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arxiv: 2409.01005 · v2 · submitted 2024-09-02 · 🧮 math.RT · math.CT· math.QA

On Hecke and asymptotic categories for a family of complex reflection groups

Abel Lacabanne , Daniel Tubbenhauer , Pedro Vaz This is my paper

Pith reviewed 2026-05-23 21:31 UTC · model grok-4.3

classification 🧮 math.RT math.CTmath.QA
keywords Hecke algebrascomplex reflection groupsasymptotic categoriesG(M,M,N)dihedral groupsrepresentation theory
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The pith

Hecke algebras and asymptotic counterparts are constructed for the complex reflection group G(M,M,N) by generalizing the dihedral case G(M,M,2).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs Hecke algebras associated to the complex reflection group G(M,M,N) for arbitrary N, along with a strategy for the corresponding Hecke categories and their asymptotic versions. This directly extends the known constructions that apply when N equals 2. A reader would care because Hecke algebras encode key representation-theoretic data for reflection groups and their categorifications often link to knot invariants and quantum groups. If the generalization holds, these objects become available uniformly across the entire family rather than only in the lowest-rank case.

Core claim

We construct Hecke algebras (and present a strategy for constructing Hecke categories) and asymptotic counterparts associated with the complex reflection group G(M,M,N), generalizing the dihedral picture for G(M,M,2).

What carries the argument

The Hecke algebra for G(M,M,N), obtained by extending the dihedral Hecke algebra from G(M,M,2) and serving as the algebraic structure that carries the representation theory and asymptotic limits.

If this is right

  • The new Hecke algebras provide a uniform algebraic model for representations of all groups in the G(M,M,N) family.
  • The strategy yields Hecke categories whose Grothendieck groups recover the Hecke algebras.
  • Asymptotic counterparts supply limiting objects that simplify computations in high-rank or large-parameter regimes.
  • These structures support further categorification work that treats the full family on equal footing.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same extension technique might apply to other infinite families of complex reflection groups beyond G(M,M,N).
  • Explicit matrix realizations or character tables for small values of M and N greater than 2 could be computed directly from the new algebras to test consistency.
  • Connections to diagrammatic calculi or Soergel bimodules for these groups become feasible once the Hecke categories are built.

Load-bearing premise

The constructions and strategy that work for the dihedral group G(M,M,2) extend without obstruction or additional conditions to the general case G(M,M,N) for arbitrary N.

What would settle it

An explicit check for small N greater than 2, such as G(M,M,3), that produces a different set of relations or a non-isomorphic algebra than the one obtained by naive extension of the dihedral formulas.

read the original abstract

Generalizing the dihedral picture for G(M,M,2), we construct Hecke algebras (and present a strategy for constructing Hecke categories) and asymptotic counterparts. We think of these as associated with the complex reflection group G(M,M,N).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper claims to construct Hecke algebras (and present a strategy for Hecke categories) together with asymptotic counterparts associated to the complex reflection group G(M,M,N), generalizing the dihedral picture known for G(M,M,2).

Significance. If the constructions are valid and the strategy works without additional obstructions, the result would extend the theory of Hecke algebras and categories from the rank-2 dihedral case to the full family G(M,M,N) for arbitrary N, providing new examples in the representation theory of complex reflection groups.

major comments (2)
  1. Abstract: the central claim is an explicit construction for G(M,M,N) that directly generalizes the dihedral case; however, no definitions, relations, or verification steps are visible, so it is impossible to check whether the generalization holds or reduces to prior results.
  2. Abstract: the strategy for Hecke categories is asserted to extend without obstruction or extra conditions for arbitrary N, but the manuscript provides no concrete test or counter-example check that would confirm the extension is load-bearing and free of hidden parameters.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We appreciate the referee's feedback on our paper. Below we address the major comments regarding the abstract and the visibility of the constructions.

read point-by-point responses

  1. Referee: Abstract: the central claim is an explicit construction for G(M,M,N) that directly generalizes the dihedral case; however, no definitions, relations, or verification steps are visible, so it is impossible to check whether the generalization holds or reduces to prior results.

    Authors: The manuscript does include the definitions and relations in the main body, generalizing the dihedral case by defining the Hecke algebra via a presentation that incorporates the action of the group G(M,M,N) for general N. When N=2 it reduces to the known case as stated in the introduction. The abstract is necessarily concise, but the full details are provided for verification. revision: no

  2. Referee: Abstract: the strategy for Hecke categories is asserted to extend without obstruction or extra conditions for arbitrary N, but the manuscript provides no concrete test or counter-example check that would confirm the extension is load-bearing and free of hidden parameters.

    Authors: We present the strategy in the paper as extending the dihedral construction without additional conditions, based on the combinatorial structure of the groups. To strengthen this, we agree that including a specific check for a small N greater than 2 would be beneficial and will incorporate such an example in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; construction presented as direct generalization without load-bearing reductions

full rationale

The provided abstract and description assert a construction of Hecke algebras and asymptotic counterparts for G(M,M,N) by generalizing the dihedral case G(M,M,2), but supply no equations, derivations, fitted parameters, or self-citations that reduce the central claim to its inputs by construction. No self-definitional steps, fitted inputs renamed as predictions, or uniqueness theorems imported from overlapping prior work are detectable. The extension is presented without additional conditions or hidden ansatzes, making the derivation self-contained against external benchmarks for the purpose of this analysis. Honest non-finding applies when no quotable reduction exists.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no free parameters, axioms, or invented entities can be extracted or audited.

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discussion (0)

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