Symmetric function

In mathematics, 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. So, if e.g. $f(\bold{x})=f(x_1,x_2,x_3)$ , the function can be symmetric on all its variables, or just on $(x_1,x_2)$ , $(x_2,x_3)$ , or $(x_1,x_3)$ . 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. 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.

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

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.

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, ISBN 978-0-521-73794-4 .

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