Definition:Symmetric Function

Definition

Absolute

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.

Cyclic

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

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 cyclic permutation of $\tuple {x_1, x_2, \ldots, x_n}$.

Then $f$ is a cyclosymmetric function.

Elementary

Let $a, b \in \Z$ be integers such that $b \ge a$.

Let $U$ be a set of $n = b - a + 1$ numbers $\set {x_a, x_{a + 1}, \ldots, x_b}$.

Let $m \in \Z_{>0}$ be a (strictly) positive integer.

An elementary symmetric function of degree $m$ is a polynomial which can be defined by the formula:

\(\ds \map {e_m} U\)

\(=\)

\(\ds \sum_{a \mathop \le j_1 \mathop < j_2 \mathop < \mathop \cdots \mathop < j_m \mathop \le b} \paren {\prod_{i \mathop = 1}^m x_{j_i} }\)

\(\ds \)

\(=\)

\(\ds \sum_{a \mathop \le j_1 \mathop < j_2 \mathop < \mathop \cdots \mathop < j_m \mathop \le b} x_{j_1} x_{j_2} \cdots x_{j_m}\)

That is, it is the sum of all products of $m$ distinct elements of $\set {x_a, x_{a + 1}, \dotsc, x_b}$.

Also defined as

Some sources use the term symmetric function to mean what $\mathsf{Pr} \infty \mathsf{fWiki}$ defines as an even function.

Also see

• Results about symmetric functions can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): symmetric function: 2.