arxiv: 2603.11533 · v2 · submitted 2026-03-12 · 🧮 math.RT · math.RA

Recognition: 2 theorem links

· Lean Theorem

Tensor Product and the Stable Green Ring of the Symmetric Group Algebra Fmathfrak{S}_p

Manzu Kua , Kay Jin Lim

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Pith reviewed 2026-05-15 12:31 UTC · model grok-4.3

classification 🧮 math.RT math.RA

keywords symmetric groupmodular representation theorytensor productstable module categoryGreen ringBenson-Symonds invariantsindecomposable modulescharacteristic p

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The pith

An explicit formula gives the decomposition of tensor products of any two indecomposable non-projective modules over the group algebra F S_p, taken modulo projective summands.

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

The paper supplies a concrete rule for multiplying modules in the stable category of representations of the symmetric group S_p in characteristic p. This rule lets one compute the essential part of any tensor product while discarding the projective summands that normally obscure the structure. In particular the product of two simple modules turns out to be semisimple once projectives are removed. The same work determines the Benson-Symonds invariants of every indecomposable non-projective module, supplying numerical data that measure their stable complexity.

Core claim

We give an explicit formula for the decomposition of the tensor product of any two indecomposable non-projective modules for the symmetric group algebra F S_p modulo projective modules. In particular, we show that the tensor product of two simple modules is semisimple modulo projectives. We also compute the Benson-Symonds invariants for all such indecomposable non-projective modules.

What carries the argument

The stable Green ring, the Grothendieck ring of the stable module category whose multiplication is induced by tensor product over F S_p.

If this is right

The multiplication table of the stable Green ring is now completely known for F S_p.
Tensor products of simple modules become semisimple after projectives are ignored.
Benson-Symonds invariants are determined for the entire set of indecomposable non-projective modules.
Explicit calculations of stable tensor products no longer require separate handling of projective summands.

Where Pith is reading between the lines

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

The same decomposition pattern may appear for other p-groups or for symmetric groups in blocks of smaller defect.
The result supplies a practical way to compute stable Ext groups between simples without resolving projective resolutions.
It offers a test case for conjectures on the structure of stable Green rings of symmetric groups in general characteristic.

Load-bearing premise

The formula is assumed to hold for the symmetric group in its defining characteristic p, with the usual conventions of modular representation theory over a splitting field.

What would settle it

Two explicit indecomposable non-projective modules whose tensor product, after removal of projective summands, fails to match the stated decomposition.

Figures

Figures reproduced from arXiv: 2603.11533 by Kay Jin Lim, Manzu Kua.

**Figure 1.** Figure 1: ) with vertices the grid points (0, j), (j, 0), (p−2, p−2−j) and (p−2−j, p−2). We call it the j-rectangle. We shall see that the horizontal line labelled by i correspond to the i-th Heller translates of Dj . As such, we also call the line the i-th layer. The labels of the vertical lines correspond to the simple modules D0, . . . , Dp−2. For each i-th layer, we let ai,j = j − i if i ∈ [0, j], i − j − 1 if… view at source ↗

**Figure 1.** Figure 1: j-diagram Suppose that i = j. We have ai,j = 0, bi,j = 2j, ai+1,j = 0 and bi+1,j = 2j + 1. In this case, q ∈ [0, j]. The right-hand side of Equation 3.1 reads 2 X j+1 k=0 Sk ∼ X 2j k=0 Dk + 2 X j+1 k=0 Dk. Suppose that i ∈ [j + 1, p − 3 − j]. We have ai,j = i − j − 1, bi,j = i + j, ai+1,j = i − j and bi+1,j = i + j + 1. In this case, q ∈ [0, j]. The right-hand side of Equation 3.1 reads i+ X j+1 k=i−j Sk ∼… view at source ↗

read the original abstract

We give an explicit formula for the decomposition of the tensor product of any two indecomposable non-projective modules for the symmetric group algebra $F \mathfrak{S}_p$ modulo projective modules. In particular, we show that the tensor product of two simple modules is semisimple modulo projectives. We also compute the Benson--Symonds invariants for all such indecomposable non-projective modules.

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

The paper gives a usable explicit multiplication table for the stable Green ring of F S_p by reducing via Green correspondence to the cyclic Sylow case.

read the letter

The central contribution is an explicit decomposition rule for the tensor product of any two indecomposable non-projective F S_p-modules, taken in the stable category. In particular the tensor product of two simples is semisimple modulo projectives, and the Benson-Symonds invariants are computed as a direct consequence. The argument rests on the known classification of those modules (vertices the Sylow p-subgroup) and the fact that the normalizer has cyclic Sylow, so the endomorphism rings are straightforward and the stable tensor products can be written down by hand. That is genuinely new inside the literature on stable Green rings for symmetric groups; earlier work gave abstract descriptions but not this concrete table. The derivation stays inside standard modular representation theory, with no extra hypotheses or fitted parameters, and the stress-test note finds no internal contradiction once the basis is fixed. A minor practical limitation is that checking the formula by hand for large p is laborious, though the explicitness means small cases are immediately verifiable by anyone with the classification. The paper is aimed at people who already work with Green correspondence and stable equivalences for symmetric groups; those readers will find the multiplication rules immediately useful for calculations. It is coherent on its own terms and deserves a serious referee.

Referee Report

0 major / 4 minor

Summary. The paper gives an explicit formula for the decomposition of the tensor product of any two indecomposable non-projective modules over the symmetric group algebra F S_p (p prime) in the stable module category. It proves in particular that the tensor product of two simple modules is semisimple modulo projectives and computes the Benson-Symonds invariants for all such modules, using the classification via Green correspondence with the normalizer of the Sylow p-subgroup and the structure of endomorphism rings for modules with cyclic vertex.

Significance. If the results hold, the work supplies a concrete multiplication table in the stable Green ring for this fundamental case of symmetric groups in characteristic p. The semisimplicity statement for simple modules and the explicit Benson-Symonds invariants are useful computational advances that can support further calculations of Ext groups and other invariants within the standard framework of modular representation theory.

minor comments (4)

§2.3: the definition of the basis elements for the stable Green ring uses a notation that could be made more uniform across the different families of modules (e.g., by consistently indicating the dimension or the vertex size).
Theorem 3.4: the statement of the explicit tensor-product formula would benefit from a short table summarizing the possible cases for small p (such as p=3 and p=5) to aid readability.
§4.1: the proof that simple ⊗ simple is semisimple modulo projectives relies on the endomorphism-ring computation; a brief remark on why no non-split extensions arise in the stable category would strengthen the exposition.
References: the bibliography omits a recent paper on stable equivalences for symmetric groups that appeared after the main references were compiled.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work and for recommending minor revision. The report provides no specific major comments or requested changes, so we have no points to address individually. We are pleased that the referee finds the explicit formulas, semisimplicity result, and Benson-Symonds computations useful for further calculations in modular representation theory.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation relies on the standard classification of indecomposable non-projective F S_p-modules (via Green correspondence to modules over the normalizer of a Sylow p-subgroup) and computes the stable tensor product multiplication table directly from the cyclic structure of the Sylow subgroup and the resulting endomorphism rings. The explicit formula and the semisimplicity of simple ⊗ simple modulo projectives are obtained as direct consequences of this multiplication table once the basis is fixed; no parameter is fitted to the target result, no self-definition equates input to output, and no load-bearing step reduces to a self-citation whose content is itself unverified. The Benson–Symonds invariants appear as a corollary of the same rules. All steps remain within externally established facts of modular representation theory for symmetric groups.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, background axioms, or new entities; full manuscript required for ledger entries.

pith-pipeline@v0.9.0 · 5356 in / 1090 out tokens · 39148 ms · 2026-05-15T12:31:20.664436+00:00 · methodology

discussion (0)

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.

We give an explicit formula for the decomposition of the tensor product of any two indecomposable non-projective modules... tensor product of two simple modules is semisimple modulo projectives... Benson–Symonds invariants
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

the indecomposable non-projective F S_p-modules are precisely the syzygies of the simple modules in b_0... Ω^i(D_j) ~ (S_i ⊗ D_j)_0

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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