Definition:Symmetric Function/Absolute

Definition

Let $f: \R^n \to \R$ be a real-valued function.

Definition $1$

Let $f$ be such that, for every pair $\tuple {x_i, x_j}$ in $\R^2$:

$\map f {x_1, x_2, \ldots, x_i, \ldots, x_j, \ldots, x_n} = \map f {x_1, x_2, \ldots, x_j, \ldots, x_i, \ldots, x_n}$

Then $f$ is an absolutely symmetric function.

Definition $2$

Let $f$ be such that, for all $\mathbf x := \tuple {x_1, x_2, \ldots, x_n} \in \R^n$:

$\map f {\mathbf x} = \map f {\mathbf y}$

where $\mathbf y$ is a permutation of $\tuple {x_1, x_2, \ldots, x_n}$.

Then $f$ is an absolutely symmetric function.

Examples

Arbitrary Example $1$

Let $f: \R^3 \to \R$ be the real-valued function defined as:

$\forall \tuple {x, y, z} \in \R^3: \map f {x, y, z} = x^2 + y^2 + 2 x y z$

Then $f$ is an absolutely symmetric function.

Arbitrary Example $2$

Let $f: \R^2 \to \R$ be the real-valued function defined as:

$\forall \tuple {x, y} \in \R^2: \map f {x, y} = x^2 + 2 x y + y^2$

Then $f$ is an absolutely symmetric function.

Also known as

An absolutely symmetric function is also known as a totally symmetric function.

Some sources use the term symmetric function,

Also see

• Equivalence of Definitions of Absolutely Symmetric Function

• Results about absolutely symmetric functions can be found here.