ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.
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 Laurent polynomials in the center

Example

For $n \geq 2$ , each elementary symmetric polynomial

e_{k} : = \sum_{1 \leq i_{1} < \dots < i_{k} \leq n} X_{i_{1}} \dots X_{i_{k}} (1 \leq k \leq n)

is central in $H_{n}^{a f f}$ , while $X_{1}$ is not central.

Facts & Assumptions

Given: $A = R_{0} [X_{1}^{\pm 1}, \dots, X_{n}^{\pm 1}]$ , the permutation action of $S_{n}$ , and the universal center theorem.

[F1]

The type-A weight lattice, Laurent ring, and divided differences: $S_{n}$ acts on $A$ by permuting the variables, and distinct Laurent monomials are $R_{0}$ -linearly independent.

[F2]

The center of the universal affine Hecke algebra of type A: $Z (H_{n}^{a f f}) = A^{S_{n}}$ .

Verification

Given: The data of the Statement.

1.1givenF1algebra

A permutation sends each monomial $X_{i_{1}} \dots X_{i_{k}}$ to the monomial indexed by the permuted $k$ -element subset; as the permutation bijects those subsets, it fixes their sum $e_{k}$ . For $n \geq 2$ , $s_{1} (X_{1}) = X_{2} \neq X_{1}$ because these are distinct Laurent basis monomials by [F1].

2.1F1F2step 1.1∎

By [F2], each invariant $e_{k}$ lies in the center. The same equality of the center with $A^{S_{n}}$ and the non-invariance of $X_{1}$ from step 1.1 show that $X_{1}$ is not central.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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