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Symmetric Function

A symmetric function on n variables x_1, ..., x_n is a function that is unchanged by any permutation of its variables. In most contexts, the term "symmetric function" refers to a polynomial on n variables with this feature (more properly called a "symmetric polynomial"). Another type of symmetric functions is symmetric rational functions, which are the rational functions that are unchanged by permutation of variables.

The symmetric polynomials (respectively, symmetric rational functions) can be expressed as polynomials (respectively, rational functions) in the elementary symmetric polynomials. This is called the fundamental theorem of symmetric functions.

A function f(x) is sometimes said to be symmetric about the y-axis if f(-x)=f(x). Examples of such functions include |x| (the absolute value) and x^2 (the parabola).

See also

Fundamental Theorem of Symmetric Functions, Rational Function, Symmetric Polynomial

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References

Bressoud, D. Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture. Cambridge, England: Cambridge University Press, 1999.Littlewood, D. E. A University Algebra, 2nd ed. London: Heinemann, 1958.Macdonald, I. G. Symmetric Functions and Hall Polynomials, 2nd ed. Oxford, England: Oxford University Press, 1995.Macdonald, I. G. Symmetric Functions and Orthogonal Polynomials. Providence, RI: Amer. Math. Soc., 1997.Petkovšek, M.; Wilf, H. S.; and Zeilberger, D. "Symmetric Function Identities." §1.7 in A=B. Wellesley, MA: A K Peters, pp. 12-13, 1996. http://www.cis.upenn.edu/~wilf/AeqB.html.

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Symmetric Function

Cite this as:

Weisstein, Eric W. "Symmetric Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SymmetricFunction.html

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