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arxiv: 2211.07069 · v8 · submitted 2022-11-14 · 🧮 math.RT

On the cocenter of the cyclotomic Hecke algebra of type G(r,1,n)

Jun Hu , Lei Shi This is my paper

Pith reviewed 2026-05-24 10:38 UTC · model grok-4.3

classification 🧮 math.RT
keywords cyclotomic Hecke algebracocentercenterG(r,1,n)integral basescomplex reflection groupsKLR algebras
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The pith

The dimensions of the cocenter and center of the cyclotomic Hecke algebra of type G(r,1,n) are independent of the characteristic of the ground field and the Hecke and cyclotomic parameters.

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

The paper constructs integral bases for the cocenter of the cyclotomic Hecke algebra of type G(r,1,n) by generalizing a method used for Iwahori-Hecke algebras of finite Weyl groups. It establishes that the dimensions of both the cocenter and the center remain the same irrespective of the field's characteristic or the specific values of the parameters. This result verifies a conjecture about polynomial coefficients for the complex reflection group of type G(r,1,n) and extends the independence property to certain cyclotomic KLR algebras of affine type A.

Core claim

By generalizing Geck and Pfeiffer's construction of bases, integral bases are built for the cocenter of the cyclotomic Hecke algebra of type G(r,1,n). These bases imply that the dimensions of the cocenter and the center do not depend on the characteristic of the ground field, the Hecke parameter, or the cyclotomic parameters. The same independence holds for the cocenters and centers of certain cyclotomic KLR algebras of affine type A, and the construction verifies Chavli-Pfeiffer's conjecture on the polynomial coefficient g_{w,C} for G(r,1,n).

What carries the argument

Integral bases for the cocenter constructed by generalizing Geck and Pfeiffer's method from Iwahori-Hecke algebras to the cyclotomic setting of type G(r,1,n).

If this is right

  • The dimensions of the cocenter and center are the same in all characteristics and for all parameters.
  • Chavli-Pfeiffer's conjecture on g_{w,C} holds for the complex reflection group of type G(r,1,n).
  • Both the cocenters and centers of certain cyclotomic KLR algebras of affine type A have dimensions independent of the characteristic of the ground field.

Where Pith is reading between the lines

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

  • The independence allows reducing questions about representations over arbitrary fields to the case of characteristic zero.
  • The construction of these bases may provide a template for studying cocenters in other classes of algebras associated to reflection groups.
  • If the bases are explicit enough, they could be used to compute the actual dimension as a function of n and r.

Load-bearing premise

The generalization of Geck and Pfeiffer's bases extends to the cyclotomic Hecke algebra without problems in the integral structure or the independence properties.

What would settle it

Finding a specific choice of n, r, characteristic p, and parameters where the dimension of the cocenter differs from the one computed over the rationals would disprove the independence claim.

read the original abstract

In this paper, we construct some integral bases for the cocenter of the cyclotomic Hecke algebra $\mathscr{H}_{n,K}$ of type $G(r,1,n)$ by generalizing Geck and Pfeiffer's work on the cocenters of the Iwahori-Hecke algebras associated to finite Weyl groups. We show that the dimensions of both the cocenter and the center of the cyclotomic Hecke algebra $\mathscr{H}_{n,K}$ are independent of the characteristic of the ground field, its Hecke parameter and cyclotomic parameters. As applications, we verify Chavli-Pfeiffer's conjecture on the polynomial coefficient $g_{w,C}$ for the complex reflection group of type $G(r,1,n)$ and also show that both the cocenters and the centers of certain cyclotomic KLR algebras of affine type $A$ are independent of the characteristic of the ground field.

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

0 major / 2 minor

Summary. The paper constructs integral bases for the cocenter of the cyclotomic Hecke algebra of type G(r,1,n) by generalizing Geck and Pfeiffer's bases for Iwahori-Hecke algebras of finite Weyl groups. It proves that the dimensions of both the cocenter and the center of H_{n,K} are independent of the characteristic of the ground field, the Hecke parameter, and the cyclotomic parameters. Applications include verifying the Chavli-Pfeiffer conjecture on the coefficients g_{w,C} for G(r,1,n) and establishing analogous independence for cocenters and centers of certain cyclotomic KLR algebras of affine type A.

Significance. If the generalization of the Geck-Pfeiffer construction holds with the claimed integral structure, the parameter-independence of the dimensions supplies a uniform computational tool across characteristics and specializations, which is valuable for the representation theory of complex reflection groups. The explicit verification of the conjecture for G(r,1,n) and the extension to KLR algebras are concrete contributions that strengthen the result.

minor comments (2)
  1. The notation for the parameters q and the cyclotomic parameters in the definition of H_{n,K} should be introduced with a brief reminder of the standard relations before the main constructions begin.
  2. In the applications section, the statement of Chavli-Pfeiffer's conjecture would benefit from an explicit citation to the original reference and a short restatement of the coefficient g_{w,C} being verified.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The report contains no major comments.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via external generalization

full rationale

The paper's core results rest on constructing integral bases for the cocenter by generalizing Geck-Pfeiffer's external construction for Weyl-group Iwahori-Hecke algebras, then proving dimension independence directly from those bases. No equations reduce a claimed prediction or dimension to a fitted input by construction, no load-bearing self-citations appear, and no uniqueness theorems are imported from the authors' prior work. The argument is a standard explicit extension whose validity is established by the proofs themselves rather than by definitional equivalence to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard properties of Hecke algebras and prior results of Geck-Pfeiffer; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (1)
  • standard math Standard algebraic properties of cyclotomic Hecke algebras and their cocenters as defined in the literature
    Invoked implicitly when generalizing the Geck-Pfeiffer construction

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Reference graph

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