# Symmetrical function

In mathematics, a symmetrical function is a function of several variables in which the variables can 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 symmetrical with respect to the exchange of particle positions. The counterpart to the symmetric functions are antisymmetric functions .

## Definitions

If and are two sets , then a multivariate function is called symmetric if for all permutations of the symmetric group and all elements${\ displaystyle X}$${\ displaystyle Y}$${\ displaystyle f \ colon X ^ {n} \ to Y}$ ${\ displaystyle \ sigma \ in S_ {n}}$ ${\ displaystyle S_ {n}}$${\ displaystyle x_ {1}, \ dotsc, x_ {n} \ in X}$

${\ displaystyle f (x_ {1}, \ dotsc, x_ {n}) = f (x _ {\ sigma (1)}, \ dotsc, x _ {\ sigma (n)})}$

applies. In practice, the sets and mostly vector spaces over the real or complex numbers are used. ${\ displaystyle X}$${\ displaystyle Y}$

This definition can be generalized to functions with a countable number of arguments as follows . A function is called -symmetric if for all permutations and all elements${\ displaystyle f \ colon X ^ {\ mathbb {N}} \ to Y}$${\ displaystyle n}$${\ displaystyle \ sigma \ in S_ {n}}$${\ displaystyle x_ {i} \ in X}$

${\ displaystyle f (x_ {1}, \ dotsc, x_ {n}, x_ {n + 1}, \ dots) = f (x _ {\ sigma (1)}, \ dotsc, x _ {\ sigma (n) }, x_ {n + 1}, \ dots)}$

applies. A symmetric function is therefore symmetrical in the first arguments. A function is called symmetric if it is -symmetric for all . ${\ displaystyle n}$${\ displaystyle n}$${\ displaystyle f \ colon X ^ {\ mathbb {N}} \ to Y}$${\ displaystyle n}$${\ displaystyle n \ in \ mathbb {N}}$

## Examples

### Concrete examples

The sum and the product

${\ displaystyle f (x_ {1}, x_ {2}) = x_ {1} + x_ {2}}$   or.   ${\ displaystyle f (x_ {1}, x_ {2}) = x_ {1} \ cdot x_ {2}}$

are symmetrical, because swapping the two operands and does not change the result. For example, a symmetric function of three variables is the discriminant${\ displaystyle x_ {1}}$${\ displaystyle x_ {2}}$

${\ displaystyle f (x_ {1}, x_ {2}, x_ {3}) = (x_ {1} -x_ {2}) ^ {2} (x_ {1} -x_ {3}) ^ {2 } (x_ {2} -x_ {3}) ^ {2}}$,

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

${\ displaystyle f (x_ {1}, x_ {2}, x_ {3}) = \ max \ {| x_ {1} -x_ {2} |, | x_ {1} -x_ {3} |, | x_ {2} -x_ {3} | \}}$.

## further criteria

To prove the symmetry of a function, not all possible permutations of the symmetric group have to be checked. ${\ displaystyle n!}$ ${\ displaystyle S_ {n}}$

### Swap two variables

After each permutation as sequential execution of transpositions of the form can be written, a function is already if and symmetrical when the function value by swapping any two variables and does not change, so ${\ displaystyle (i ~ j)}$${\ displaystyle x_ {i}}$${\ displaystyle x_ {j}}$

${\ displaystyle f (\ dotsc, x_ {i}, \ dotsc, x_ {j}, \ dotsc) = f (\ dotsc, x_ {j}, \ dotsc, x_ {i}, \ dotsc)}$

for with is. ${\ displaystyle i, j \ in \ {1, \ ldots, n \}}$${\ displaystyle i

### Interchanges of neighboring variables

Since every transposition can also be written as a sequential execution of neighboring interchanges of the form , it is even sufficient to only consider successive variables and . So it just has to be for the existence of symmetry ${\ displaystyle (i ~ i + 1)}$${\ displaystyle x_ {i}}$${\ displaystyle x_ {i + 1}}$

${\ displaystyle f (\ dotsc, x_ {i}, x_ {i + 1}, \ dotsc) = f (\ dotsc, x_ {i + 1}, x_ {i}, \ dotsc)}$

for apply. ${\ displaystyle i = 1, \ ldots, n-1}$

### Swap with a fixed variable

Alternatively, one can also consider the transpositions of the form ; a function is therefore symmetrical if and only if the first variable can be exchanged with the -th variable without the function value changing. To prove the symmetry it is sufficient if ${\ displaystyle (1 ~ i)}$${\ displaystyle i}$

${\ displaystyle f (x_ {1}, \ dotsc, x_ {i}, \ dotsc) = f (x_ {i}, \ dotsc, x_ {1}, \ dotsc)}$

for applies. Instead of the first variable, you can also select any variable and swap it with all other variables. ${\ displaystyle i = 2, \ ldots, n}$

### Minimum criterion

The two permutations and represent a minimal generating system of the symmetric group . Therefore a function is already symmetric if and only if the two conditions ${\ displaystyle S_ {n}}$${\ displaystyle (1 ~ 2 ~ \ ldots ~ n)}$${\ displaystyle (1 ~ 2)}$

${\ displaystyle f (x_ {1}, x_ {2}, \ dotsc, x_ {n}) = f (x_ {2}, \ ldots, x_ {n}, x_ {1})}$

and

${\ displaystyle f (x_ {1}, x_ {2}, \ dotsc, x_ {n}) = f (x_ {2}, x_ {1}, \ ldots, x_ {n})}$

are fulfilled. The pair and can be replaced by any cycle of length and any transposition of successive elements in this cycle. ${\ displaystyle (1 ~ 2 ~ \ ldots ~ n)}$${\ displaystyle (1 ~ 2)}$${\ displaystyle n}$

## properties

The symmetric functions form a subspace in the vector space of all functions from to (with the component-wise addition and scalar multiplication ), that is ${\ displaystyle X ^ {n}}$${\ displaystyle Y}$

• a scalar multiple of a symmetric function is again a symmetric function and
• the sum of two symmetric functions is also symmetric again,

where the null function is trivially symmetric.

## Symmetrization

Through symmetrization, that is, through summation over all possible permutations

${\ displaystyle Sf (x_ {1}, \ dotsc, x_ {n}) = {\ frac {1} {n!}} \ sum _ {\ sigma \ in S_ {n}} f (x _ {\ sigma ( 1)}, \ dotsc, x _ {\ sigma (n)})}$,

an associated symmetric function can be assigned to each non- symmetric function . The symmetrization operator carries out a projection onto the subspace of the symmetric functions. ${\ displaystyle f}$${\ displaystyle Sf}$${\ displaystyle S}$