DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-16

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Power sums $p_{k}$ and complete homogeneous symmetric polynomials $h_{k}$

Definition

For $k \geq 1$ , the $k$ -th power-sum symmetric polynomial is

$p_{k} : = x_{1}^{k} + \dots + x_{n}^{k} .$

For $k \geq 0$ , the $k$ -th complete homogeneous symmetric polynomial is

$h_{k} : = \sum_{a_{1} + \dots + a_{n} = k} x_{1}^{a_{1}} \dots x_{n}^{a_{n}} .$

Thus $h_{0} = 1$ , while for $n = 0$ one has $h_{k} = 0$ when $k > 0$ . Both families are fixed by every permutation of the variables.

Depends on

Used by

Power sums fail to span integrally in degree two Counterexample
Colourings, weight functions, and the pattern inventory Definition
A disconnected skew Schur function factors Example
Cauchy kernel through bidegree three Example
The five standard symmetric-function bases in degree three Example
Power sums form a rational but not integral stable basis Proposition
The generating-series identity E(-t)H(t)=1 Proposition
Elementary and complete families freely generate the stable ring Theorem
Jacobi–Trudi and dual Jacobi–Trudi identities Theorem
Newton's identities: k eₖ=∑ᵢ₌₁ᵏ(-1)ⁱ⁻¹eₖ₋ᵢpᵢ Theorem
Power-sum, complete, and Schur expansions of the Cauchy kernel Theorem
Skew Jacobi–Trudi and tableau expansion Theorem

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

D. Grinberg, An Introduction to Algebraic Combinatorics, Chapter 7, Sections 7.1-7.2 (standard reference, not scraped)

Power sums pkpk​ and complete homogeneous symmetric polynomials hkhk​