Alphabeta MathHow 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.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
The elementary symmetric polynomials
Definition
For , the -th elementary symmetric polynomial in is
The empty product gives . We put for . Each is symmetric because a permutation of the variables merely permutes the -element index sets in the sum.
Depends on
Used by
- A nonzero stable Schur function can vanish in too few variables Counterexample
- Reducing a symmetric polynomial in two variables to a polynomial in e₁ and e₂ Example
- The five standard symmetric-function bases in degree three Example
- The symmetric polynomials as the invariant ring of the symmetric group, seen through Noether's finiteness theorem Example
- The leading multidegree of e₁^b₁⋯ eₙ^bₙ is (b₁+⋯+bₙ,b₂+⋯+bₙ,…,bₙ) with coefficient one Lemma
- 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
- Vieta expansion: ∏ᵢ₌₁ⁿ(t-xᵢ)=∑ₖ₌₀ⁿ(-1)ᵏ eₖ tⁿ⁻ᵏ Theorem
Dependency tree · two levels
6 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
- 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)