TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

The higher-level Schur-Weyl double centralizer and Schur quotient

Statement

Assume the Axiom of Choice. Fix the pyramid $λ$ with columns $q_{1} \geq \dots \geq q_{l}$ , an origin $c$ , a degree $d \geq 0$ , and let $V_{c}^{\otimes d}$ be the truncated tensor space with its finite W-algebra image $W_{d}$ and cyclotomic Hecke image $B_{d}$ . Then:

$W_{d}$ and $B_{d}$ are mutual centralizers on $V_{c}^{\otimes d}$ , for every complex origin $c$ ; this is the raw double centralizer.
For admissible origins, $V_{c}^{\otimes d}$ is a progenerator and ${Hom}_{W_{d}} (V_{c}^{\otimes d}, -)$ is a Morita equivalence between $W_{d}$ -mod and $B_{d}$ -mod.
The genuine Whittaker functor followed by this equivalence is the exact Schur Serre quotient killing exactly the off-standard simples; it is fully faithful on all projectives of the relevant parabolic summand and on all tiltings, and its left adjoint is right exact and recovers $add (T)$ but need not recover every projective.
For a nonempty pyramid, in the stable range $q_{l} \geq d$ the algebra $B_{d}$ is the full cyclotomic algebra $H_{d} (a)$ ; outside the stable range and for admissible origins the central-corner identification $B_{d} = f H_{d} (a) f$ is used.

Mutual centralizers alone do not give the Morita statement: the nonadmissible origin example with $c = (0, 1)$ , $q = (1, 1)$ and $d = 2$ has a nonprojective tensor module while the mutual centralizer identities still hold.

Facts & Assumptions

Given: The pyramid, the origin $c$ , the tensor space $V_{c}^{\otimes d}$ , the image algebras $W_{d}$ , $B_{d}$ , and the tensor object $T = P_{c} \otimes V^{\otimes d}$ of the relevant parabolic summand.

[F1]

The raw filtered double centralizer holds for every complex origin; for admissible origins the tensor space is a progenerator and ${Hom}_{W_{d}} (V_{c}^{\otimes d}, -)$ is a Morita equivalence $W_{d} -mod \to B_{d} -mod$ ; the nonadmissible counterexample with $c = (0, 1)$ , $q = (1, 1)$ , $d = 2$ separates the two assertions (The finite W and degenerate cyclotomic Hecke actions satisfy a double centralizer).

[F2]

${End}_{g} (T)^{o p} = B_{d}$ and, componentwise, ${End}_{g} (T_{α})^{o p} = B_{α}$ for admissible origins; the map $H_{d} (a) ↠ B_{d}$ is surjective with injectivity exactly in the nonempty stable range and also in the trivial empty branch (The higher-level Schur-Weyl endomorphism isomorphism).

[F3]

The Whittaker quotient $π = {Hom}_{g} (T, -)$ is exact, its kernel is the Serre subcategory of off-standard simples, it is fully faithful on all projectives and on tiltings, its left adjoint $π^{*} = T \otimes_{B_{d}} -$ is right exact, and $π^{*} π$ recovers $add (T)$ but need not recover every projective of the ambient category (The higher-level Schur-Weyl projective generator and Schur functor, The Whittaker functor is exact and fully faithful on the required projectives).

[F4]

The stable range is $q_{l} \geq d$ ; in it $B_{d} = H_{d} (a)$ , and outside it the admissible-origin image is the central block corner $f H_{d} (a) f$ (Stable range and block idempotent corner for higher-level Schur-Weyl duality, The degenerate cyclotomic Hecke algebra).

Proof

technique · direct

Given: The mutual centralizer and Morita inputs of [F1], the endomorphism isomorphism of [F2], the Schur quotient properties of [F3], and the stable-range/corner dictionary of [F4].

1.1F1algebra

The raw double centralizer. For arbitrary complex origin the filtered leading-symbol lifting argument of [F1] gives $W_{d} = B_{d}^{'}$ and $B_{d} = W_{d}^{'}$ on $V_{c}^{\otimes d}$ , with associated graded images the Vust centralizers; this is the all-origin clause and does not use admissibility.

1.2F1F2algebra

Admissible Morita equivalence. Assume $c$ admissible. By the progenerator clause of [F1], $V_{c}^{\otimes d}$ is a progenerator for $W_{d}$ and ${Hom}_{W_{d}} (V_{c}^{\otimes d}, -)$ is a Morita equivalence; by the component clause of [F2] its endomorphism algebra is $B_{d} = {End}_{W_{d}} (V_{c}^{\otimes d})^{o p}$ , so the equivalence is the standard one attached to the $(W_{d}, B_{d})$ -bimodule $V_{c}^{\otimes d}$ , identifying $W_{d}$ -mod with $B_{d}$ -mod.

2.1F2F3step 1.2algebra

The Schur Serre quotient. The composite of the genuine Whittaker functor with the equivalence of step 1.2 is $π = {Hom}_{g} (T, -)$ by [F2] and [F3]; it is exact and its kernel is the Serre subcategory generated by the off-standard simples, so it is the exact Schur Serre quotient killing exactly those simples.

3.1F3step 2.1given

Full faithfulness and the left adjoint. By [F3] the quotient is fully faithful on all projectives of the relevant parabolic summand and on all tiltings, while its left adjoint $T \otimes_{B_{d}} -$ is only right exact and $π^{*} π$ recovers $add (T)$ but need not recover every projective; these boundaries are part of the statement and are not repaired by changing conventions.

4.1F2F4step 1.1algebra∎

Stable range and central corner. For the empty pyramid degree zero is the scalar identity and positive degree is the zero-algebra identity, so all comparisons hold directly with $f = 1$ in degree zero and $f = 0$ in positive degree. For a nonempty pyramid, if $q_{l} \geq d$ then $B_{d} = H_{d} (a)$ by [F4] and the component form of step 2.1 uses the full cyclotomic algebra; outside the stable range, and for admissible origins, the image is the central block corner $B_{d} = f H_{d} (a) f$ of [F4], with the component corners $B_{α} = f_{α} H_{α} (a) f_{α}$ of [F2]. For nonadmissible origins only the raw mutual centralizer statement of step 1.1 remains.

Remarks

The statement separates three levels of structure that are often conflated: the all-origin mutual centralizer (step 1.1), the admissible-origin Morita equivalence (step 1.2), and the stable-range simplification $B_{d} = H_{d} (a)$ (step 4.1). Each consumes a different supplier.
The left adjoint $π^{*}$ is used only where it is exact enough: recovering $add (T)$ on standard-labelled prinjectives, never for arbitrary projectives.

Depends on

Used by

Dependency tree · two levels

40 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

Jonathan Brundan and Alexander Kleshchev, Schur-Weyl duality for higher levels, Selecta Math. (N.S.) 14 (2008) 1-57 (complete 50-page arXiv article) (standard reference, not scraped)
Jonathan Brundan and Alexander Kleshchev, The degenerate analogue of Ariki's categorification theorem, Math. Z. 266 (2010) 877-919 (complete 44-page arXiv article) (standard reference, not scraped)
Jonathan Brundan and Alexander Kleshchev, Representations of shifted Yangians and finite W-algebras, Mem. Amer. Math. Soc. 196 (2008), no. 918 (complete 109-page arXiv memoir) (standard reference, not scraped)