[it:Funzione simmetrica]]

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

[[he:פונקציה סימטרית]]

* 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}}.

[[pl:Funkcja symetryczna]]

[[fi:Symmetrinen funktio]]

{{Tensors}}

[[Category:Combinatorics]]

[[Category:Symmetric functions| ]]

[[Category:Properties of binary operations]]