Symmetric polynomial

A polynomial $f$ with coefficients in a field or a commutative associative ring $K$ with a unit, which is a 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).

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 [Pr, Thm. 3.1.2] 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 $\Delta_n^2=\operatorname{Dis}(s_1,\ldots,s_n)$.

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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., [Li, 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.

Let $A_n \subset S_n$ be the 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 $u \in k$.

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