# Symmetric function

In mathematics, a symmetric function of n variables is one whose value given n arguments is the same no matter the order of the arguments. For example, if $f=f(x_{1},x_{2})$ is a symmetric function, then $f(x_{1},x_{2})=f(x_{2},x_{1})$ for all pairs $(x_{1},x_{2})$ in the domain of $f$ . While this notion can apply to any type of function whose n arguments have the same domain set, it is most often used for polynomial functions, in which case these are the functions given by symmetric polynomials. A related notion is that of the alternating polynomials, who 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.

## Examples

• Consider the real function
$f(x_{1},x_{2},x_{3})=(x-x_{1})(x-x_{2})(x-x_{3})$ By definition, a symmetric function with n variables has the property that
$f(x_{1},x_{2},...,x_{n})=f(x_{2},x_{1},...,x_{n})=f(x_{3},x_{1},...,x_{n},x_{n-1})$ etc.
In general, the function remains the same for every permutation of its variables. This means that, in this case,
$(x-x_{1})(x-x_{2})(x-x_{3})=(x-x_{2})(x-x_{1})(x-x_{3})=(x-x_{3})(x-x_{1})(x-x_{2})$ and so on, for all permutations of $x_{1},x_{2},x_{3}$ • Consider the function
$f(x,y)=x^{2}+y^{2}-r^{2}$ If x and y are interchanged the function becomes
$f(y,x)=y^{2}+x^{2}-r^{2}$ which yields exactly the same results as the original f(x,y).
• Consider now the function
$f(x,y)=ax^{2}+by^{2}-r^{2}$ If x and y are interchanged, the function becomes
$f(y,x)=ay^{2}+bx^{2}-r^{2}.$ This function is obviously not the same as the original if ab, which makes it non-symmetric.

## 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.