DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge 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.

The rank-one DAHA polynomial representation

Definition

The polynomial representation is the action of $H_{q, t} (A_{1})$ on $P = K [z, z^{- 1}]$ specified by sending $Z$ to multiplication by $z$ , $Z^{- 1}$ to multiplication by $z^{- 1}$ , $π$ to $f (z) \mapsto f (q^{- 1} z^{- 1})$ , and $T_{i}$ to

(T_{i} f) (z) = t f (s_{i} z) + (t - t^{- 1}) \frac{f (s_{i} z) - f (z)}{z^{- α_{i}} - 1}, i = 0, 1,

with $s_{1} z = z^{- 1}$ , $s_{0} z = q^{- 2} z^{- 1}$ , $z^{- α_{1}} = z^{- 2}$ , and $z^{- α_{0}} = q^{2} z^{2}$ . The quotient is interpreted by the finite Laurent sums in The rank-one DAHA Bernstein quotients are Laurent polynomials. Well-definedness as an algebra action is proved in The rank-one DAHA difference-reflection operators satisfy the presentation ↗.

Depends on

Used by

Dependency tree · two levels

4 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

Alexander A. Kirillov Jr., Lectures on the affine Hecke algebras and Macdonald conjectures (standard reference, not scraped)
Pavel Etingof and Xiaoguang Ma, Lecture Notes on Cherednik Algebras (standard reference, not scraped)