ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

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 trivial reflection group gives the first Weyl algebra

Example

Take $W = {1}$ and $\dim h = 1$ . Then $H_{1, 0} (W, h)$ is the first Weyl algebra, with polynomial representation $C [x]$ on which $x$ acts by multiplication and $y$ acts by $d / d x$ .

Facts & Assumptions

Given: The one-dimensional reflection datum with trivial group.

[F1]

The rational Cherednik algebra H_{t,c}(W,h): $H_{t, c}$ is generated by $x, y$ with $[y, x] = t x (y) - \sum_{s} c_{s} α_{s} (y) α_{s}^{\lor} (x) s$ .

[F2]

The PBW theorem for rational Cherednik algebras: the ordered monomials $x^{a} w y^{b}$ form a basis of $H_{t, c}$ .

[F3]

The polynomial representation of a rational Cherednik algebra: $P_{c} = Δ_{c} (1)$ has underlying vector space $C [x]$ .

[F4]

Dunkl operators realize the polynomial standard module: on $P_{c}$ , $y$ acts by $\partial_{y} - \sum_{s} c_{s} α_{s} (y) α_{s}^{- 1} (1 - s)$ .

Proof

technique · direct

Given: [F1]–[F4] and a basis $x \in h^{*}$ with dual $y \in h$ .

1.1F1F2algebra

Since $S = \emptyset$ for $W = {1}$ and $x (y) = 1$ , [F1] reduces to $[y, x] = 1$ with no other generators. This is the defining presentation of the first Weyl algebra; [F2] also gives its basis ${x^{a} y^{b} : a, b \geq 0}$ .

1.2F3F4algebra

In [F4] the reflection sum is empty, so $y = \partial_{y} = d / d x$ and $x$ is multiplication by $x$ on [F3]. Directly, $(d / d x) (x f) - x (d / d x) f = f$ for every polynomial $f$ , verifying the Weyl relation in this representation.

2.1F1F2F3F4step 1.1step 1.2∎

Thus the trivial-group rational Cherednik algebra is the first Weyl algebra and its polynomial standard module is the usual polynomial representation.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

Pavel Etingof and Xiaoguang Ma, Lecture Notes on Cherednik Algebras (standard reference, not scraped)
Victor Ginzburg, Nicolas Guay, Eric Opdam, and Raphael Rouquier, On the category O for rational Cherednik algebras (standard reference, not scraped)