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arxiv: 2408.09036 · v1 · submitted 2024-08-16 · 🧮 math.RT

On commutative tensor factors of group algebras

Diego Garc\'ia-Lucas , \'Angel del R\'io , Taro Sakurai This is my paper

Pith reviewed 2026-05-23 22:06 UTC · model grok-4.3

classification 🧮 math.RT
keywords group algebrastensor productscommutative factorsmodular group algebrasprime fieldsdirect product decompositions
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The pith

Any tensor product factorization of a modular group algebra over a prime field with a commutative factor arises from a direct product decomposition of the group.

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

The paper proves that for a finite group G and its modular group algebra over a prime field, every tensor product splitting in which one factor is commutative must come from writing G as a direct product of subgroups. This gives a complete description of such algebraic factorizations in terms of the group's internal structure. The result extends the commutative case already known and settles some open questions about when these splittings exist. A reader would care because it turns an algebra problem into a direct statement about groups.

Core claim

For a finite group G and a prime field F whose characteristic divides the order of G, if the group algebra FG is isomorphic to a tensor product A ⊗_F B with A commutative, then G decomposes as a direct product H × K such that A is isomorphic to the group algebra FH and B to FK.

What carries the argument

The reduction of tensor factorizations of the group algebra to direct product decompositions of the underlying group.

If this is right

  • All commutative tensor factors of such group algebras are themselves group algebras of subgroups.
  • The only possible commutative tensor factors are those coming from direct factors of G.
  • The result classifies all such factorizations completely in the modular prime-field setting.

Where Pith is reading between the lines

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

  • The same conclusion may fail when the field is not prime or when the algebra is not modular.
  • One could test whether the statement survives for infinite groups or for algebras over rings rather than fields.

Load-bearing premise

The base field is a prime field and the group algebra is modular.

What would settle it

An explicit finite group G, prime p dividing |G|, and a tensor factorization FG ≅ A ⊗ B over the field with p elements, where A is commutative but no subgroups H and K exist with G = H × K, A ≅ FH, and B ≅ FK.

read the original abstract

We prove that any tensor product factorization with a commutative factor of a modular group algebra over a prime field comes from a direct product decomposition of the group basis. This extends previous work by Carlson and Kov\'acs for the commutative case and answers a question of them in some cases.

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 manuscript proves that if k is a prime field of characteristic p dividing |G| for a finite group G, and kG ≅ A ⊗_k B as k-algebras with B commutative, then the factorization arises from a direct product decomposition G = H × K such that the factors correspond to kH and kK. This extends the commutative-both-factors result of Carlson–Kovács and answers one of their questions under the stated hypotheses.

Significance. If the result holds, it is a clean structural theorem on tensor factors of modular group algebras. The argument removes the commutativity assumption on A while retaining the prime-field and modular hypotheses, yielding a parameter-free statement that directly extends prior published work without introducing fitted parameters or self-referential reductions.

minor comments (2)
  1. The abstract states the result for 'modular group algebra' but does not explicitly record that G is finite; while standard, a single sentence clarifying finiteness would improve readability for readers outside the immediate subfield.
  2. Notation for the isomorphism kG ≅ A ⊗ B is used without a preliminary sentence fixing the base field k and the characteristic hypothesis; adding this in the first paragraph of the introduction would prevent any momentary ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our results, the assessment of significance, and the recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; direct proof extending independent prior work

full rationale

The manuscript is a pure existence-and-uniqueness theorem in modular representation theory. Its central statement is proved directly from the definitions of group algebras, tensor products, and commutativity, extending the Carlson–Kovács result (distinct authors) without any fitted parameters, self-referential definitions, or load-bearing self-citations. No equation reduces to an input by construction, no ansatz is smuggled via citation, and the argument does not rename an empirical pattern. The derivation chain is therefore self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no information on free parameters, axioms, or invented entities; the claim appears to rest on standard properties of group algebras and tensor products over fields.

pith-pipeline@v0.9.0 · 5561 in / 945 out tokens · 20848 ms · 2026-05-23T22:06:02.146546+00:00 · methodology

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Reference graph

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