LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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Distinct rank-one q-affine substitutions are linearly independent

Statement

Let $G$ be the group of substitutions of $K [z, z^{- 1}]$ generated by $s_{0}, s_{1}, π$ from The rank-one root datum and parameter conventions for DAHA. If $g_{1}, \dots, g_{N} \in G$ are distinct and $f_{1}, \dots, f_{N} \in K (z)$ satisfy

\sum_{j = 1}^{N} f_{j} (z) g_{j} (h) (z) = 0 for every h \in K [z, z^{- 1}],

then every $f_{j}$ is zero. This holds over $K = Q (q, t)$ because $q$ is indeterminate; it is not asserted when distinct translations collapse at a root of unity.

Facts & Assumptions

Given: The field $K = Q (q, t)$ , the Laurent variable $z$ , and the substitutions $s_{0} (z) = q^{- 2} z^{- 1}$ , $s_{1} (z) = z^{- 1}$ , and $π (z) = q^{- 1} z^{- 1}$ .

[F1]

The rank-one root datum and parameter conventions for DAHA: the extended affine substitutions act by $z \mapsto q^{r} z$ or $z \mapsto q^{r} z^{- 1}$ for integers $r$ , and $q$ is an indeterminate.

Proof

1.1F1givenalgebra

Write $γ_{j} = g_{j} (z) \in K (z)$ . Distinct elements of $G$ give distinct $γ_{j}$ : if the signs differ, one is a nonzero scalar multiple of $z$ and the other of $z^{- 1}$ ; if the signs agree, equality would give $q^{r} = q^{s}$ , hence $r = s$ because $q$ is indeterminate. Thus all differences $γ_{j} - γ_{i}$ for $i \neq j$ are nonzero in $K (z)$ .

2.1step 1.1algebra∎

Apply the assumed operator relation to $h = 1, z, \dots, z^{N - 1}$ . The resulting linear system for $(f_{j})$ has matrix $(γ_{j}^{k})_{0 \leq k < N, 1 \leq j \leq N}$ , whose determinant is the Vandermonde product $\prod_{i < j} (γ_{j} - γ_{i}) \neq 0$ by step 1.1. The matrix is invertible over $K (z)$ , so every $f_{j} = 0$ .

Depends on

The rank-one root datum and parameter conventions for DAHA

Used by

The generic rank-one DAHA polynomial representation is faithful Theorem

Dependency tree · one level

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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)