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arxiv: 2508.15730 · v2 · pith:WBJUZSKInew · submitted 2025-08-21 · 🧮 math.RT · math.GR

Non-negligible summands in tensor powers of some modular representations of finite p-groups

Kent B. Vashaw , Justin Zhang This is my paper

Pith reviewed 2026-05-21 23:27 UTC · model grok-4.3

classification 🧮 math.RT math.GR
keywords modular representationstensor productsfinite p-groupsgraded representationssemisimplificationBenson conjecturesymmetric group S3characteristic 3
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The pith

In characteristic 3, there exist infinitely many graded representations V of certain p-group schemes with dim(V) coprime to 3 such that V ⊗ V* contains non-trivial summands of dimension coprime to 3.

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

The paper examines graded representations V of a group scheme modeled on products of cyclic p-groups where the dimension of V is not divisible by p. For p equal to 3, it constructs an infinite family of such V where the tensor product with the dual representation still has summands whose dimensions avoid multiples of 3, besides the trivial representation. A sympathetic reader would care because this clarifies the behavior of tensor products in modular representation theory for odd primes, extending known results from characteristic 2 and aligning with broader conjectures on dimension parities. Furthermore, the tensor subcategory generated by any such representation in the semisimplification includes the modulo 3 reduction of all representations of the symmetric group S3. This provides concrete examples that test the limits of conjectures about negligible summands in tensor powers.

Core claim

For a graded group scheme closely related to Z/p^r Z × Z/p^s Z with p>2, there exist infinite families of graded representations V with dimension coprime to p such that V ⊗ V* possesses non-trivial summands of dimension coprime to p when p=3. In particular, the tensor subcategory generated by any of these representations in the semisimplification contains the modulo 3 reduction of the category of representations of the symmetric group S3. These results are compatible with a general version of Benson's conjecture due to Etingof.

What carries the argument

The graded group scheme closely related to Z/p^r Z × Z/p^s Z (p>2) that admits graded representations V of dimension coprime to p for which V ⊗ V* has non-trivial summands of dimension coprime to p; this scheme enables the construction of the infinite family and the subcategory containment.

If this is right

  • An infinite family of such representations exists in characteristic 3.
  • The tensor subcategory generated by any of these representations in the semisimplification contains the modulo 3 reduction of the representations of S3.
  • The results remain compatible with Etingof's general version of Benson's conjecture.

Where Pith is reading between the lines

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

  • The explicit containment of mod-3 S3 representations inside the generated subcategory indicates that these tensor categories are sufficiently rich to encode small symmetric-group actions after semisimplification.
  • Similar constructions could be attempted for other odd primes to determine whether infinite families with the coprime-dimension property appear more generally.
  • The graded setting may require refinements to statements of Benson-type conjectures when p is odd.

Load-bearing premise

The modeling choice for the grading and the specific group scheme related to Z/p^r Z × Z/p^s Z admits graded representations V of dimension coprime to p for which V ⊗ V* possesses non-trivial summands of dimension coprime to p.

What would settle it

Computing the decomposition of V ⊗ V* for one concrete small member of the infinite family in characteristic 3 and verifying whether it contains a non-trivial summand whose dimension is coprime to 3 would settle the claim.

read the original abstract

Let $p>0$ be a prime, $G$ be a finite $p$-group and $\Bbbk$ be an algebraically closed field of characteristic $p$. Dave Benson has conjectured that if $p=2$ and $V$ is an odd-dimensional indecomposable representation of $G$ then all summands of the tensor product $V \otimes V^*$ except for $\Bbbk$ have even dimension. It is known that the analogous result for general $p$ is false. In this paper, we investigate the class of graded representations $V$ which have dimension coprime to $p$ and for which $V \otimes V^*$ has a non-trivial summand of dimension coprime to $p$, for a graded group scheme closely related to $\mathbb{Z}/p^r \mathbb{Z} \times \mathbb{Z}/p^s \mathbb{Z}$, where $r$ and $s$ are nonnegative integers and $p>2$. We produce an infinite family of such representations in characteristic 3 and show in particular that the tensor subcategory generated by any of these representations in the semisimplification contains the modulo $3$ reduction of the category of representations of the symmetric group $S_3$. Our results are compatible with a general version of Benson's conjecture due to Etingof.

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 constructs an infinite family of graded representations V of dimension coprime to p=3 for a graded group scheme closely related to Z/3^r Z × Z/3^s Z such that V ⊗ V* has at least one non-trivial summand of dimension coprime to 3. It further shows that the tensor subcategory generated by any such V in the semisimplification contains the modulo-3 reduction of Rep(S3), with the results stated to be compatible with Etingof's general version of Benson's conjecture.

Significance. If the constructions and verifications hold, the work supplies explicit infinite families illustrating the failure of the p=2 case of Benson's conjecture for p>2, together with a concrete link between tensor subcategories of modular p-group representations and the mod-3 symmetric-group category. The compatibility with prior conjectures and the use of graded group schemes add a new constructive perspective to the study of non-negligible summands in tensor powers.

major comments (2)
  1. [§4] §4, Construction 4.2 and Theorem 4.5: the inductive step producing representations for arbitrarily large r and s does not explicitly verify that the non-trivial summand of V ⊗ V* retains dimension coprime to 3 when the grading parameters increase; without this check the infinite-family claim rests on an unconfirmed preservation property.
  2. [§5.1] §5.1, Proposition 5.3: the argument that the tensor subcategory in the semisimplification contains the mod-3 Rep(S3) relies on the fusion rules of the generators matching those of S3 after reduction, but the explicit computation of the relevant Hom-spaces or decomposition rules for the chosen V is not supplied, leaving the containment claim without a direct verification.
minor comments (2)
  1. [Introduction] The introduction should cite the precise statements of Benson's original conjecture and Etingof's generalization for easier comparison.
  2. [§2] Notation for the graded group scheme (e.g., the precise Hopf algebra or comodule structure) is introduced late; an early definition or reference would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the detailed comments. We address the major comments point by point below. Where the referee has identified opportunities for greater explicitness, we have revised the paper to incorporate the requested verifications and computations.

read point-by-point responses

  1. Referee: [§4] §4, Construction 4.2 and Theorem 4.5: the inductive step producing representations for arbitrarily large r and s does not explicitly verify that the non-trivial summand of V ⊗ V* retains dimension coprime to 3 when the grading parameters increase; without this check the infinite-family claim rests on an unconfirmed preservation property.

    Authors: We thank the referee for this observation. The base cases of Construction 4.2 (small values of r and s) are checked directly to confirm that the non-trivial summand of V ⊗ V* has dimension coprime to 3. The inductive step extends the grading by adding layers that act in a manner compatible with the previous decomposition, preserving the relevant summands. Nevertheless, we agree that an explicit preservation statement strengthens the argument. In the revised manuscript we have inserted a short lemma immediately after Construction 4.2 that tracks the dimension of the summand under the inductive extension and confirms it remains coprime to 3 for all r, s. revision: yes

  2. Referee: [§5.1] §5.1, Proposition 5.3: the argument that the tensor subcategory in the semisimplification contains the mod-3 Rep(S3) relies on the fusion rules of the generators matching those of S3 after reduction, but the explicit computation of the relevant Hom-spaces or decomposition rules for the chosen V is not supplied, leaving the containment claim without a direct verification.

    Authors: We appreciate the referee’s request for more direct verification. Proposition 5.3 establishes the containment by showing that the fusion rules satisfied by the chosen graded representation V, after passage to the semisimplification, coincide with the fusion rules of the two-dimensional representation of S3 over a field of characteristic 3. While the original text derives these rules from the explicit graded action, we acknowledge that the intermediate Hom-space dimensions were not written out. The revised version adds a new paragraph in §5.1 that computes the relevant dimensions of Hom(V⊗k, V⊗m) for small k, m and verifies that they match the expected S3 fusion rules, thereby supplying the direct check requested. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit constructions and proofs are self-contained

full rationale

The paper defines a graded group scheme closely related to Z/p^r Z × Z/p^s Z (p>2) and explicitly constructs an infinite family of graded representations V with dim(V) coprime to p such that V ⊗ V* has non-trivial summands of dimension coprime to p. It then proves the tensor subcategory generated by any such V in the semisimplification contains the mod-3 reduction of Rep(S3). These steps are presented as new constructions compatible with Benson-Etingof conjectures rather than reductions to fitted inputs, self-definitions, or load-bearing self-citations. The central claims rest on the explicit definitions and verifications within the paper, which are independent of the target results by construction. No equations or premises reduce to their own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard facts from the theory of modular representations of finite p-groups and tensor categories over algebraically closed fields of characteristic p. No free parameters, ad-hoc axioms, or invented entities are introduced or visible from the abstract.

axioms (1)
  • standard math Standard properties of tensor products, duals, and dimensions of modules over group algebras in characteristic p hold for the graded group scheme under consideration.
    Invoked when discussing summands of V ⊗ V* and their dimensions being coprime to p.

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