Inmathematics, afunction of${\displaystyle n}$ variables issymmetric if its value is the same no matter the order of itsarguments. For example, a function${\displaystyle f\left(x_1,x_2\right)}$ of two arguments is a symmetric function if and only if${\displaystyle f\left(x_1,x_2\right)=f\left(x_2,x_1\right)}$ for all${\displaystyle x_1}$ and${\displaystyle x_2}$ such that${\displaystyle \left(x_1,x_2\right)}$ and${\displaystyle \left(x_2,x_1\right)}$ are in thedomain of${\displaystyle f.}$ The most commonly encountered symmetric functions arepolynomial functions, which are given by thesymmetric polynomials.

A related notion isalternating polynomials, which change sign under an interchange of variables. Aside from polynomial functions,tensors that act as functions of several vectors can be symmetric, and in fact the space of symmetric${\displaystyle k}$-tensors on avector space${\displaystyle V}$ isisomorphic to the space ofhomogeneous polynomials of degree${\displaystyle k}$ on${\displaystyle V.}$ Symmetric functions should not be confused witheven and odd functions, which have a different sort of symmetry.

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

• Consider thereal function$${\displaystyle f(x_1,x_2,x_3)=(x-x_1)(x-x_2)(x-x_3).}$$By definition, a symmetric function with${\displaystyle n}$ variables has the property that$${\displaystyle f(x_1,x_2,\dots ,x_n)=f(x_2,x_1,\dots ,x_n)=f(x_3,x_1,\dots ,x_n,x_{n-1}),\text{ etc.}}$$In general, the function remains the same for everypermutation of its variables. This means that, in this case,$${\displaystyle (x-x_1)(x-x_2)(x-x_3)=(x-x_2)(x-x_1)(x-x_3)=(x-x_3)(x-x_1)(x-x_2)}$$and so on, for all permutations of${\displaystyle x_1,x_2,x_3.}$
• Consider the function$${\displaystyle f(x,y)=x^2+y^2-r^2.}$$If${\displaystyle x}$ and${\displaystyle y}$ are interchanged the function becomes$${\displaystyle f(y,x)=y^2+x^2-r^2,}$$which yields exactly the same results as the original${\displaystyle f(x,y).}$
• Consider now the function$${\displaystyle f(x,y)=a{x}^2+b{y}^2-r^2.}$$If${\displaystyle x}$ and${\displaystyle y}$ are interchanged, the function becomes$${\displaystyle f(y,x)=a{y}^2+b{x}^2-r^2.}$$This function is not the same as the original if${\displaystyle a\ne b,}$ which makes it non-symmetric.

Instatistics, an${\displaystyle n}$-sample statistic (a function in${\displaystyle n}$ variables) that is obtained bybootstrapping symmetrization of a${\displaystyle k}$-sample statistic, yielding a symmetric function in${\displaystyle n}$ variables, is called aU-statistic. Examples include thesample mean andsample variance.

• Alternating polynomial
• Elementary symmetric polynomial – Mathematical function
• Even and odd functions – Functions such that f(–x) equals f(x) or –f(x)
• Exchangeable random variables – Concept in statistics
• Quasisymmetric function
• Ring of symmetric functions
• Symmetrization – process that converts any function in n variables to a symmetric function in n variablesPages displaying wikidata descriptions as a fallback
• Vandermonde polynomial – determinant of Vandermonde matrixPages displaying wikidata descriptions as a fallback

• 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 H. Yan (2009)Combinatorics: The Rota Way, §5.1 Symmetric functions, pp 222–5, Cambridge University Press,ISBN 978-0-521-73794-4.

• Combinatorics
• Symmetric functions
• Properties of binary operations

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• Short description matches Wikidata
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