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arxiv: 2604.06166 · v3 · submitted 2026-04-07 · 🧮 math.RT · math.GR

The Galois Alperin weight conjecture for finite category algebras

Xin Huang This is my paper

Pith reviewed 2026-05-10 18:06 UTC · model grok-4.3

classification 🧮 math.RT math.GR
keywords Galois Alperin weight conjecturefinite category algebrasp-orbit categoryEI-categoriessimple modulesweightsequivariant bijectionmodular representation theory
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The pith

The Galois Alperin weight conjecture for finite category algebras reduces to the version for finite groups.

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

The paper formulates the Galois Alperin weight conjecture in the setting of algebras of finite categories over an algebraic closure of the prime field. It requires a bijection that matches isomorphism classes of simple modules for the category algebra to the weights of the algebra of the associated p-orbit category, while preserving the natural actions of the Galois group and the category automorphisms. The central result is a reduction theorem establishing that the conjecture for arbitrary finite categories holds precisely when the original conjecture holds for finite groups. A parallel blockwise formulation is given for EI-categories together with its own reduction to groups. This links the representation theory of categories to the well-studied case of groups, so that any future verification or counterexample in the group setting transfers directly to the larger class of category algebras.

Core claim

The paper defines a version of the Galois Alperin weight conjecture for a finite category C by requiring a Γ × Aut(C)-equivariant bijection between the isomorphism classes of simple kC-modules and the weights of kO_C, where O_C is the p-orbit category of C defined by Linckelmann. It proves that this statement for finite categories reduces to the corresponding statement for finite groups. For an EI-category C it partitions the weights of kO_C according to the blocks of kC, formulates the blockwise Galois Alperin weight conjecture, and reduces that version to finite groups as well.

What carries the argument

The p-orbit category O_C of the finite category C, on whose algebra the weights are defined so that they stand in equivariant bijection with the simple modules of the category algebra kC.

Load-bearing premise

The p-orbit category of a finite category behaves sufficiently like the orbit category of a finite group that weights and simple modules correspond under the required restrictions and equivariant maps.

What would settle it

An explicit finite category C together with its p-orbit category O_C for which no Γ × Aut(C)-equivariant bijection exists between the simple kC-modules and the weights of kO_C would falsify the formulated conjecture.

read the original abstract

Let $p$ be a prime, $k$ an algebraic closure of $\mathbb{F}_p$ and $\Gamma$ the Galois group ${\rm Gal}(k/\mathbb{F}_p)$. Let $\mathcal{C}$ be a finite category and $\mathcal{O}_{\mathcal{C}}$ the $p$-orbit category of $\mathcal{C}$ defined by Linckelmann. We formulate a version of the Galois Alperin weight conjecture (GAWC) for finite category algebras stating that there exists a $\Gamma\times {\rm Aut}(\mathcal{C})$-equivariant bijection between the set of isomorphism classes of simple $k\mathcal{C}$-modules and that of the weights of $k\mathcal{O}_{\mathcal{C}}$. We reduce the GAWC for finite categories to finite groups. For $\mathcal{C}$ an EI-category, we give a partition of weights of $k\mathcal{O}_{\mathcal{C}}$ with respect to blocks of $k\mathcal{C}$ and then formulate a blockwise Galois Alperin weight conjecture (BGAWC) for $\mathcal{C}$. Similarly, we reduce the BGAWC for finite EI-categories to finite groups.

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

1 major / 1 minor

Summary. The paper formulates a Galois Alperin weight conjecture (GAWC) for finite category algebras, asserting the existence of a Γ×Aut(C)-equivariant bijection between the isomorphism classes of simple kC-modules and the weights of kO_C (the p-orbit category of C). It reduces the GAWC for finite categories to the corresponding conjecture for finite groups. For EI-categories, it partitions the weights of kO_C with respect to blocks of kC, formulates a blockwise version (BGAWC), and reduces that to the group case as well.

Significance. If the reduction is valid, the work would meaningfully extend the Galois Alperin weight conjecture from groups to category algebras by showing that the category case follows from the group case. This could allow group-theoretic results to imply statements for categories, which is a useful bridge in the representation theory of finite categories and their algebras, building on the p-orbit category construction.

major comments (1)
  1. [The reduction step (following the formulation of GAWC)] The reduction of GAWC (and BGAWC) to finite groups assumes that weights of kO_C—pairs (P,S) with P a p-subgroup in the category sense and S a simple module for the normalizer in O_C—biject with standard Alperin weights of the automorphism groups of objects in C, while preserving Γ×Aut(C) equivariance, and that simple kC-modules arise as inflations or restrictions from those group algebras. This correspondence is load-bearing for the central claim that the category conjecture follows from the group one; if non-isomorphism morphisms in C induce extra relations in kC not present in O_C, or if the p-subgroup structure in O_C deviates from group normalizers, the implication fails. The manuscript must supply an explicit lemma or proposition verifying this bijection and equivariance preservation.
minor comments (1)
  1. [Abstract] The abstract states the reduction but does not reference the theorem number or section containing the proof; adding this would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need to make the reduction step more explicit. We address the major comment below and will revise the paper accordingly.

read point-by-point responses

  1. Referee: [The reduction step (following the formulation of GAWC)] The reduction of GAWC (and BGAWC) to finite groups assumes that weights of kO_C—pairs (P,S) with P a p-subgroup in the category sense and S a simple module for the normalizer in O_C—biject with standard Alperin weights of the automorphism groups of objects in C, while preserving Γ×Aut(C) equivariance, and that simple kC-modules arise as inflations or restrictions from those group algebras. This correspondence is load-bearing for the central claim that the category conjecture follows from the group one; if non-isomorphism morphisms in C induce extra relations in kC not present in O_C, or if the p-subgroup structure in O_C deviates from group normalizers, the implication fails. The manuscript must supply an explicit lemma or proposition verifying this bijection and equivariance preservation.

    Authors: We agree that the reduction of GAWC and BGAWC relies on a precise correspondence between the weights of kO_C and the Alperin weights of the automorphism groups of objects in C, together with preservation of Γ×Aut(C)-equivariance and the appropriate inflation/restriction maps for simple modules. While the manuscript describes this reduction by appealing to the definition of the p-orbit category O_C and the structure of category algebras, we acknowledge that the key bijection and equivariance are not isolated in a single lemma. We will add an explicit lemma (or proposition) in the revised version that verifies: (i) the bijection between weights (P,S) of kO_C and the standard Alperin weights of the automorphism groups, (ii) the Γ×Aut(C)-equivariance of this bijection, and (iii) the correspondence of simple kC-modules via inflation from the relevant group algebras. The lemma will also address why morphisms in C that are not isomorphisms do not introduce additional relations affecting the weight count or the reduction. This addition will make the argument self-contained and directly address the referee's concern. revision: yes

Circularity Check

0 steps flagged

No significant circularity; reduction uses external orbit category definition

full rationale

The paper formulates the GAWC for finite category algebras by direct reference to the externally defined p-orbit category O_C (Linckelmann) and states an equivariant bijection between simple kC-modules and weights of kO_C. The reduction to the finite-group case proceeds by establishing correspondences between these objects and the standard Alperin weights of automorphism groups of objects in C, using the given category structure and block partitions for the EI-case. No equation or step redefines a quantity in terms of itself, renames a fitted input as a prediction, or relies on a load-bearing self-citation whose content is unverified outside the paper. The central claims remain independent of the target conjecture and rest on the external definition of O_C together with standard module-theoretic arguments.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the prior definition of the p-orbit category and standard notions of blocks and weights in modular representation theory; no free parameters are fitted and no new entities are postulated.

axioms (1)
  • domain assumption The p-orbit category O_C of a finite category C is defined as in Linckelmann's work.
    This definition is required to state the weights of kO_C and to formulate the bijection.

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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