View source for Symmetric polynomial

{{TEX|done}}
A polynomial $f$ with coefficients in a field or a commutative associative ring $K$ with a unit, which is a [[Symmetric function|symmetric function]] in its variables, that is, is invariant under all permutations of the variables:

\[
\label{symm}
f(x_1,\ldots,x_n) = f(\pi(x_1),\ldots,\pi(x_n)).
\]

The symmetric polynomials form the algebra $S(x_1,\ldots,x_n)$ over $K$.

The most important examples of symmetric polynomials are the elementary symmetric polynomials

\[
s_k(x_1,\ldots,x_n) = \sum_{1 \leq i_1 < \ldots < i_k \leq n} x_{i_1} \ldots x_{i_k}
\]

and the power sums

\[
p_k(x_1,\ldots,x_n) = x_1^k + \ldots + x_n^k.
\]

The latter can be expressed in terms of elementary symmetric polynomials by recurrence formulas, called Newton's formulas:

\[
\begin{aligned}
p_k - p_{k-1} s_1 + p_{k-2} s_2 + \ldots  + (-1)^{k-1} p_1 s_{k-1}+  (-1)^{k} k s_{k} &= 0 \quad &\text{if }1 \leq k \leq n,
\\
p_k - p_{k-1} s_1 + p_{k-2} s_2 + \ldots  + (-1)^{n-1} p_{k-n+1} s_{n-1}+  (-1)^{n} p_{k-n} s_{n} &= 0 \quad &\text{if }k > n.
\end{aligned}
\]

For the elementary symmetric polynomials $s_1,\ldots,s_k$ ($1 \leq k \leq n$) of the roots of an arbitrary polynomial in one variable with leading coefficient 1, $x^n+\alpha_1 x^{n-1}+ \ldots  + \alpha_n$, one has $\alpha_k=(-1)^k s_k$ (see [[Viète theorem|Viète theorem]]).

The fundamental theorem on symmetric polynomials: Every symmetric polynomial is a polynomial in the elementary symmetric polynomials, and this representation is unique. In other words, the elementary symmetric polynomials are a set of free generators for the algebra $S(x_1,\ldots,x_n)$. If the field has characteristic 0, then the polynomials $p_1,\ldots,p_n$ also form a set of free generators of this algebra.

A skew-symmetric, or alternating, polynomial is a polynomial $f(x_1,\ldots,x_n)$ satisfying the relation \ref{symm} if $\pi$ is even and the relation

\[
f(x_1,\ldots,x_n) = -f(\pi(x_1),\ldots,\pi(x_n))
\]

if $\pi$ is odd. Any skew-symmetric polynomial can be written in the form $\Delta_n g$, where $g$ is a symmetric polynomial and

\[
\Delta_n = \prod_{i<j} (x_i-x_j).
\]

This representation is not unique, in view of the relation $\Delta_n^2=\operatorname{Dis}(s_1,\ldots,s_n)$.

====References====
{|
||{{Ref|Ku}}||  A.G. Kurosh, "Higher algebra", MIR  (1972)  (Translated from Russian)
|-
||{{Ref|Ko}}||  A.I. Kostrikin, "Introduction to algebra", Springer  (1982)  (Translated from Russian)
|-
||{{Ref|MP}}||  A.P. Mishina, I.V. Proskuryakov, "Higher algebra. Linear algebra, polynomials, general algebra", Pergamon (1965)  (Translated from Russian)
|-
||{{Ref|Pr}}|| V. V. Prasolov, "Polynomials", vol. 11 of "Algorithms and Computation in Mathematics", Springer (2004).
|}

====Comments====
Another important set of symmetric polynomials, which appear in the representations of the symmetric group, are the Schur polynomials ($S$-functions) $S_{\{\lambda\}}$. These are defined for any partition $\{\lambda_1,\ldots,\lambda_p\}=\lambda$, and include as special cases the above functions, e.g. $S_{\{1,\ldots,1\}}=s_k$,  $S_{\{ k \}}=p_k$ (see, e.g., [[#References|[a4]]], Chapt. VI).

In general, the discriminant of the polynomial $a_0 x^n + \ldots + a_n $  with roots $x_1,\ldots,x_n$ is defined as $D=a_0^{2n-2} \prod_{1 \leq i < j \leq n} (x_i-x_j)^2$, and satisfies

\[
D = a_0^{2n-2} \begin{vmatrix}
p_0 & \ldots & p_{n-1}
\\
p_1 &\ldots & p_n
\\
\ldots & \ldots & \ldots
\\
p_{n-1} & \ldots & p_{2n-2}
\end{vmatrix},
\]

with $p_0(x_1,\ldots,x_n)=n$.

See [[Discriminant|Discriminant]].

Let $A_n \subset S_n$ be the [[Alternating group|alternating group]], consisting of the even permutations. The ring of polynomials $k[x_1,\ldots,x_n]^{A_n}$ of polynomials over a field $k$ obviously contains the elementary symmetric functions $s_1,\ldots,s_n$ and $\Delta_n=\prod_{i<j} (x_i-x_j)$. If $k$ is not of characteristic $2$, the ring of polynomials is generated by $s_1,\ldots,s_n$ and $\Delta$, and the ideal of relations is generated by $\Delta_n^2=\operatorname{Dis}(s_1,\ldots,s_n)$. The condition $\operatorname{char} k \neq 2$ is also necessary for the statement that every skew-symmetric polynomial is of the form $\Delta_n g$  with $g$ symmetric. More precisely, what is needed for this is that $2u=0$ implies $u=0$ for $u \in k$.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Jacobson,   "Basic algebra" , '''1''' , Freeman  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.G. Kurosh,   "An introduction to algebra" , MIR  (1971)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B.L. van der Waerden,   "Algebra" , '''1''' , Springer  (1967)  (Translated from German)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D.E. Littlewood,   "The theory of group characters and matrix representations of groups" , Clarendon Press  (1950)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  V. Poénaru,   "Singularités <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091700/s09170053.png" /> en présence de symmétrie" , Springer  (1976)  pp. 14ff</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P.M. Cohn,   "Algebra" , '''1''' , Wiley  (1982)  pp. 181</TD></TR></table>