Symmetric function: Difference between revisions

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{{About| general properties of symmetric functions of several real or complex variable|the ring of symmetric functions in algebraic combinatorics|ring of symmetric functions|symmetric functions on elements of a vector space |symmetric tensor}}

In [[mathematics]], a '''symmetric function of ''n'' variables''' is one whose value given n [[argument of a function|arguments]] is the same no matter the order of the arguments. For example, if <math>f=f(x_1,x_2)</math> is a symmetric function, then <math>f(x_1,x_2)=f(x_2,x_1)</math> for all pairs <math>(x_1,x_2)</math> in the ___domain of <math>f</math>. While this notion can apply to any type of function whose ''n'' arguments have the same ___domain set, it is most often used for [[polynomial function]]s, in which case these are the functions given by [[symmetric polynomials]]. ThereA related notion is verythat littleof systematicthe theory[[Alternating polynomial|alternating polynomials]], who change sign under an interchange of symmetricvariables. Aside from non-polynomial functions, [[Symmetric tensor|tensors]] that act as functions of several vectors can be symmetric, and in fact the space of symmetric ''nk''-tensors variables,on soa thisvector sensespace ''V'' is little-usedisomorphic to the space of homogeneous polynomials of degree ''k'' on ''V.'' Symmetric functions should not be confused with [[even and odd functions]], exceptwhich ashave a generaldifferent sort of definitionsymmetry.

== Symmetrization ==

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==See also==

* [[Symmetrization]]
*[[Elementary symmetric polynomial]]

*[[Alternating polynomial|Alternating polynomials]]

*[[Vandermonde polynomial]]

* [[Quasisymmetric function]]

* [[Ring of symmetric functions]]