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arxiv: 1907.10353 · v1 · pith:U6JAMDKPnew · submitted 2019-07-24 · 🧮 math.RT · math.GR

Bounds on the number of simple modules in blocks of finite groups of Lie type

Ruwen Hollenbach This is my paper

Pith reviewed 2026-05-24 16:46 UTC · model grok-4.3

classification 🧮 math.RT math.GR
keywords finite groups of Lie typeblockssimple modulesquasi-isolated blocksexceptional typebad primesmodular representation theory
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The pith

Upper bounds are given for the number of simple modules in quasi-isolated ℓ-blocks of exceptional groups of Lie type when ℓ is bad for the group.

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

The paper considers simple simply connected algebraic groups G of exceptional type over finite fields, with associated finite groups G^F. It derives upper bounds on the number of simple modules lying in the quasi-isolated ℓ-blocks of the group algebra of G^F, and likewise for the quotient by the center. The bounds are stated specifically for the case in which the prime ℓ is bad for G. A sympathetic reader would care because the count of simple modules controls the size and structure of the modular representation theory of these groups, and explicit bounds make that theory more tractable for the exceptional families.

Core claim

In this work we give upper bounds on the number of simple modules in the quasi-isolated ℓ-blocks of G^F and G^F/Z(G^F) when ℓ is bad for G, where G is a simple, simply connected linear algebraic group of exceptional type defined over F_q with Frobenius endomorphism F.

What carries the argument

Quasi-isolated ℓ-blocks of G^F for bad primes ℓ, in groups of exceptional Lie type.

Load-bearing premise

The blocks under consideration are quasi-isolated and the groups are of exceptional type, which depends on prior classifications of blocks and bad primes in the literature on groups of Lie type.

What would settle it

An explicit quasi-isolated ℓ-block with ℓ bad for an exceptional group that contains strictly more simple modules than the upper bound stated in the paper.

read the original abstract

Let $G$ be a simple, simply connected linear algebraic group of exceptional type defined over $\mathbb{F}_q$ with Frobenius endomorphism $F: G \to G$. In this work we give upper bounds on the number of simple modules in the quasi-isolated $\ell$-blocks of $G^F$ and $G^F/Z(G^F)$ when $\ell$ is bad for $G$.

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 / 2 minor

Summary. The manuscript claims to establish explicit upper bounds on the number of simple modules lying in the quasi-isolated ℓ-blocks of G^F and of G^F/Z(G^F), where G is a simple simply-connected algebraic group of exceptional type over F_q and ℓ is a bad prime for G.

Significance. If the stated bounds are correctly derived from the known classifications of quasi-isolated blocks, the result supplies concrete numerical information that can be used in further work on the modular representation theory of finite groups of Lie type in non-defining characteristic.

major comments (2)
  1. [Main results / case analysis] The central claim is obtained by case-by-case enumeration over the exceptional types (E6, E7, E8, F4, G2) using prior classifications of quasi-isolated blocks; the manuscript must therefore verify, in the section containing the main results, that every relevant block for each bad prime is covered by those external lists and that no additional blocks exist.
  2. [Introduction / references to prior classifications] The argument inherits any incompleteness in the cited classifications of quasi-isolated blocks; a load-bearing gap in one of those references would invalidate the claimed upper bounds for the corresponding type.
minor comments (2)
  1. [Abstract] The abstract states the existence of bounds without indicating the numerical values or the method; the introduction should state the explicit bounds obtained.
  2. [Notation section] Notation for the finite groups G^F and the quotient by the center should be introduced once and used consistently throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the identification of points that will improve the clarity of the manuscript. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Main results / case analysis] The central claim is obtained by case-by-case enumeration over the exceptional types (E6, E7, E8, F4, G2) using prior classifications of quasi-isolated blocks; the manuscript must therefore verify, in the section containing the main results, that every relevant block for each bad prime is covered by those external lists and that no additional blocks exist.

    Authors: We agree that an explicit verification statement is required. In the revised manuscript the main results section will contain a short paragraph confirming that the external classifications cited for each exceptional type and each bad prime ℓ are exhaustive, with a table or list cross-referencing the blocks appearing in those references against the possible bad primes for the given group. This will make the coverage explicit without adding new computations. revision: yes

  2. Referee: [Introduction / references to prior classifications] The argument inherits any incompleteness in the cited classifications of quasi-isolated blocks; a load-bearing gap in one of those references would invalidate the claimed upper bounds for the corresponding type.

    Authors: This observation is correct. We will add one sentence in the introduction (and a corresponding remark in the main results section) stating that the stated upper bounds are conditional on the completeness of the cited classifications of quasi-isolated blocks. This makes the logical dependence transparent while leaving the numerical bounds unchanged. revision: yes

Circularity Check

0 steps flagged

No circularity; bounds rest on external classifications of blocks

full rationale

The paper enumerates or bounds simple modules in quasi-isolated ℓ-blocks for exceptional types by invoking prior independent classifications of such blocks and bad primes from the literature on groups of Lie type. These classifications are external results (typically by other authors) rather than self-citations or self-definitions internal to this work. No equations or steps in the abstract reduce a claimed prediction or bound to a fitted parameter or prior result by the same authors; the derivation chain therefore remains non-circular and self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information on free parameters, axioms or invented entities is present in the abstract.

pith-pipeline@v0.9.0 · 5587 in / 934 out tokens · 31800 ms · 2026-05-24T16:46:58.699777+00:00 · methodology

discussion (0)

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Reference graph

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