Symmetric function: Difference between revisions

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{{about|symmetric functions which are not necessarily polynomials|symmetric functions which are polynomials|symmetric polynomials}}

In [[mathematics]], the term "symmetric function" can mean two different concepts.

A '''symmetric function of ''n'' variables''' is one whose value at any ''n''-tuple of arguments is the same as its value at any permutation of that ''n''-tuple. While this notion can apply to any type of function whose ''n'' arguments live in the same set, it is most often used for [[polynomial function]]s, in which case these are the functions given by '''[[symmetric polynomial]]spolynomials'''. There is very little systematic theory of symmetric non-polynomial functions of ''n'' variables, so this sense is little-used, except as a general definition.

In [[algebra]] and in particular in [[algebraic combinatorics]], the term "symmetric function" is often used instead to refer to elements of the '''[[ring of symmetric functions]]''', where that ring is a specific limit of the rings of symmetric polynomials in ''n'' indeterminates, as ''n'' goes to infinity. This ring serves as universal structure in which relations between symmetric polynomials can be expressed in a way independent of the number ''n'' of indeterminates (but its elements are neither polynomials nor functions). Among other things, this ring plays an important role in the [[representation theory of the symmetric group]]s.

For these specific uses, see the articles [[symmetric polynomial]]s and [[ring of symmetric functions]]; the remainder of this article addresses general properties of symmetric functions in ''n'' variables.