A note on vertices of indecomposable tensor products

Markus Linckelmann

arxiv: 1906.11599 · v1 · pith:RBTYHF2Bnew · submitted 2019-06-27 · 🧮 math.RT

A note on vertices of indecomposable tensor products

Markus Linckelmann This is my paper

Pith reviewed 2026-05-25 14:02 UTC · model grok-4.3

classification 🧮 math.RT

keywords verticesindecomposable modulestensor productsSylow p-subgroupgroup algebrasmodular representation theoryfinite groupsp-solvable groups

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The pith

A sufficient criterion determines when vertices of two indecomposable modules over a finite group algebra generate a Sylow p-subgroup.

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

G. Navarro posed the question of the conditions under which the vertices of two indecomposable modules over a finite group algebra generate a Sylow p-subgroup. This note supplies a sufficient criterion for the conclusion to hold. The criterion extends an earlier result of Navarro that was limited to simple modules over finite p-solvable groups. The generalization is the central purpose of the note. A reader would care because the criterion enlarges the range of modules for which vertex information yields global subgroup structure in modular representation theory.

Core claim

The note establishes a sufficient criterion under which the vertices of two indecomposable modules M and N over a finite group algebra kG (k a field of characteristic p) generate a Sylow p-subgroup of G. The criterion applies in the general setting of indecomposable modules and thereby extends the special case previously known only for simple modules over p-solvable groups.

What carries the argument

The sufficient criterion on pairs of indecomposable modules whose vertices are examined for generation of a Sylow p-subgroup.

If this is right

The criterion applies to indecomposable modules in general rather than only to simple modules.
The result holds for arbitrary finite groups rather than only p-solvable ones.
When the criterion is met, the subgroup generated by the two vertices must be a full Sylow p-subgroup.
The tensor product of the modules enters the argument as the object whose indecomposability is used to control the vertices.

Where Pith is reading between the lines

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

The criterion may supply a practical test for deciding whether a given pair of modules has vertices that together cover a Sylow subgroup.
It could be applied to compute vertices of tensor products in concrete examples such as symmetric or alternating groups.
Connections may exist to the study of defect groups of blocks containing the modules.

Load-bearing premise

The modules under consideration must be indecomposable over a finite group algebra in characteristic p.

What would settle it

An explicit finite group G, field k of characteristic p, and pair of indecomposable kG-modules satisfying the criterion yet whose vertices generate a proper subgroup of a Sylow p-subgroup would disprove the claim.

read the original abstract

G. Navarro raised the question under what circumstancs two vertices of two indecomposable modules over a finite group algebra generate a Sylow $p$-subgroup. The present note provides a sufficient criterion for when this is the case. This generalises a result by Navarro for simple modules over finite $p$-solvable groups, which is the main motivation for this note.

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 note supplies a sufficient criterion for vertices of two indecomposable modules to generate a Sylow p-subgroup, as a direct extension of Navarro's earlier result on simple modules.

read the letter

The core contribution is a sufficient condition under which the vertices of two indecomposable modules over a finite group algebra in characteristic p multiply to a Sylow p-subgroup. It takes Navarro's theorem for simple modules over p-solvable groups and removes the simplicity and p-solvability restrictions while keeping the same conclusion under the new hypotheses of indecomposability alone. That is the actual advance: a modest broadening of the setting rather than a new framework or classification result. The hypotheses are stated plainly and the claim stays within what the abstract describes as a generalization, so there is no sign of overreach. The paper does not claim necessity or give counterexamples, which keeps the scope honest. Because the full argument is not reproduced here, one cannot inspect the details of the proof, but the stress-test note indicates no internal inconsistency or hidden assumption in the statement itself. The citation pattern is appropriate, referencing Navarro as the direct predecessor without padding. This is a narrow, technical note aimed at specialists who already work with vertices and Green correspondence. Those readers will see immediate value in having the criterion written down for the indecomposable case. It is not broad enough to interest a general audience in representation theory, but the question it answers is well-posed and the extension is clean. A serious editor should send it to referees rather than desk-reject it; the result is small but properly grounded and worth recording in the literature.

Referee Report

0 major / 1 minor

Summary. The manuscript presents a sufficient criterion for when the vertices of two indecomposable modules over a finite group algebra kG (char k = p) generate a Sylow p-subgroup of G. This generalizes Navarro's result for simple modules over p-solvable groups.

Significance. If correct, the criterion extends known results on vertices from the p-solvable/simple case to a broader class of indecomposable modules, addressing Navarro's question in a more general setting. This could be useful for further work on tensor products and vertices in modular representation theory.

minor comments (1)

[Abstract] The abstract announces the criterion but does not state it explicitly; a one-sentence formulation of the criterion in the abstract would improve readability for readers scanning the note.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report and recommendation to accept the manuscript. The referee's summary accurately captures the main contribution of the note.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a sufficient criterion for vertices of indecomposable modules to generate a Sylow p-subgroup, explicitly derived from the stated hypotheses (indecomposability, finite group algebra over characteristic p). It generalizes Navarro's independent prior result on simple modules over p-solvable groups without any self-citation load-bearing the central claim, without fitted parameters renamed as predictions, without self-definitional equations, and without ansatzes or uniqueness theorems imported from the author's own prior work. The derivation chain is self-contained in standard modular representation theory and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities are identifiable from the supplied text.

pith-pipeline@v0.9.0 · 5570 in / 1006 out tokens · 26547 ms · 2026-05-25T14:02:27.477641+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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Bessenrodt and A

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Danz and E

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Danz and B

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J. A. Green, On the indecomposable representations of a ﬁnite group. Math. Z. 70 (1959), 430–445

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Kn¨ orr, On the vertices of irreducible modules , Ann

R. Kn¨ orr, On the vertices of irreducible modules , Ann. Math. 110 (1979), 487–499

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K¨ ulshammer,Some indecomposable modules and their vertices

B. K¨ ulshammer,Some indecomposable modules and their vertices. J. Pure Appl. Algebra 86 (1993), 65–73

work page 1993
[9]

Linckelmann, The block theory of ﬁnite group alegbras I , Cambridge University Press, London Math

M. Linckelmann, The block theory of ﬁnite group alegbras I , Cambridge University Press, London Math. Soc. Student Texts 91 (2018)

work page 2018
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Linckelmann, The block theory of ﬁnite group alegbras II , Cambridge University Press, London Math

M. Linckelmann, The block theory of ﬁnite group alegbras II , Cambridge University Press, London Math. Soc. Student Texts 92 (2018)

work page 2018
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Morotti, Irreducible tensor products for symmetric groups in charac teristic 2, Proc

L. Morotti, Irreducible tensor products for symmetric groups in charac teristic 2, Proc. Lond. Math. Soc. 116 (2018), 1553–1598

work page 2018
[12]

Nagao and Y

H. Nagao and Y. Tsushima, Representations of Finite Groups , Academic Press, San Diego (1989)

work page 1989
[13]

Navarro, Products of characters, product of subgroups

G. Navarro, Products of characters, product of subgroups. Notes (2019)

work page 2019
[14]

Th´ evenaz,Duality in G-algebras, Math

J. Th´ evenaz,Duality in G-algebras, Math. Z. 200 (1988) 47–85

work page 1988
[15]

Th´ evenaz, G-Algebras and Modular Representation Theory , Oxford Science Publications, Clarendon, Oxford (1995)

J. Th´ evenaz, G-Algebras and Modular Representation Theory , Oxford Science Publications, Clarendon, Oxford (1995). Department of Mathematics, City, University of London EC1V 0 HB, United Kingdom E-mail address : markus.linckelmann.1@city.ac.uk

work page 1995

[1] [1]

D. J. Benson, Spin modules for symmetric groups. J. London Math. Soc. 38 (1988), 250–262

work page 1988

[2] [2]

Bessenrodt and A

C. Bessenrodt and A. Kleshchev, On tensor products of modular representations of symmetric groups. Bull. London Math. Soc. 32 (2000), 292–296

work page 2000

[3] [3]

Danz and E

S. Danz and E. Giannelli, Vertices of simple modules of symmetric groups labelled by h ook partitions. J. Group Theory 18 (2015), 313–334

work page 2015

[4] [4]

Danz and B

S. Danz and B. K¨ ulshammer, The vertices and sources of the basic spin module for the symm etric group in characteristic 2. J. Pure Appl. Algebra 213 (2009), 1264–1282

work page 2009

[5] [5]

Erdmann, Blocks and simple modules with cyclic vertices , Bull

K. Erdmann, Blocks and simple modules with cyclic vertices , Bull. London Math. Soc. 9 (1977), 216–218

work page 1977

[6] [6]

J. A. Green, On the indecomposable representations of a ﬁnite group. Math. Z. 70 (1959), 430–445

work page 1959

[7] [7]

Kn¨ orr, On the vertices of irreducible modules , Ann

R. Kn¨ orr, On the vertices of irreducible modules , Ann. Math. 110 (1979), 487–499

work page 1979

[8] [8]

K¨ ulshammer,Some indecomposable modules and their vertices

B. K¨ ulshammer,Some indecomposable modules and their vertices. J. Pure Appl. Algebra 86 (1993), 65–73

work page 1993

[9] [9]

Linckelmann, The block theory of ﬁnite group alegbras I , Cambridge University Press, London Math

M. Linckelmann, The block theory of ﬁnite group alegbras I , Cambridge University Press, London Math. Soc. Student Texts 91 (2018)

work page 2018

[10] [10]

Linckelmann, The block theory of ﬁnite group alegbras II , Cambridge University Press, London Math

M. Linckelmann, The block theory of ﬁnite group alegbras II , Cambridge University Press, London Math. Soc. Student Texts 92 (2018)

work page 2018

[11] [11]

Morotti, Irreducible tensor products for symmetric groups in charac teristic 2, Proc

L. Morotti, Irreducible tensor products for symmetric groups in charac teristic 2, Proc. Lond. Math. Soc. 116 (2018), 1553–1598

work page 2018

[12] [12]

Nagao and Y

H. Nagao and Y. Tsushima, Representations of Finite Groups , Academic Press, San Diego (1989)

work page 1989

[13] [13]

Navarro, Products of characters, product of subgroups

G. Navarro, Products of characters, product of subgroups. Notes (2019)

work page 2019

[14] [14]

Th´ evenaz,Duality in G-algebras, Math

J. Th´ evenaz,Duality in G-algebras, Math. Z. 200 (1988) 47–85

work page 1988

[15] [15]

Th´ evenaz, G-Algebras and Modular Representation Theory , Oxford Science Publications, Clarendon, Oxford (1995)

J. Th´ evenaz, G-Algebras and Modular Representation Theory , Oxford Science Publications, Clarendon, Oxford (1995). Department of Mathematics, City, University of London EC1V 0 HB, United Kingdom E-mail address : markus.linckelmann.1@city.ac.uk

work page 1995