Editing Symmetric polynomial (section)

{{Short description|Polynomial invariant under variable permutations}}
{{About|individual symmetric polynomials|the ring of symmetric polynomials|ring of symmetric functions}}
In [[mathematics]], a '''symmetric polynomial''' is a [[polynomial]] {{math|''P''(''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''n''</sub>)}} in {{math|''n''}} variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, {{math|''P''}} is a ''symmetric polynomial'' if for any [[permutation]] {{math|σ}} of the subscripts {{math|1, 2, ..., ''n''}} one has {{math|''P''(''X''<sub>σ(1)</sub>, ''X''<sub>σ(2)</sub>, ..., ''X''<sub>σ(''n'')</sub>)&nbsp;{{=}}&nbsp;''P''(''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''n''</sub>)}}.

Symmetric polynomials arise naturally in the study of the relation between the [[root of a polynomial|roots of a polynomial]] in one variable and its [[coefficient]]s, 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 polynomial]]s 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 symmetric polynomial|complete homogeneous]], [[power sum symmetric polynomial|power sum]], and [[Schur polynomial]]s 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]].