 Symmetric function

In mathematics, the term "symmetric function" can mean two different concepts.
A symmetric function of n variables is one whose value at any ntuple of arguments is the same as its value at any permutation of that ntuple. 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 functions, in which case these are the functions given by symmetric polynomials. There is very little systematic theory of symmetric nonpolynomial functions of n variables, so this sense is littleused, 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 groups.
For these specific uses, see the corresponding articles; the remainder of this article addresses general properties of symmetric functions in n variables.
Contents
Symmetrization
Main article: SymmetrizationGiven 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 antisymmetric function can be constructed by summing over even permutations and subtracting the sum over odd permutations. 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 antisymmetrization are known is when n = 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 antisymmetrization.
Applications
Ustatistics
Main article: UstatisticIn statistics, an nsample statistic (a function in n variables) that is obtained by bootstrapping symmetrization of a ksample statistic, yielding a symmetric function in n variables, is called a Ustatistic. Examples include the sample mean and sample variance.
See also
 Ring of symmetric functions
 Quasisymmetric function
Categories: Symmetric functions
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