Definition:Symmetric Function/Elementary

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Definition

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 $\set {x_a, x_{a + 1}, \dotsc, x_b}$.


Notation

Let $U$ be established such that $\size U = n$.

It is commonplace for $U$ to be presented in the form $\set {x_1, x_2, \ldots, x_n}$.

Consequently, the elementary symmetric function of degree $m$ is often presented $\map {e_m} {\set {x_1, x_2, \ldots, x_n} }$, which is unwieldy.


Hence $\mathsf{Pr} \infty \mathsf{fWiki}$ introduces the notation:

$E_{\tuple {m, n} } := \map {e_m} {\set {x_1, x_2, \ldots, x_n} }$


The term $E_{\tuple {m, n} }$ was invented by $\mathsf{Pr} \infty \mathsf{fWiki}$ in order to streamline the presentation of resources concerning elementary symmetric functions.

As such, it is not generally expected to be seen in this context outside $\mathsf{Pr} \infty \mathsf{fWiki}$.


Examples

Example: $m = 0$

$\map {e_0} {\set {x_1, x_2, \ldots, x_n} } = 1$


Example: $m = 1$

\(\ds e_1 \left({\left\{ {x_1, x_2, \ldots, x_n}\right\} }\right)\) \(=\) \(\ds x_1 + x_2 + \cdots + x_n\)


Example: $m = 2$

\(\ds e_2 \left({\left\{ {x_1, x_2, \ldots, x_n}\right\} }\right)\) \(=\) \(\ds x_1 x_2 + x_1 x_3 + \cdots + x_1 x_n\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds x_2 x_3 + \cdots + x_2 x_n\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \cdots\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds x_{n - 1} x_n\)


Example: $m = n$

\(\ds \map {e_n} {\set {x_1, x_2, \ldots, x_n} }\) \(=\) \(\ds x_1 x_2 \cdots x_n\)


Example: $m > n$

Let $m > n$.

Then:

\(\ds \map {e_m} {\set {x_1, x_2, \ldots, x_n} }\) \(=\) \(\ds 0\)


Example: Monic polynomial coefficients

Let $\set {x_1, x_2, \ldots, x_n}$ be a set of real or complex values, not required to be unique.

The expansion of the monic polynomial in variable $x$ with roots $\set {x_1, x_2, \ldots, x_n}$ has coefficients which are parity operators times an elementary symmetric function:

$\ds \prod_{j \mathop = 1}^n \paren {x - x_j} = x^n - \map {e_1} {\set {x_1, \ldots, x_n} } x^{n - 1} + \map {e_2} {\set {x_1, \ldots, x_n} } x^{n - 2} - \dotsb + \paren {-1}^n \map {e_n} {\set {x_1, \ldots, x_n} }$


Also known as

An elementary symmetric function is also known as an elementary symmetric polynomial.


Also see

  • Results about elementary symmetric functions can be found here.


Sources

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