LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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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.

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Rank-one DAHA affine normal words span

Statement

In the algebra of The equal-parameter double-affine Hecke algebra of type A1, every element is a finite $K$ -linear combination of words

Z^{m} T_{i_{1}} \dots T_{i_{r}} π^{ϵ},

where $m \in Z$ , $ϵ \in {0, 1}$ , $r \geq 0$ , and adjacent indices in the string $i_{1}, \dots, i_{r} \in {0, 1}$ are distinct. This is a spanning statement only; no linear independence is asserted here.

Facts & Assumptions

Given: The generator-and-relation algebra $H_{q, t} (A_{1})$ , its Bernstein relations, and its length-zero involution.

[F1]

The equal-parameter double-affine Hecke algebra of type A1: $π^{2} = 1$ , $π T_{0} π = T_{1}$ , $(T_{i} - t) (T_{i} + t^{- 1}) = 0$ , $π Z π = q^{- 1} Z^{- 1}$ , and $T_{i} Z^{m} - s_{i} (Z^{m}) T_{i} = (t - t^{- 1}) B_{i, m} (Z)$ for all integers $m$ .

Proof

1.1F1givenalgebra

From $π^{2} = 1$ and $π T_{0} π = T_{1}$ one obtains $π T_{0} = T_{1} π$ and $π T_{1} = T_{0} π$ . Repeatedly use these identities to move every occurrence of $π$ in an affine word to the right, and then use $π^{2} = 1$ to leave at most one $π$ . The quadratic relation gives $T_{i}^{2} = (t - t^{- 1}) T_{i} + 1$ , so induction on affine-word length reduces any word in $T_{0}, T_{1}$ to a finite linear combination of alternating strings. This proves the asserted form for words containing no $Z$ letters.

2.1F1step 1.1algebra∎

From $π^{2} = 1$ and $π Z π = q^{- 1} Z^{- 1}$ , conjugation gives $π Z^{m} π = q^{- m} Z^{- m}$ for every integer $m$ , hence $π Z^{m} = q^{- m} Z^{- m} π$ . In a generator word, move each lattice letter left across $π$ by this identity and across $T_{i}$ by $T_{i} Z^{m} = s_{i} (Z^{m}) T_{i} + (t - t^{- 1}) B_{i, m} (Z)$ . Use lexicographic induction on (the number of $T_{0}, T_{1}$ letters, the number of out-of-order lattice/reflection pairs): a main term decreases the second coordinate, while a correction term is a finite Laurent sum with one fewer Hecke letter and decreases the first. After all lattice letters are on the left, move every $π$ right using $π T_{0} = T_{1} π$ , $π T_{1} = T_{0} π$ and $π^{2} = 1$ ; each move decreases the number of $π$ -before-Hecke pairs. Finally apply step 1.1 to reduce the remaining Hecke words to alternating strings. The process terminates and gives the asserted spanning form.

Depends on

Used by

The generic rank-one DAHA polynomial representation is faithful Theorem

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)