Symmetric function

This article is about general properties of symmetric functions. For the ring of symmetric functions in algebraic combinatorics, see ring of symmetric functions.

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. ${\displaystyle f({\mathbf {x}})=f(x_{1},x_{2},x_{3})}$, the function can be symmetric on all its variables, or just on ${\displaystyle (x_{1},x_{2})}$, ${\displaystyle (x_{2},x_{3})}$, or ${\displaystyle (x_{1},x_{3})}$. 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

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

${\displaystyle f(x_{1},x_{2},x_{3})=(x-x_{1})(x-x_{2})(x-x_{3})}$ 

By definition, a symmetric function with n variables has the property that

${\displaystyle f(x_{1},x_{2},...,x_{n})=f(x_{2},x_{1},...,x_{n})=f(x_{3},x_{1},...,x_{n},x_{n-1})}$  etc.

In general, the function remains the same for every permutation of its variables. This means that, in this case,

${\displaystyle (x-x_{1})(x-x_{2})(x-x_{3})=(x-x_{2})(x-x_{1})(x-x_{3})=(x-x_{3})(x-x_{1})(x-x_{2})}$ 

and so on, for all permutations of ${\displaystyle x_{1},x_{2},x_{3}}$ 

• Consider the function

${\displaystyle f(x,y)=x^{2}+y^{2}-r^{2}}$ 

If x and y are interchanged the function becomes

${\displaystyle f(y,x)=y^{2}+x^{2}-r^{2}}$ 

which yields exactly the same results as the original f(x,y).

• Consider now the function

${\displaystyle f(x,y)=ax^{2}+by^{2}-r^{2}}$ 

If x and y are interchanged, the function becomes

${\displaystyle f(y,x)=ay^{2}+bx^{2}-r^{2}.}$ 

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

• Elementary symmetric polynomial
• Quasisymmetric function
• Ring of symmetric functions

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 .