In mathematics, a symmetric polynomial is a polynomial P(X1, X2, ..., Xn) in n variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, P is a symmetric polynomial if for any permutation σ of the subscripts 1, 2, ..., n one has P(Xσ(1), Xσ(2), ..., Xσ(n)) = P(X1, X2, ..., Xn). Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients, since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting. From this point of view the elementary symmetric polynomials are the most fundamental symmetric polynomials. Indeed, a theorem called the fundamental theorem of symmetric polynomials states that any symmetric polynomial can be expressed in terms of elementary symmetric polynomials. This implies that every symmetric polynomial expression in the roots of a monic polynomial can alternatively be given as a polynomial expression in the coefficients of the polynomial. Symmetric polynomials also form an interesting structure by themselves, independently of any relation to the roots of a polynomial. In this context other collections of specific symmetric polynomials, such as complete homogeneous, power sum, and Schur polynomials play important roles alongside the elementary ones. The resulting structures, and in particular the ring of symmetric functions, are of great importance in combinatorics and in representation theory.
Examples The following polynomials in two variables X1 and X2 are symmetric:
X 1 3 + X 2 3 − 7 {\displaystyle X_{1}^{3}+X_{2}^{3}-7}
4 X 1 2 X 2 2 + X 1 3 X 2 + X 1 X 2 3 + ( X 1 + X 2 ) 4 {\displaystyle 4X_{1}^{2}X_{2}^{2}+X_{1}^{3}X_{2}+X_{1}X_{2}^{3}+(X_{1}+X_{2})^{4}}
as is the following polynomial in three variables X1, X2, X3:
X 1 X 2 X 3 − 2 X 1 X 2 − 2 X 1 X 3 − 2 X 2 X 3 {\displaystyle X_{1}X_{2}X_{3}-2X_{1}X_{2}-2X_{1}X_{3}-2X_{2}X_{3}}
There are many ways to make specific symmetric polynomials in any number of variables (see the various types below). An example of a somewhat different flavor is
∏ 1 ≤ i < j ≤ n ( X i − X j ) 2 {\displaystyle \prod _{1\leq i<j\leq n}(X_{i}-X_{j})^{2}}
where first a polynomial is constructed that changes sign under every exchange of variables, and taking the square renders it completely symmetric (if the variables represent the roots of a monic polynomial, this polynomial gives its discriminant). On the other hand, the polynomial in two variables
X 1 − X 2 {\displaystyle X_{1}-X_{2}}
is not symmetric, since if one exchanges X 1 {\displaystyle X_{1}} and X 2 {\displaystyle X_{2}} one gets a different polynomial, X 2 − X 1 {\displaystyle X_{2}-X_{1}} . Similarly in three variables
X 1 4 X 2 2 X 3 + X 1 X 2 4 X 3 2 + X 1 2 X 2 X 3 4 {\displaystyle X_{1}^{4}X_{2}^{2}X_{3}+X_{1}X_{2}^{4}X_{3}^{2}+X_{1}^{2}X_{2}X_{3}^{4}}
has only symmetry under cyclic permutations of the three variables, which is not sufficient to be a symmetric polynomial. However, the following is symmetric:
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