Power sum symmetric polynomial

In mathematics, specifically in commutative algebra, the power sum symmetric polynomials are a type of basic building block for symmetric polynomials, in the sense that every symmetric polynomial can be expressed as a sum and difference of products of power sum symmetric polynomials.

Definition

The power sum symmetric polynomial of degree k in $n$ variables x₁, ..., x_n, written p_k for k = 0, 1, 2, ..., is the sum of all kth powers of the variables. Formally,

$p_{k}(x_{1},x_{2},\dots ,x_{n})=\sum _{i=1}^{n}x_{i}^{n}\,.$

The first few of these polynomials are

$p_{0}(x_{1},x_{2},\dots ,x_{n})=n,$

$p_{1}(x_{1},x_{2},\dots ,x_{n})=x_{1}+x_{2}+\cdots +x_{n}\,,$

$p_{2}(x_{1},x_{2},\dots ,x_{n})=x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\,,$

$p_{3}(x_{1},x_{2},\dots ,x_{n})=x_{1}^{3}+x_{2}^{3}+\cdots +x_{n}^{3}\,.$

Thus, for each nonnegative integer $k$, there exists exactly one power sum symmetric polynomial of degree $k$ in $n$ variables.

The polynomial ring formed by taking all integral linear combinations of products of the power sum symmetric polynomials is a commutative ring.

Examples

The following lists the $n$ power sum symmetric polynomials of positive degrees up to n for the first three positive values of $n.$ In every case, $p_{0}=n$ is one of the polynomials. The list goes up to degree n because the power sum symmetric polynomials of degrees 1 to n are basic in the sense of the Main Theorem stated below.

For n = 1:

$h_{1}=x_{1}\,.$

For n = 2:

$h_{1}=x_{1}+x_{2}\,$ and

$h_{2}=x_{1}^{2}+x_{2}^{2}\,.$

For n = 3:

$h_{1}=x_{1}+x_{2}+x_{3}\,,$

$h_{2}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\,,$ and

$h_{3}=x_{1}^{3}+x_{2}^{3}+x_{3}^{3}\,,$

Properties

The set of complete homogeneous symmetric polynomials of degrees 1, 2, ..., n in n variables generates the ring of symmetric polynomials in n variables. More specifically (Main Theorem), the ring of symmetric polynomials with rational coefficients equals the integral polynomial ring $\mathbb {Q} [h_{0},h_{1},\ldots ,h_{n}].$ The same is true if the coefficients are taken in any field F. However, it is not true if the coefficients must be integers. For example, for n = 2, the symmetric polynomial

$P(x_{1},x_{2})=x_{1}^{2}x_{2}+x_{1}x_{2}^{2}+x_{1}x_{2}$

has the expression

$P(x_{1},x_{2})={\frac {p_{1}^{3}-p_{1}p_{2}}{2}}+{\frac {p_{1}^{2}-p_{2}}{2}}\,,$

which involves fractions. According to the Main Theorem this is the only way to represent $P(x_{1},x_{2})$ in terms of p₁ and p₂.

Partial sketch of proof of the Main Theorem: Represent the elementary symmetric polynomials as polynomial functions of the power sum symmetric polynomials of degrees 1, ..., n. Since the elementary symmetric polynomials are an algebraic basis for all symmetric polynomials with coefficients in a field, it follows that every symmetric polynomial in n variables is a polynomial function $f(p_{1},\ldots ,p_{n})$ of the power sum symmetric polynomials p₁, ..., p_n. (This does not show how to prove the polynomial f is unique.) For more information about the relationship between the power sum and elementary symmetric polynomials see Newton's identities.

We cannot come to the same conclusion if we take integral coefficients, since the elementary symmetric polynomials e_k, expressed as polynomials in the power sum polynomials, do not all have integral coefficients. For instance,

$e_{2}:=\sum {1\leq i<j\leq n}x_{i}x_{j}={\frac {p_{1}^{2}-p_{2}}{2}}\,.$

For another system of symmetric polynomials with similar properties see complete homogeneous symmetric polynomials.

References

• Macdonald, I.G. (1979), Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs. Oxford: Clarendon Press.
• Macdonald, I.G. (1995), Symmetric Functions and Hall Polynomials, second ed. Oxford: Clarendon Press. ISBN 0-19-850450-0 (paperback, 1998).
• Richard P. Stanley (1999), Enumerative Combinatorics, Vol. 2. Camridge: Cambridge University Press. ISBN 0-521-56069-1

See also

• Representation theory
• Newton's identities