Symmetric function

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 antisymmetrization 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 antisymmetrization. == 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,\ldots,x_n) = f(x_2,x_1,\ldots,x_n) = f(x_3,x_1,\ldots,x_n,x_{n-1}), \quad \text{ 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 not the same as the original if a \neq b, 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. ==See also==