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 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

[1]	A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian)
[2]	A.I. Kostrikin, "Introduction to algebra" , Springer (1982) (Translated from Russian)
[3]	A.P. Mishina, I.V. Proskuryakov, "Higher algebra. Linear algebra, polynomials, general algebra" , Pergamon (1965) (Translated from Russian)

Comments

Another important set of symmetric polynomials, which appear in the representations of the symmetric group, are the Schur polynomials (-functions) . These are defined for any partition , and include as special cases the above functions, e.g. , (see, e.g., [a4], Chapt. VI).

In general, the discriminant of the polynomial with roots is defined as , and satisfies

with .

See Discriminant.

Let be the alternating group, consisting of the even permutations. The ring of polynomials of polynomials over a field obviously contains the elementary symmetric functions and . If is not of characteristic , the ring of polynomials is generated by and , and the ideal of relations is generated by . The condition is also necessary for the statement that every skew-symmetric polynomial is of the form with symmetric. More precisely, what is needed for this is that implies for .

References

[a1]	N. Jacobson, "Basic algebra" , 1 , Freeman (1974)
[a2]	A.G. Kurosh, "An introduction to algebra" , MIR (1971) (Translated from Russian)
[a3]	B.L. van der Waerden, "Algebra" , 1 , Springer (1967) (Translated from German)
[a4]	D.E. Littlewood, "The theory of group characters and matrix representations of groups" , Clarendon Press (1950)
[a5]	V. Poénaru, "Singularités en présence de symmétrie" , Springer (1976) pp. 14ff
[a6]	P.M. Cohn, "Algebra" , 1 , Wiley (1982) pp. 181