Content deleted Content added
add a ref about antisymmetric functions (OA paper) |
|||
Line 2:
In [[mathematics]], a [[Function (mathematics)|function]] of ''n'' variables is '''symmetric''' if its value is the same no matter the order of its [[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 <math>x_1</math> and <math>x_2</math> such that <math>(x_1,x_2)</math> and <math>(x_2,x_1)</math> are in the [[Domain of a function|___domain]] of ''f''. The most commonly encountered symmetric functions are [[polynomial function]]s, which are given by the [[symmetric polynomial]]s.
A related notion is [[alternating polynomial]]s, which 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. Antisymmetric functions are not the opposite of symmetric functions: it exists functions which are both symmetric and antisymmetric.<ref>{{cite journal | author = Petitjean, M. | title = Symmetry, antisymmetry and chirality: Use and misuse of terminology | journal = Symmetry | year = 2021 | volume = 13 |issue = 4 | doi = 10.3390/sym13040603 | doi-access = free }}</ref>
== Symmetrization ==
| |||