Symmetric Function

a function of two or more variables that remains unchanged under all permutations of the variables. Examples are and 4x₁x₂x₃

Of particular importance in algebra are symmetric polynomials, especially elementary symmetric polynomials—that is, the functions

where the summations are extended over all combinations of unequal numbers k, l, …. The sums are linear in each of the variables. According to the Vieta formulas, x₁, x₂,…,x_n are the roots of the equation

xⁿ – f₁ x^n-1 + f₂ x^n-2 – … + (–1)ⁿf_n = 0

The fundamental theorem of the theory of symmetric polynomials states that any symmetric polynomial can be represented in one, and only one, way as a polynomial in the elementary symmetric polynomials: F (x₁, x₂,…. x_n) = G (f₁, f₂,…, f_n). If all the coefficients of Fare integers, then so are the coefficients of G. Thus, every symmetric polynomial with integer coefficients on the roots of an equation can be expressed as a polynomial with integer coefficients on the coefficients of that equation; for example.

Another important class of symmetric functions are the power sums

The relation between such power sums and elementary symmetric polynomials is given by the Newton formulas:

and

These formulas permit the f_k to be expressed successively in terms of the s_m, and vice versa.

A function is said to be skew-symmetric, or alternating, if it remains unchanged under even permutations of x₁, x₂, …., x_n and changes sign under odd permutations. Such functions can be rationally expressed in terms of f₁, f₂,…, f_n and the product (seeDISCRIMINANT)

the square of which is a symmetric function and thus can be rationally expressed in terms of f₁, f₂,…,f_n