Editing Ring of symmetric functions (section)

== Symmetric polynomials ==

{{main | Symmetric polynomial }}

The study of symmetric functions is based on that of symmetric polynomials. In a [[polynomial ring]] in some finite set of indeterminates, a polynomial is called ''symmetric'' if it stays the same whenever the indeterminates are permuted in any way. More formally, there is an [[group action|action]] by [[ring automorphism]]s of the [[symmetric group]] ''S<sub>n</sub>'' on the polynomial ring in ''n'' indeterminates, where a [[permutation]] acts on a polynomial by simultaneously substituting each of the indeterminates for another according to the permutation used. The [[Invariant (mathematics)#Unchanged under group action|invariants]] for this action form the [[subring]] of symmetric polynomials. If the indeterminates are ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>, then examples of such symmetric polynomials are

: <math>X_1+X_2+\cdots+X_n, \, </math>
: <math>X_1^3+X_2^3+\cdots+X_n^3, \, </math>

and

:<math>X_1X_2\cdots X_n. \, </math>

A somewhat more complicated example is
''X''<sub>1</sub><sup>3</sup>''X''<sub>2</sub>''X''<sub>3</sub> + ''X''<sub>1</sub>''X''<sub>2</sub><sup>3</sup>''X''<sub>3</sub> + ''X''<sub>1</sub>''X''<sub>2</sub>''X''<sub>3</sub><sup>3</sup> + ''X''<sub>1</sub><sup>3</sup>''X''<sub>2</sub>''X''<sub>4</sub> + ''X''<sub>1</sub>''X''<sub>2</sub><sup>3</sup>''X''<sub>4</sub> + ''X''<sub>1</sub>''X''<sub>2</sub>''X''<sub>4</sub><sup>3</sup> + ...
where the summation goes on to include all products of the third power of some variable and two other variables. There are many specific kinds of symmetric polynomials, such as [[elementary symmetric polynomial]]s, [[power sum symmetric polynomial]]s, [[monomial symmetric polynomial]]s, [[complete homogeneous symmetric polynomial]]s, and [[Schur polynomial]]s.