Symmetric function: Difference between revisions

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

{{otheruses4|symmetric functions which are not necessarily polynomials|symmetric functions which are polynomials|symmetric polynomials}}

{{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}}

{{for|the ring of symmetric functions|ring of symmetric functions}}

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.

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

== Symmetrization ==

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 polynomial]]s'''. 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.

{{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 antisymmetrization are known is when <math>n = 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 antisymmetrization.

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.

== Examples==

For these specific uses, see the articles [[symmetric polynomial]]s and [[ring of symmetric functions]]; the remainder of this article addresses general properties of symmetric functions in ''n'' variables.

<ul>

== Symmetrization ==

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

{{main|Symmetrization}}

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

Given any function f in ''n'' variables with values in an abelian group, it can be made into a symmetric function by averaging over permutations. Similarly, it can be made into an anti-symmetric function by averaging over [[even permutation]]s and subtracting the average over [[odd permutation]]s.

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

<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) = 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) = ax^2+by^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>

This function is not the same as the original if <math>a \neq b,</math> which makes it non-symmetric.

</li>

</ul>

== Applications ==

=== U-statistics ===

{{main|U-statistic}}

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

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}}

* {{annotated link|Elementary symmetric polynomial}}

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

* {{annotated link|Exchangeable random variables}}

* {{annotated link|Quasisymmetric function}}

* {{for|the ring of symmetricannotated functionslink|ringRing 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:Combinatorics]]

[[Category:Symmetric functions| ]]

[[Category:Properties of binary operations]]