DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Symmetric polynomials as the invariants of variable permutations

Definition

Let $R$ be a commutative ring and let $R [x_{1}, \dots, x_{n}]$ be the iterated polynomial ring. Every permutation $σ \in Sym ({1, \dots, n})$ acts on it by

$σ \cdot f (x_{1}, \dots, x_{n}) = f (x_{σ (1)}, \dots, x_{σ (n)}) .$

A polynomial $f \in R [x_{1}, \dots, x_{n}]$ is symmetric when $σ \cdot f = f$ for every permutation $σ$ . The symmetric polynomials form the fixed subset

$R [x_{1}, \dots, x_{n}]^{{Sym}_{n}} : = {f : σ \cdot f = f for every σ \in {Sym}_{n}} .$

For $n = 0$ this means the coefficient ring $R$ itself.

Depends on

Used by

Monomial symmetric polynomials indexed by partitions Definition
Power sums pₖ and complete homogeneous symmetric polynomials hₖ Definition
The elementary symmetric polynomials e₀,e₁,…,eₙ Definition
The leading multidegree of a symmetric polynomial is weakly decreasing Lemma
The square of the Vandermonde polynomial is symmetric Proposition
The symmetric polynomials form a subring Proposition
Monomial symmetric polynomials form an R-basis of the symmetric-polynomial ring Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 21 results over 15 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

K. Conrad, Symmetric Polynomials, Sections 1-5 (standard reference, not scraped)
D. Grinberg, An Introduction to Algebraic Combinatorics, Chapter 7, Sections 7.1-7.2 (standard reference, not scraped)