Symmetric function: Difference between revisions

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{{About|general properties of symmetric functions|the ring of symmetric functions in algebraic combinatorics|ring of symmetric functions}}

{{technical|date=March 2013}}

In [[mathematics]], a '''symmetric function of ''n'' variables''' is one whose value atgiven any ''n''-[[tuple]] of [[argument of a function|arguments]] is the same asno itsmatter valuethe at any [[permutation]]order of thatthe ''n''-tuplearguments. So, if e.g. <math>f(\bold{x})=f(x_1,x_2,x_3)</math>, the function can be symmetric on all its variables, or just on <math>(x_1,x_2)</math>, <math>(x_2,x_3)</math>, or <math>(x_1,x_3)</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]]. 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.

== Symmetrization ==

{{main|Symmetrization}}

Given any function ''f'' in ''n'' variables with values in an [[abelian group]], a symmetric function can be constructed by summing values of ''f'' over all permutations of the arguments. Similarly, an anti-symmetric function can be constructed by summing over [[even permutation]]s and subtracting the sum over [[odd permutation]]s. These operations are of course not invertible, and could well result in a function that is identically zero for nontrivial functions ''f''. The only general case where ''f'' can be recovered if both its symmetrization and anti-symmetrization are known is when ''n''&nbsp;=&nbsp;2 and the abelian group admits a division by 2 (inverse of doubling); then ''f'' is equal to half the sum of its symmetrization and its anti-symmetrization.

== Examples ==

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* [[Quasisymmetric function]]

* [[Ring of symmetric functions]]

* [[Even and odd functions]]

==References==