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Fundamental Theorem of Symmetric Functions

The fundamental theorem of symmetric functions states that every symmetric polynomial in variables X_1, ..., X_n over a commutative ring R with identity is a polynomial over R in the elementary symmetric polynomials on those variables. Over a field, every symmetric rational function is a rational function in the elementary symmetric polynomials.

There is a generalization of this theorem to polynomial invariants of permutation groups G, which states that any polynomial invariant f in R[X_1,...,X_n] can be represented as a finite linear combination of special G-invariant orbit polynomials with symmetric functions as coefficients, i.e.,

f=sum_(tspecial)p_t(sigma_1,...,sigma_n)orbit_G(t),

where p_t in R[X_1,...,X_n],

orbit_G(t)=sum_(s in {pi(t)|pi in G})s,

and sigma_1, ..., sigma_n are the elementary symmetric polynomials, and t=X_1^(e_1)...X_n^(e_n) ranges over special terms. Furthermore, any special term t has a total degree <=n(n-1)/2 and a maximal variable degree <=n-1.

See also

Elementary Symmetric Polynomial, Permutation Group, Symmetric Polynomial

Portions of this entry contributed by Manfred Goebel

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References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 2, 1959.Göbel, M. "Computing Bases for Rings of Permutation-Invariant Polynomials." J. Symb. Comput. 19, 285-291, 1995.Göbel, M. "On the Number of Special Permutation-Invariant Orbits and Terms." Appl. Algebra Eng. Comm. Comput. 8, 505-509, 1997.Herstein, I. N. Noncommutative Rings. Washington, DC: Math. Assoc. Amer., 1968.

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Fundamental Theorem of Symmetric Functions

Cite this as:

Weisstein, Eric W., with contributions by Manfred Goebel. "Fundamental Theorem of Symmetric Functions." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FundamentalTheoremofSymmetricFunctions.html

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