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.
Symmetric polynomials as the invariants of variable permutations
Definition
Let be a commutative ring and let be the iterated polynomial ring. Every permutation acts on it by
A polynomial is symmetric when for every permutation . The symmetric polynomials form the fixed subset
For this means the coefficient ring itself.
Depends on
Used by
- A group acting on a ring by automorphisms and its invariant subring Definition
- 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 stable graded ring of symmetric functions Definition
- The symmetric polynomials as the invariant ring of the symmetric group, seen through Noether's finiteness theorem Example
- 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 · two levels
9 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)