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I added an example because the article was a little too technical for people not familiar with group theory. Let me know what you think! |
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{{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'' = 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.
== Example ==
Let's consider the following real function:
<math>f(x_1,x_2,x_3)=(x-x_1)(x-x_2)(x-x_3)</math>
By definition, a symmetric function with n variables has the property that
<math>f(x_1,x_2,...,x_n)=f(x_2,x_1,...,x_n)=f(x_3,x_1,...,x_n,x_{n-1})</math> etc.
In general, the function remains the same for every [[permutation]] of its variables. This means that, in this case:
<math> (x-x_1)(x-x_2)(x-x_3)=(x-x_2)(x-x_1)(x-x_3)=(x-x_3)(x-x_1)(x-x_2)</math>, and so on, for all permutations of <math>x_1,x_2,x_3</math>
== Applications ==
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