# Symmetric function

A function that does not change under any permutation of its independent variables. The following are examples of symmetric functions: $x _ {1} + \dots + x _ {n}$, $x _ {1} \cdots x _ {n}$,

$$\sum _ {1 \leq i < j \leq n } x _ {i} x _ {j} ,\ \ \max ( x _ {1}, \dots, x _ {n} ),$$

$$x _ {1} + \dots + x _ {n} ( \mathop{\rm mod} m),$$

the sum in decimal notation of an arbitrary set of one-digit numbers, a "voting function" , which is characterized by its independent variables taking only two values 1 ( "for" ) and 0 ( "against" ), and the function itself is put equal to 1 if more than half of its independent variables are 1 and is put equal to 0 otherwise. Trivial examples of symmetric functions are constant functions and a function of one variable.

Any non-constant symmetric function is essentially dependent on all its variables. Thus, the addition of inessential variables other than constants makes a function non-symmetric, and their removal may make it symmetric. Thus, the concept of a symmetric function relies on an exact indication of all its variables. A simple criterion for the symmetry of a function $f ( x _ {1}, \dots, x _ {n} )$ is that the following two equations hold simultaneously:

$$f ( x _ {1} , x _ {2} , x _ {3}, \dots, x _ {n} ) = \ f ( x _ {2} , x _ {1} , x _ {3}, \dots, x _ {n} ),$$

$$f ( x _ {1} , x _ {2} , x _ {3}, \dots, x _ {n} ) = f ( x _ {n} , x _ {1} , x _ {2}, \dots, x _ {n - 1 } )$$

or, equivalently, that $n- 1$ of the following equations hold:

$$f ( x _ {1}, \dots, x _ {i} , x _ {i + 1 }, \dots, x _ {n} ) = \ f ( x _ {1}, \dots, x _ {i + 1 } , x _ {i}, \dots, x _ {n} ),$$

together with

$$f ( x _ {1} , x _ {2}, \dots, x _ {n - 1 } , x _ {n} ) = \ f ( x _ {n} , x _ {2}, \dots, x _ {n - 1 } , x _ {1} ).$$

Symmetric functions are related to symmetric polynomials (cf. Symmetric polynomial). Every rational symmetric function (over a field of characteristic 0) is the quotient of two symmetric polynomials. Any Boolean symmetric function takes equal values on sets of its arguments containing an equal number of identities. These functions play a major role in mathematical cybernetics and its applications and, in particular, they crop up in the schematic realization of arithmetical and other operations.

How to Cite This Entry:
Symmetric function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symmetric_function&oldid=52132
This article was adapted from an original article by V.M. Khrapchenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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