Symmetric Function

A Symmetric Function is a function that remains unchanged under any permutation of its input variables, meaning its value is the same regardless of the order of its arguments.

• Context:
  • It can satisfy the condition that for any permutation [math]\displaystyle{ \sigma }[/math], [math]\displaystyle{ f(x_1, x_2, \ldots, x_n) = f(x_{\sigma(1)}, x_{\sigma(2)}, \ldots, x_{\sigma(n)}) }[/math].
  • It can be represented by symmetric polynomials in the context of algebra.
  • It can appear in the study of invariant theory and group theory, particularly involving the symmetric group.
  • It can model physical systems where particles are indistinguishable, as in quantum mechanics with bosons.
  • It can be used in defining symmetric tensors in tensor analysis.
  • It can play a role in statistics when dealing with functions of identically distributed random variables.
  • It can generalize to symmetric functions in infinite variables within symmetric function theory.
  • It can be contrasted with alternating functions, which change sign under odd permutations.
  • It can be applied in combinatorics and representation theory.
  • ...
• Example(s):
  • A Distance Metric Function like the Euclidean distance [math]\displaystyle{ d(x, y) = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2} }[/math], which is symmetric in [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math].
  • The function [math]\displaystyle{ f(x, y) = x + y }[/math], satisfying [math]\displaystyle{ f(x, y) = f(y, x) }[/math].
  • The symmetric polynomial [math]\displaystyle{ f(x, y, z) = x y + y z + x z }[/math].
  • The function [math]\displaystyle{ f(x_1, x_2, x_3) = x_1 x_2 x_3 }[/math], which remains the same under any permutation of [math]\displaystyle{ x_1, x_2, x_3 }[/math].
  • ...
• Counter-Example(s):
  • A Non-Symmetric Function such as [math]\displaystyle{ f(x, y) = x - y }[/math], where [math]\displaystyle{ f(x, y) \neq f(y, x) }[/math].
  • An Anti-Symmetric Function like [math]\displaystyle{ f(x, y) = x y - y x }[/math], which satisfies [math]\displaystyle{ f(x, y) = -f(y, x) }[/math].
  • Functions like the Kullback-Leibler (KL) Divergence, which are not symmetric in their arguments.
• See: Symmetric Relation, Symmetric Operation, Alternating Function, Even Function, Odd Function, Permutation Group, Symmetric Group

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References

2024

• (Wikipedia, 2024) ⇒ https://en.wikipedia.org/wiki/Symmetric_function Retrieved:2024-9-13.
  • In mathematics, a function of [math]\displaystyle{ n }[/math] variables is symmetric if its value is the same no matter the order of its arguments. For example, a function [math]\displaystyle{ f\left(x_1,x_2\right) }[/math] of two arguments is a symmetric function if and only if [math]\displaystyle{ f\left(x_1,x_2\right) = f\left(x_2,x_1\right) }[/math] for all [math]\displaystyle{ x_1 }[/math] and [math]\displaystyle{ x_2 }[/math] such that [math]\displaystyle{ \left(x_1,x_2\right) }[/math] and [math]\displaystyle{ \left(x_2,x_1\right) }[/math] are in the domain of [math]\displaystyle{ f. }[/math] The most commonly encountered symmetric functions are polynomial functions, 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 [math]\displaystyle{ k }[/math] -tensors on a vector space [math]\displaystyle{ V }[/math] is isomorphic to the space of homogeneous polynomials of degree [math]\displaystyle{ k }[/math] on [math]\displaystyle{ V. }[/math] Symmetric functions should not be confused with even and odd functions, which have a different sort of symmetry.

2009

• …
  • In mathematics, the term "symmetric function" can mean two different things. A symmetric function of n variables is one whose value at any n-tuple of arguments is the same as its value at any permutation of that n-tuple. While this notion can apply to any type of function whose n arguments live in the same set, it is most often used for polynomial functions, in which case these are the functions given by symmetric polynomials. There is hardly any systematic theory of symmetric non-polynomial functions of n variables, which are therefore not considered in this article.

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