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} | \}}

.

More general examples

every constant function is symmetric
a commutative two-digit combination is a symmetric function of the two operands
the mean of a set of given values is a symmetric function of these values
a symmetric multilinear map is a symmetric function that is linear in each argument
a symmetric polynomial is a symmetric polynomial function

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. ${\ displaystyle i, j \ in \ {1, \ ldots, n \}}$ ${\ displaystyle i <j}$

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. ${\ displaystyle f}$ ${\ displaystyle Sf}$ ${\ displaystyle S}$

literature

Christian Karpfinger , Kurt Meyberg: Algebra: Groups - Rings - Body . Springer, 2008, ISBN 3-8274-2018-0 .

Web links

VM Khrapchenko: Symmetric function . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
Eric W. Weisstein : Symmetric Function . In: MathWorld (English).