Symmetric function

In mathematics, a function of ${\displaystyle n}$ variables is symmetric if its value is the same no matter the order of its arguments. 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 the domain of ${\displaystyle f.}$ 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 ${\displaystyle k}$-tensors on a vector space ${\displaystyle V}$ is isomorphic to the space of homogeneous polynomials of degree ${\displaystyle k}$ on ${\displaystyle V.}$ Symmetric functions should not be confused with even and odd functions, which have a different sort of symmetry.