Symmetric function

In mathematics, a function of "n" variables is symmetric if its value is the same no matter the order of its arguments. For example, if formula_1 is a symmetric function, then formula_2 for all formula_3 and formula_4 such that formula_5 and formula_6 are in the domain of "f". The most commonly encountered symmetric functions are polynomial functions, which are given by the symmetric polynomials.
A related notion is alternating 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 "k"-tensors on a vector space "V" is isomorphic to the space of homogeneous polynomials of degree "k" on "V." Symmetric functions should not be confused with even and odd functions, which have a different sort of symmetry.
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 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 anti-symmetrization 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 anti-symmetrization.
Applications.
U-statistics.
In statistics, an "n"-sample statistic (a function in "n" variables) that is obtained by 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.