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In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its
arguments An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f\left(x_2,x_1\right) for all x_1 and x_2 such that \left(x_1,x_2\right) and \left(x_2,x_1\right) are in the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Do ...
of f. The most commonly encountered symmetric functions are
polynomial function In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An examp ...
s, which are given by the
symmetric polynomial In mathematics, a symmetric polynomial is a polynomial in variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, is a ''symmetric polynomial'' if for any permutation of the subscripts one ha ...
s. A related notion is
alternating polynomial In algebra, an alternating polynomial is a polynomial f(x_1,\dots,x_n) such that if one switches any two of the variables, the polynomial changes sign: :f(x_1,\dots,x_j,\dots,x_i,\dots,x_n) = -f(x_1,\dots,x_i,\dots,x_j,\dots,x_n). Equivalently, if o ...
s, which change sign under an interchange of variables. Aside from polynomial functions,
tensors In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tens ...
that act as functions of several vectors can be symmetric, and in fact the space of symmetric k-tensors on a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
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 In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power s ...
, which have a different sort of symmetry.


Symmetrization

Given any function f in n variables with values in an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...
, 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 In mathematics, when ''X'' is a finite set with at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total ...
s and subtracting the sum over
odd permutation In mathematics, when ''X'' is a finite set with at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total or ...
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 antisymmetrization 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 antisymmetrization.


Examples


Applications


U-statistics

In statistics, an n-sample statistic (a function in n variables) that is obtained by
bootstrapping In general, bootstrapping usually refers to a self-starting process that is supposed to continue or grow without external input. Etymology Tall boots may have a tab, loop or handle at the top known as a bootstrap, allowing one to use fingers ...
symmetrization of a k-sample statistic, yielding a symmetric function in n variables, is called a U-statistic. Examples include the
sample mean The sample mean (or "empirical mean") and the sample covariance are statistics computed from a sample of data on one or more random variables. The sample mean is the average value (or mean value) of a sample of numbers taken from a larger po ...
and
sample variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...
.


See also

* * * * * * *


References

*
F. N. David Florence Nightingale David, also known as F. N. David (23 August 1909 – 23 July 1993) was an English statistician. She was head of the Statistics Department at the University of California, Riverside between 1970 – 77 and her research inte ...
,
M. G. Kendall Sir Maurice George Kendall, FBA (6 September 1907 – 29 March 1983) was a prominent British statistician. The Kendall tau rank correlation is named after him. Education and early life Maurice Kendall was born in Kettering, Northampton ...
& D. E. Barton (1966) ''Symmetric Function and Allied Tables'',
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambr ...
. * Joseph P. S. Kung,
Gian-Carlo Rota Gian-Carlo Rota (April 27, 1932 – April 18, 1999) was an Italian-American mathematician and philosopher. He spent most of his career at the Massachusetts Institute of Technology, where he worked in combinatorics, functional analysis, prob ...
, & Catherine H. Yan (2009) '' Combinatorics: The Rota Way'', §5.1 Symmetric functions, pp 222–5, Cambridge University Press, . {{Tensors Combinatorics Properties of binary operations