In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a
function of
variables is symmetric if its value is the same no matter the order of its
arguments. For example, a function
of two arguments is a symmetric function if and only if
for all
and
such that
and
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
* ...
of
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 exa ...
s, which are given by the
symmetric polynomials.
A related notion is
alternating polynomials, which change sign under an interchange of variables. Aside from polynomial functions,
tensors that act as functions of several vectors can be symmetric, and in fact the space of symmetric
-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 can ...
is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the space of
homogeneous polynomials of degree
on
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 ser ...
, which have a different sort of symmetry.
Symmetrization
Given any function
in
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 comm ...
, a symmetric function can be constructed by summing values of
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 o ...
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 o ...
s. These operations are of course not invertible, and could well result in a function that is identically zero for nontrivial functions
The only general case where
can be recovered if both its symmetrization and antisymmetrization are known is when
and the abelian group admits a division by 2 (inverse of doubling); then
is equal to half the sum of its symmetrization and its antisymmetrization.
Examples
- Consider the real function
By definition, a symmetric function with variables has the property that
In general, the function remains the same for every
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
of its variables. This means that, in this case,
and so on, for all permutations of
- Consider the function
If and are interchanged the function becomes
which yields exactly the same results as the original
- Consider now the function
If and are interchanged, the function becomes
This function is not the same as the original if which makes it non-symmetric.
Applications
U-statistics
In
statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, an
-sample statistic (a function in
variables) that is obtained by
bootstrapping symmetrization of a
-sample statistic, yielding a symmetric function in
variables, is called a
U-statistic
In statistical theory, a U-statistic is a class of statistics that is especially important in estimation theory; the letter "U" stands for unbiased. In elementary statistics, U-statistics arise naturally in producing minimum-variance unbiased es ...
. 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 popu ...
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 numbe ...
.
See also
*
*
*
*
*
*
*
References
*
F. N. David,
M. G. Kendall & 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 King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer.
Cambridge University Pr ...
.
* Joseph P. S. Kung,
Gian-Carlo Rota, &
Catherine H. Yan (2009) ''
Combinatorics: The Rota Way'', §5.1 Symmetric functions, pp 222–5, Cambridge University Press, .
{{Tensors
Combinatorics
Properties of binary operations