Ring of symmetric functions: Difference between revisions

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== Symmetric polynomials ==

{{ main article | 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 homomorphism|ring automorphism]]s of the [[symmetric group]] ''S_n'' 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 Invariant_(mathematics)#Unchanged under group actionUnchanged_under_group_action|invariants]] for this action form the subring of symmetric polynomials. If the indeterminates are ''X''₁,…,''X''_''n'', then examples of such symmetric polynomials are

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