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symmetric function
In mathematics, 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 have the same domain set, it is most often used for polynomial functions, 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 ==
(詳細はeven permutations and subtracting the sum over odd permutations. 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'' = 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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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