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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. A polynomial with this property is a symmetric polynomial. A symmetric rational function is a rational function unchanged by permutation of its variables.

Over a field, 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), the defining property of an even function. Examples include f(x)=|x| (the absolute value) and f(x)=x^2.

See also

Elementary Symmetric Polynomial, Even Function, 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, England: 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. https://www2.math.upenn.edu/~wilf/AeqB.html.

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Symmetric Function

Cite this as:

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

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