Symmetric function

A symmetric function is a mathematical function of several variables, in which the variables may be interchanged without changing the function value. Important special cases of symmetric functions are symmetric multilinear forms and symmetric polynomials. In quantum mechanics bosons are precisely those particles whose wave function is symmetric with respect to the exchange of particle positions. The counterpart of the symmetric functions are antisymmetric functions.

• 3.1 permutations of two variables
• 3.2 permutations of adjacent variables
• 3.3 permutations with a fixed variable
• 3.4 minimum criterion

Definition

Are and two quantities, it means a multivariate function symmetric if for all permutations of the symmetric group and all elements

Applies. In practice usually amounts and vector spaces over the real or complex numbers can be used.

Examples

Concrete examples

The sum and product

Are symmetric, since by interchanging the two operands and the result does not change. A symmetric function of three variables, for example, the discriminant

An example of a symmetric function that is not a polynomial function is

More general examples

• Any constant function is symmetric
• A commutative binary operation is a symmetric function of the two operands
• The average of a quantity given values ​​is a symmetric function of these values
• A symmetrical multi- linear mapping is a symmetric function that is linear in each case
• A symmetric polynomial is a symmetric polynomial

Style

For proof of the symmetry of a function does not need to be checked all the possible permutations of the symmetric group.

Permutations of two variables

After each permutation can be written as a consecutive execution of transpositions of the form, a function is already symmetric if and only if the function value by the interchange of any two variables and does not change, so

Is with.

Permutations of adjacent variables

Since each transposition can be written as a consecutive execution of Nachbarvertauschungen the form, it is sufficient even to consider only consecutive variables and. So there must be for the presence of symmetry only

Apply for.

Permutations with a fixed variable

Alternatively, one can also consider the transpositions of the form and function is thus symmetrical if and only if the first one can be swapped with the -th variable, without changing the function. Thus, for the detection of the symmetry, it is sufficient if

For true. Instead of the first variable, you can also select any variable and these commute with all other variables.

Minimum criterion

A minimal system of generators of the symmetric group, the two permutations and, and therefore there is already a function if and only symmetric if the two conditions

And

Are fulfilled. The couple and may be replaced by any of the cycle length and any transposition successive elements in this cycle.

Properties

The symmetric functions form a subspace in the vector space of all functions from to ( with componentwise addition and scalar multiplication ), that is

• A scalar multiple of a symmetric function again is a symmetric function and
• The sum of two symmetric functions is also symmetric again

The zero function is trivially symmetric.

Symmetrization

By symmetrization, that is, by summing over all possible permutations

Can be any non-symmetric function assign an associated symmetric function. The Symmetrisierungsoperator leads by a projection on the subspace of symmetric functions.