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arxiv: 2606.03759 · v2 · pith:UTMFOCCOnew · submitted 2026-06-02 · 🧮 math.RT

On Cellularity of Hecke Algebras for Wreath Products

Berta Hudak , Chun-Ju Lai This is my paper

Pith reviewed 2026-06-28 07:54 UTC · model grok-4.3

classification 🧮 math.RT
keywords Hecke algebrascellular algebraswreath productsHu algebratype Dbipartitionscomplex reflection groupsrepresentation theory
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The pith

The generalized Hu algebra admits a cellular basis for d=2 that realizes simple modules for the type D Hecke algebra via bipartitions of size (m,m).

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

The paper constructs a unified basis for the generalized Hu algebra, a quantization of the wreath product of symmetric groups, and proves this algebra is cellular when d equals 2. The cellular structure supplies a combinatorial description of the algebra's representations through cell modules. Because the Hu algebra's representation theory controls the Hecke algebra of the complex reflection group G(d,d,md), the result gives an explicit, elementary realization of the simple modules for the Hecke algebra of type D_{2m} that are indexed by bipartitions of size (m,m). A sympathetic reader cares because cellular algebras turn abstract module classification into concrete combinatorial data for these quantized wreath-product algebras.

Core claim

The authors construct a unified basis for the (generalized) Hu algebra and establish its cellular algebra structure in the case d = 2. As an application, our construction provides an elementary realization of the simple modules for the Hecke algebra of type D_{2m} that are parameterized by bipartitions of size (m,m).

What carries the argument

The unified basis that equips the generalized Hu algebra with a cellular structure for d=2.

If this is right

  • The cell modules give a basis for the simple modules of the Hu algebra when d=2.
  • The same basis transfers via the control relation to realize the simple modules of the type D_{2m} Hecke algebra indexed by bipartitions of size (m,m).
  • The explicit combinatorial basis permits direct computation of representation invariants such as dimensions and decomposition numbers.
  • The cellularity result specializes the general theory of Hu algebras to the d=2 case with concrete data.

Where Pith is reading between the lines

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

  • The same basis-construction technique could be tested for other small values of d if analogous combinatorial elements can be defined.
  • The explicit basis may simplify calculations of Ext groups or decomposition matrices for these Hecke algebras.
  • Similar cellularity statements might hold for other quantized wreath products that arise from complex reflection groups.
  • The construction could be used to compare the representation theory of type D Hecke algebras with that of other classical types through shared cellular data.

Load-bearing premise

The representation theory of the generalized Hu algebra controls the Hecke algebra of the complex reflection group G(d,d,md) so that cellularity transfers to the simple modules of type D.

What would settle it

An explicit low-rank calculation for m=2 or m=3 in which the dimension or composition factors of a module labeled by a bipartition (m,m) fail to match the predictions of the cellular basis.

read the original abstract

The (generalized) Hu algebra is a nontrivial quantization of the wreath product $\Sigma_m \wr \Sigma_d$ between symmetric groups, whose representation theory controls the Hecke algebra of the complex reflection group $G(d,d,md)$. In this paper, we construct a unified basis for this algebra and establish its cellular algebra structure in the case $d = 2$. As an application, our construction provides an elementary realization of the simple modules for the Hecke algebra of type $D_{2m}$ that are parameterized by bipartitions of size $(m,m)$.

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 constructs a unified basis for the generalized Hu algebra, a quantization of the wreath product Σ_m ≀ Σ_d, and proves that this algebra is cellular when d=2. As an application, the cellular structure is used to give an elementary realization of the simple modules of the Hecke algebra of type D_{2m} that are labeled by bipartitions of size (m,m), via the control of the representation theory of the Hu algebra over the Hecke algebra of G(d,d,md).

Significance. If the basis construction and cellularity verification hold, the work supplies an explicit cellular datum for these wreath-product Hecke algebras in the d=2 case, which is a useful tool for computing decomposition numbers and characters. The elementary realization of the indicated type-D simple modules is a concrete payoff that could simplify existing approaches based on more abstract cellular or diagrammatic methods.

major comments (2)
  1. [Abstract] The abstract invokes that 'the representation theory of the (generalized) Hu algebra controls the Hecke algebra of G(d,d,md)' as the link to the type-D application, but no explicit statement of the precise functor, Morita equivalence, or parameter specialization that realizes this control appears in the provided description; this step is load-bearing for the claimed application to bipartitions of size (m,m).
  2. [Abstract] The cellularity claim for d=2 rests on verifying the three cellular axioms (anti-involution, cell datum, and positivity) for the constructed unified basis, yet the abstract supplies neither the explicit form of the basis elements nor the multiplication rules needed to check the axioms; without these, the central claim cannot be assessed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting points that affect the clarity of the abstract. We respond to each major comment below and indicate where revisions will be made.

read point-by-point responses

  1. Referee: [Abstract] The abstract invokes that 'the representation theory of the (generalized) Hu algebra controls the Hecke algebra of G(d,d,md)' as the link to the type-D application, but no explicit statement of the precise functor, Morita equivalence, or parameter specialization that realizes this control appears in the provided description; this step is load-bearing for the claimed application to bipartitions of size (m,m).

    Authors: The control is realized by a concrete parameter specialization together with a surjective algebra homomorphism from the generalized Hu algebra onto a quotient of the Hecke algebra of type D_{2m}; the kernel acts trivially on the modules labeled by bipartitions of shape (m,m). This construction is spelled out in the introduction and the first section of the paper. We agree that a single clarifying sentence in the abstract would make the link self-contained and will add it in the revised version. revision: yes

  2. Referee: [Abstract] The cellularity claim for d=2 rests on verifying the three cellular axioms (anti-involution, cell datum, and positivity) for the constructed unified basis, yet the abstract supplies neither the explicit form of the basis elements nor the multiplication rules needed to check the axioms; without these, the central claim cannot be assessed.

    Authors: The abstract is intentionally concise. The unified basis is defined explicitly in Section 3 (as linear combinations of the standard wreath-product basis elements with coefficients in the ground ring), the anti-involution is the natural one induced by the group inversion, and the cell datum together with the verification that the structure constants satisfy the cellular axioms (including the positivity condition on the bilinear form) are proved in Sections 4 and 5. The multiplication rules follow directly from the quantized wreath-product relations. The full manuscript therefore contains all information needed to check the claim; no change to the abstract itself is required. revision: no

Circularity Check

0 steps flagged

No significant circularity; construction is self-contained

full rationale

The paper presents an explicit construction of a unified basis for the generalized Hu algebra together with direct verification of the cellular axioms (anti-involution, cell datum, positivity) for the d=2 case. This construction does not reduce to any fitted parameter, self-citation chain, or definitional renaming of its own output. The link to simple modules of the type-D Hecke algebra is stated as an application that follows from the cellularity result, not as a premise that defines the basis. No equations or steps in the provided description equate a claimed prediction or uniqueness statement back to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper builds on standard concepts in the theory of Hecke algebras and cellular algebras from the literature.

axioms (2)
  • domain assumption The (generalized) Hu algebra is a nontrivial quantization of the wreath product Σ_m ≀ Σ_d
    This is the definition used to set up the algebra whose properties are studied.
  • domain assumption The representation theory of the Hu algebra controls the Hecke algebra of G(d,d,md)
    This link is used to apply the cellularity result to the Hecke algebra of type D.

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Reference graph

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