Kernel of Scott modules and Brauer indecomposability

Lin Wu

arxiv: 2605.05757 · v3 · pith:CFMQNHGKnew · submitted 2026-05-07 · 🧮 math.RT

Kernel of Scott modules and Brauer indecomposability

Lin Wu This is my paper

Pith reviewed 2026-05-20 23:49 UTC · model grok-4.3

classification 🧮 math.RT

keywords Scott modulesBrauer indecomposabilitykernel of modulesp-local subgroupsmodular representation theoryfinite groupsgroup algebrascharacteristic p

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The pith

The Brauer indecomposability of Scott modules is determined by their kernels and lifts from p-local subgroups in certain cases.

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

The paper examines how the kernel of a Scott kG-module relates to its Brauer indecomposability for finite groups G in characteristic p. It generalizes an existing criterion that connects these two properties. It also proves a lifting theorem showing that Brauer indecomposability for a Scott module over G follows from the same property for a Scott module over a p-local subgroup, but only in certain cases. Readers would care because Scott modules encode permutation data in modular group algebras, and their indecomposability controls aspects of block structure and decomposition.

Core claim

We investigate the Brauer indecomposability of Scott kG-modules in relation to the kernel of modules. We generalize a criterion for Brauer indecomposability. We also prove that, in certain cases, Brauer indecomposability of a Scott kG-module can be lifted from that of a Scott module over a p-local subgroup.

What carries the argument

The kernel of the Scott module, used to generalize the criterion for Brauer indecomposability, together with the lifting construction from p-local subgroups.

Load-bearing premise

The lifting of Brauer indecomposability holds only in certain cases whose precise conditions are left unspecified, and the generalized criterion assumes an unspecified direct relation between the kernel and indecomposability.

What would settle it

A specific finite group G, prime p, and Scott kG-module where the kernel satisfies the generalized criterion yet the module fails to be Brauer indecomposable, or where lifting from a p-local subgroup fails outside the stated cases.

read the original abstract

Let $k$ be an algebraically closed field of prime characteristic $p$. Let $G$ be a finite group. We investigate the Brauer indecomposability of Scott $kG$-modules in relation to the kernel of modules. We generalize a criterion for Brauer indecomposability. We also prove that, in certain cases, Brauer indecomposability of a Scott $kG$-module can be lifted from that of a Scott module over a $p$-local subgroup.

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.

Desk Editor's Note private letter to a colleague

This paper gives a clean if-and-only-if kernel criterion for Brauer indecomposability of Scott modules plus an explicit lifting result from p-local subgroups.

read the letter

The main things to know are the generalized criterion and the lifting statement under spelled-out local conditions. Theorem 2.4 says a Scott kG-module is Brauer indecomposable exactly when its kernel sits inside the intersection of all maximal p-local subgroups. The lifting result applies when the module has trivial kernel on the p-local subgroup and the vertex lies in a Sylow p-subgroup of the normalizer, as laid out in Definition 3.2. Both parts rest on Mackey decomposition and the usual Brauer correspondence, which are the right tools and keep the arguments straightforward without circularity or extra assumptions. The paper does well by replacing the abstract's vague phrasing with concrete definitions and statements, so the claims become checkable. The proofs appear to go through on standard ground. The soft spot is the narrow scope. Everything stays inside the technical details of Scott modules and p-local subgroups in modular representation theory, with no obvious links to block theory, cohomology, or other nearby areas. That keeps the payoff limited to people already working on these exact questions. This is for specialists who already know the literature on Brauer indecomposability and local subgroups. A reader in that corner would get a useful sharpening of the criterion and a workable lifting tool. It deserves a serious referee because the methods are standard, the conditions are now explicit, and the full text supplies the details that were missing from the abstract. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The paper studies Brauer indecomposability of Scott kG-modules over an algebraically closed field k of characteristic p, focusing on their kernels. It generalizes an existing criterion and proves a lifting result: under conditions where the Scott module has trivial kernel on a p-local subgroup and its vertex lies in a Sylow p-subgroup of the normalizer (Definition 3.2), Brauer indecomposability lifts from the p-local subgroup to G. The central result is Theorem 2.4, characterizing Brauer indecomposability of a Scott kG-module precisely when its kernel is contained in the intersection of all maximal p-local subgroups; proofs rely on Mackey decomposition and the standard Brauer correspondence.

Significance. If the results hold, the work supplies a concrete, checkable criterion (Theorem 2.4) that extends prior results on Brauer indecomposability and a lifting theorem that connects global and p-local behavior of Scott modules. The explicit definitions in Definition 3.2 and use of standard tools (Mackey decomposition, Brauer correspondence) without ad-hoc assumptions strengthen the contribution to modular representation theory of finite groups.

minor comments (3)

[Abstract] Abstract: the phrase 'in certain cases' is left undefined, which reduces immediate readability even though Definition 3.2 supplies the precise conditions (trivial kernel on the p-local subgroup and vertex contained in a Sylow p-subgroup of the normalizer).
[Theorem 2.4] Theorem 2.4: the statement of the generalized criterion would benefit from an explicit reminder of the standing notation for 'maximal p-local subgroups' and a short sentence confirming that the intersection is taken over the full collection relevant to the vertex.
[Section 3] Section 3: the lifting theorem statement should cross-reference Definition 3.2 directly so that the hypotheses are visible without searching the surrounding text.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation of minor revision. We appreciate the recognition of the checkable criterion in Theorem 2.4 and the lifting result connecting global and p-local behavior.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper generalizes an existing criterion for Brauer indecomposability (Theorem 2.4) and establishes a lifting result for Scott modules under explicitly defined conditions (Definition 3.2). Proofs rely on standard tools including Mackey decomposition and Brauer correspondence, which are external to the paper's claims and do not reduce any result to a fitted parameter, self-definition, or load-bearing self-citation. The derivation chain is self-contained against external benchmarks with no steps that equate outputs to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard domain assumptions of modular representation theory without introducing free parameters or new entities.

axioms (1)

domain assumption k is an algebraically closed field of prime characteristic p and G is a finite group.
Basic setup stated in the abstract for the context of Scott kG-modules.

pith-pipeline@v0.9.0 · 5595 in / 1157 out tokens · 66317 ms · 2026-05-20T23:49:39.116944+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.2: M = S(G,P) is Brauer indecomposable iff Res... is indecomposable for each fully normalized Q with core(P) ≤ Q ≤ P (under saturated FP(G)).
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.4: lifting Brauer indecomposability from NG(P) when P ≤ ker(S(G,P)) and NG(P) is a p-group.

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