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 (mathematics)|function]] of ''n'' variables is '''symmetric''' isif one whoseits value given n [[argument of a function|arguments]] is the same no matter the order of theits [[argument of a function|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</math> and <math>x_2</math> such that <math>(x_1,x_2)</math> in the ___domain ofand <math>f(x_2,x_1)</math>. Whileare thisin notionthe can[[Domain applyof toa any typefunction|___domain]] of function whose ''nf''. argumentsThe havemost thecommonly sameencountered ___domainsymmetric set,functions it is most often used forare [[polynomial function]]s, in which case these are the functions given by the [[symmetric polynomialspolynomial]]s.

A related notion is that of the [[Alternatingalternating polynomial|alternating polynomials]]s, whowhich change sign under an interchange of variables. Aside from polynomial functions, [[Symmetric tensor|tensors]] that act as functions of several vectors can be symmetric, and in fact the space of symmetric ''k''-tensors on a [[vector space]] ''V'' is [[isomorphic]] to the space of [[homogeneous polynomials]] of degree ''k'' on ''V.'' Symmetric functions should not be confused with [[even and odd functions]], which have a different sort of symmetry.

== Symmetrization ==

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== Examples==

* Consider the [[Real number|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>

* Consider the function

::<math>f(x,y) = x^2+y^2-r^2</math>

:If ''x'' and ''y'' are interchanged the function becomes

::<math>f(y,x) = y^2+x^2-r^2</math>

:which yields exactly the same results as the original ''f''(''x'',''y'').

* Consider now the function

::<math>f(x,y) = ax^2+by^2-r^2</math>

:If ''x'' and ''y'' are interchanged, the function becomes

::<math>f(y,x) = ay^2+bx^2-r^2.</math>

:This function is obviously not the same as the original if {{nowrap|1=''a'' ≠ ''b''}}, which makes it non-symmetric.