variables is symmetric if its value is the same no matter the order of its arguments.
For example, a function
of two arguments is a symmetric function if and only if
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
-tensors on a vector space
is isomorphic to the space of homogeneous polynomials of degree
Symmetric functions should not be confused with even and odd functions, which have a different sort of symmetry.
variables with values in an abelian group, a symmetric function can be constructed by summing values of
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
The only general case where
can be recovered if both its symmetrization and antisymmetrization are known is when
and the abelian group admits a division by 2 (inverse of doubling); then
is equal to half the sum of its symmetrization and its antisymmetrization.
-sample statistic (a function in
variables) that is obtained by bootstrapping symmetrization of a
-sample statistic, yielding a symmetric function in
variables, is called a U-statistic.
Examples include the sample mean and sample variance.