Symmetric function

In mathematics, the term "symmetric function" can mean two different concepts.

A symmetric function of n variables is one whose value at any n-tuple of arguments is the same as its value at any permutation of that n-tuple. While this notion can apply to any type of function whose n arguments live in the same set, it is most often used for polynomial functions, in which case these are the functions given by symmetric polynomials. There is very little systematic theory of symmetric non-polynomial functions of n variables, so this sense is little-used, except as a general definition.

In algebra and in particular in algebraic combinatorics, the term "symmetric function" is often used instead to refer to elements of the ring of symmetric functions, where that ring is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity. This ring serves as universal structure in which relations between symmetric polynomials can be expressed in a way independent of the number n of indeterminates (but its elements are neither polynomials nor functions). Among other things, this ring plays an important role in the representation theory of the symmetric groups.

For these specific uses, see the corresponding articles; the remainder of this article addresses general properties of symmetric functions in n variables.

Symmetrization

Main article: 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 a ≠ b, which makes it non-symmetric.

Applications

U-statistics

Main article: U-statistic

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.

See also

• Ring of symmetric functions
• Quasisymmetric function