ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

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.

A positive p-adic parameter and a finite root-of-unity parameter

Example

Take the unnormalized split p-adic Iwahori parameter $q_{F} = 3$ (for instance, the residue-field cardinality of $Q_{3}$ ) and the finite type-A Hecke parameter $ξ = i \in C^{\times}$ . The first is positive real and greater than $1$ ; the second is a primitive fourth root of unity. Their quantum characteristics differ, so the parameters belong to different regimes. This calculation does not assert that the resulting affine and finite Hecke algebras are nonisomorphic or non-Morita-equivalent.

Facts & Assumptions

Given: The affine parameter $q_{F} = 3$ and the finite type-A parameter $ξ = i$ .

[F1]

Every real affine parameter $q > 1$ has $e (C, q) = \infty$ (A real affine-Hecke parameter greater than one has infinite quantum characteristic).

[F2]

For a field $F$ and $u \in F^{\times}$ , the quantum characteristic is the least $n \geq 2$ with $1 + u + \dots + u^{n - 1} = 0$ , or $\infty$ if no such $n$ exists (The quantum characteristic of an invertible Hecke parameter).

Verification

Given: $q_{F} = 3$ and $ξ = i$ as above.

1.1F1givenalgebra

Since $3 > 1$ , [F1] gives $e (C, 3) = \infty$ .

1.2F2givenalgebra

The first candidate geometric sums for $i$ are $1 + i \neq 0$ , $1 + i + i^{2} = i \neq 0$ , and $1 + i + i^{2} + i^{3} = 0$ . By the least-exponent definition [F2], $e (C, i) = 4$ .

2.1step 1.1step 1.2algebra∎

Thus the p-adic positive-real parameter has infinite quantum characteristic and the finite type-A root-of-unity parameter has finite quantum characteristic $4$ . Since $∣ 3 ∣ = 3$ and $∣ i ∣ = 1$ , they are different scalars. These calculations establish only the stated parameter distinction.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Maarten Solleveld, Affine Hecke Algebras and Their Representations (standard reference, not scraped)