In mathematics, a function variables is symmetric if its value is the same no matter the order of its arguments are in the domain 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-tensors on a vector space is isomorphic to the space of homogeneous polynomials Symmetric functions should not be confused with even and odd functions, which have a different sort of symmetry.

Symmetrization

variables with values in an abelian group 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 is equal to half the sum of its symmetrization and its antisymmetrization.

Examples

Consider the real In general, the function remains the same for every permutation

which makes it non-symmetric.

Applications

U-statistics

In statistics variables) that is obtained by bootstrapping variables, is called a U-statistic. Examples include the sample mean and sample variance.

See also

References

• 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, .