Periodicity shadows II. Computational aspects

Adam Skowyrski; Jerzy Bia{\l}kowski

REVIEW 3 major objections 4 minor 6 references

Periodicity shadows II. Computational aspects

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper claims that a recursive row-by-row algorithm enumerates all tame periodicity shadows up to size 6, with counts 5, 12, 65, and 516.

desk verdict A useful computational census with a real reproducibility gap: the PS3 filter is explicitly skipped and the n=6 shade count is inconsistent (1260 vs 1290), so the exhaustive lists are not yet certified. read the letter →

arxiv 2411.19682 v3 pith:RIRMBLUM submitted 2024-11-29 math.RT

Jerzy Bia{l}kowski , Adam Skowyrski This is my paper · ORCID

classification math.RT MSC 05E1616D5016E2016G2016Z05

keywords periodicityshadowstamealgebrassymmetricperiodicmodulesgeneralizedquaterniontypeGabrielquiverskew-symmetricmatricesrecursiveenumeration

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that a recursive algorithm, which builds skew-symmetric matrices row by row and keeps only canonical representatives under row/column permutation and sign reversal, generates every basic shade of a given size. Filtering those shades by the existence of a symmetric natural-coefficient solution to AC = 0 then yields exactly the tame periodicity shadows. The authors assert that for n = 3, 4, 5, 6 the counts of basic tame periodicity shadows are 5, 12, 65, and 516, with the essential shadows (those that can belong to a tame symmetric algebra with period-4 simples) numbering 4, 7, 26, and 223. The point is to make the combinatorial classification of Gabriel quivers of small tame symmetric algebras computationally explicit and complete.

What carries the argument

The central object is a periodicity shadow: a singular skew-symmetric integer matrix A whose rows do not have nonzero entries of one sign and for which some symmetric matrix C with natural entries and nonzero columns satisfies $AC = 0$, subject to the tame entry bounds T1-T3. The machinery that makes enumeration possible is the total order $\preceq$ on matrices, comparing entries lexicographically after rows and columns are listed in a fixed order. For each orbit under simultaneous row/column permutation the algorithm keeps the minimal representative $A^{\prec}$, and since $A^T = -A$ has the same nullspace, it also identifies a matrix with its negative; line 8 of SetRow returns $\{M\}$ only when $M$ is its own canonical representative and $M \preceq (-M)^{\prec}$. This canonical filter runs inside a recursion that builds matrices row by row, collapsing the search space to the 5, 12, 138, 1290 basic shades for $n = 3, 4, 5, 6$. PS3 is then decided from the parametrised nullspace of $A$.

What would settle it

Run an independent implementation of MatricesSatisfyingTPS(n) for n = 3, 4, 5, 6 together with a certified linear-inequality solver for PS3, and compare the outputs with the published lists: the claim stands only if the counts 5, 12, 65, 516, the essential counts 4, 7, 26, 223, and every listed matrix (A, x, C) match exactly, with all C entries nonnegative and no zero columns.

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Extended reading notes

Core claim

The central claim is that the recursive procedure MatricesSatisfyingTPS(n), whose core is the function SetRow, generates one representative from each orbit of shades under simultaneous row/column permutation, with opposites identified. The filter at line 8 of SetRow keeps a matrix M only when M = M^prec and M is no larger than the canonical representative of -M, so every generated matrix is the minimal representative of its orbit. For each surviving shade A, the paper solves AC = 0 with C = C^T and tests whether a generic nullspace vector admits a symmetric natural-coefficient solution; triples whose vector has a zero entry, opposite parameter entries, or a nonnegative linear relation are deleted, because each such configuration forces every solution to have a zero column or a negative entry. The paper concludes that the remaining matrices form the complete set of basic tame periodicity shadows, and that the tables in Sections 3-5 and the Appendix are complete for n <= 6.

Load-bearing premise

The completeness of the published shadow lists depends on two unverified implementation details: the automatic check that a symmetric matrix with positive integer entries sits in the nullspace, and the omitted recursive procedure that tests whether a matrix is its own canonical representative under permutations; a bug in either would make the counts 5, 12, 65, 516 incomplete.

Editorial extensions

If this is right

For n <= 6, every Gabriel quiver of a tame symmetric algebra with simple modules of period 4 is, modulo loops and 2-cycles, one of the essential shadows listed in Sections 3-5.
Any algebra whose periodicity shadow fails the essentiality conditions PS4 or PS5 is wild, so the search for tame algebras can be restricted to the essential lists.
For n = 3 and 4 the sets of shades and shadows coincide, while for n = 5 and 6 the appendix lists the 73 and 774 shades that are not tame periodicity shadows, respectively.
The algorithm terminates for n = 7 and n = 8 only with impractical runtime, and n >= 9 is intractable; the published tables therefore cover exactly the feasible range n <= 6.
From the complete lists one can read off the generic symmetric Cartan-matrix solutions C for every shadow, which is the data needed to reconstruct candidate algebras.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the PS3 filter is independently reimplemented and verified, the same counts would certify the completeness of the published tables; until then the counts 65 and 516 are best read as computational claims awaiting independent confirmation.
The canonical-representative ordering used here is a general principle: any class of matrices closed under simultaneous permutation and negation can be enumerated by row-by-row generation with the same filtering, so the technique may transfer to other nullspace-defined combinatorial classes.
The listed essential shadows for n <= 6 give a testbed for conjectures about n >= 7 stability: one could check whether every larger shadow decomposes into smaller ones, which would make the apparent stability at n >= 7 a consequence of the small cases.
Because the paper leaves the natural-solution check to external software, a natural extension is to replace it with an explicit linear-inequality certificate per shadow, turning the tables into machine-checkable proofs.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The real new content is the recursive row-by-row generator with a correctness proof for basic shades, and the first complete tables for n=5 and n=6, including the 26 and 223 essential shadows. That is useful, concrete material for anyone trying to classify tame symmetric algebras with period-four simples or algebras of generalized quaternion type. The generator proof is sound in structure: the total order on matrices, the M≺ and (-M)≺ orbit filtering, and the argument that every shade appears up to permutation and sign are all coherent. The Appendix, with full shade lists, is a substantial data dump and appears organized. The soft spots are real but localized. The paper explicitly skips the PS3 check for the surviving triples: verifying C in M_n(N) is described as doable in Maple, then not done. No code, certificates, or output files are provided. For a computational census whose main claim is exhaustiveness, that is a material gap. A second, smaller gap is the omitted recursive detail for testing M=M≺. And there is a worrying count discrepancy: Section 3's table says 1260 shades for n=6, while the text in Section 5 and the Appendix say |S(6)|=1290. I did not find a decisive derivation error in the mathematics, and the reduction to essential shadows is plausibly imported from Part I and [3]. The citation pattern is appropriate; the authors flag their dependence. The paper deserves referee time, but only with a request for code/data or certificates for the final PS3 filtering, and a fix of the count mismatch. As it stands, the complete lists should not be accepted as certified. For a specialist in self-injective algebras the data is still worth having, and the appendix is a useful reference. I would not cite it in a proof without re-running the computation.

Referee Report

3 major / 4 minor

Summary. The paper is the computational companion to arXiv:2411.17381. It defines periodicity shadows as skew-symmetric integer matrices satisfying PS1–PS3 plus tameness conditions T1–T3, and presents a recursive row-by-row generator, MatricesSatisfyingTPS, which is intended to produce all basic shades (shades modulo row/column permutation and transposition) for a given n. A correctness proof is given for the generator: every shade is eventually generated, and the canonical-representative filter in SetRow removes duplicates up to permutation and sign. The authors then filter shades by the PS3 condition, i.e. existence of a symmetric matrix C with natural, non-zero columns satisfying AC = 0, and list the resulting tame periodicity shadows and essential shadows for n ≤ 6. The announced counts are 5, 12, 65 and 516 tame shadows for n = 3, 4, 5, 6, with 4, 7, 26, 223 essential shadows, and the Appendix contains the full lists of basic shades, tame shadows and non-shadow shades.

Significance. If the enumeration is correct, this is a useful exhaustive census of the small cases that are directly relevant to Gabriel quivers of tame symmetric algebras with periodic simples of period four. The recursive generation scheme is a genuine algorithmic contribution, and the proof in Section 2 that the generator produces every shade up to permutation and sign is plausible and covers the orbit deduplication. The displayed triples (A, x, C) provide explicit nullspace data and symmetric solution matrices for every essential shadow, which is valuable for further structural study. However, the paper stops short of certifying the final PS3 filtering step and contains a direct numerical inconsistency in the central counts, so the census as published is not yet fully verifiable.

major comments (3)

[Section 2 (PS3 filtering)] The completeness of the published census for n = 5 and n = 6 rests on the PS3 verification, but this verification is explicitly not carried out in the paper. After listing three sufficient deletion conditions, the authors state that for each remaining triple one can check C ∈ M_n(N) 'automatically via standard computational environments like Maple, but we skip this for simplicity' (Section 2, end). No code, Maple output, or certificates are supplied, and the deletion rules are only sufficient conditions for failure. Therefore the claim that Sections 4–5 and the Appendix list all tame periodicity shadows S(5) and S(6) is not certified by the manuscript; a bug in the omitted feasibility check could either include non-shadows or exclude genuine shadows.
[Section 3 table vs Section 5 and Appendix] There is a direct numerical inconsistency in the central data. The table in Section 3 gives 1260 basic shades for n = 6, while Section 5 states |S(6)| = 1290 basic shades, and the Appendix repeats the figure 1290 and lists items (1)–(1290). Since these counts are the main quantitative output of the paper, the discrepancy must be resolved; if 1290 is the correct count, the table entry 1260 is wrong, and if 1260 is correct, the Appendix and Section 5 overcount.
[Section 2, SetRow line 8] The correctness proof for MatricesSatisfyingTPS depends on the test M = M≺ for canonical representatives, but the paper states that this test 'can be efficiently verified by a recursive procedure (we omit the details)' and no implementation is given. Together with the omitted PS3 check, the algorithm as presented is therefore not fully reproducible from the text, although the high-level correctness argument for the recursive enumeration is otherwise coherent.

minor comments (4)

[Abstract and Section 2] There are several typographical errors that should be corrected, including 'cosiderations' in the abstract and 'partial oreder' in Section 2, as well as the entry '−0' in the description of ComposeRow.
[Author and address lines] The name 'BIA/suppress lkowski' appears corrupted, and 'Skowyrski' is used inconsistently with 'Skowroński' in the references; the LaTeX source appears to have a suppression artifact that must be fixed before publication.
[Appendix data] Given the extreme length of the lists for n = 6, the authors should provide machine-readable data (e.g. an ancillary file) so that readers can verify the counts and reuse the matrices; this also mitigates the omitted-certificate problem.
[Section 3 quiver diagrams] The hand-drawn quiver diagrams for n = 4 are very hard to read; vector graphics or a structured description would be preferable.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor self-citation and one omitted computational verification; the enumeration itself is not circular.

self citation load bearing [Section 2, PS4) and PS5) conditions; essential shadow reduction.]
"If Λ = KQ/I is a tame symmetric algebra with simple modules of period 4 and SΛ ⁄= M, then its shadow SΛ must satisfy PS4). This is a consequence of arguments presented in [3, Lemma 5.2] which can be repeated in general. Alternatively, one can prove this using [6, Lemmas 3.2 and 3.3] ..."

The reduction to 'essential shadows' — the central object of the paper's main tables — is not proved in this paper but imported by citation from the authors' own companion paper [6] and from [3], which is co-authored by the same research group. The paper says the arguments 'can be repeated in general' but does not repeat them. Thus the main claim about essential shadows rests on a self-citation chain. However, the quoted result concerns the algebra interpretation, not the purely combinatorial enumeration, which is independently described by the algorithm in Section 2.

full rationale

The paper's central computational claim — that the recursive procedure MatricesSatisfyingTPS(n) generates all basic shades, and that the listed matrices are all tame periodicity shadows for n ≤ 6 — is self-contained. The algorithm is defined explicitly in terms of the combinatorial conditions PS1)–PS3) and T1)–T3), and the proof of correctness (Section 2) is a direct induction on rows. The PS3 deletion step is described with explicit sufficient conditions (zero entry, opposite entries, nonnegative relation) and the remaining feasibility check is stated to be an automatic linear-inequality test; the paper says this check is skipped for simplicity, which is an omitted verification step, not a circular one. The reader's concern about the unverified PS3 check is a correctness/certification issue, not a circularity issue. The paper's own text even flags the omission explicitly. The only genuine (though mild) circularity-adjacent feature is the reliance on the authors' own prior work [6] (and [3]) for the reduction to essential shadows and for the claim that non-essential shadows are wild. That reduction is load-bearing for the presentation of Sections 4–5 as 'all essential shadows', but it is imported background theory rather than a prediction derived from the present paper's own fitted values. Since the enumeration itself stands on its own definitions and the self-citation is not itself the source of the numerical results, the appropriate score is 2.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted constants appear. The central premises are the definitions and algebra-to-shadow correspondence imported from the companion paper [6], plus essential-shadow wildness criteria from [3] and [6]. The algorithm also relies on elementary linear algebra facts about skew-symmetric matrices and on the assumed totality of the lexicographic order on shades. The most fragile imported premise is the unprovided PS3 verification step.

assumptions (6)

domain assumption Definitions of periodicity shadow, tame shadow, shade, and essential shadow are taken from [6] without reproof.
Section 1 and Section 2 import PS1-PS5, T1-T3, and the Markov shadow from the companion paper; all subsequent computation is relative to these definitions.
domain assumption Every tame symmetric algebra with all simple modules periodic of period 4 gives rise to a tame periodicity shadow S_Lambda = Ad_QLambda.
Justifies the matrix census as relevant to the algebra classification. It is attributed to [6, Theorem 2.1] and not reproved here.
domain assumption A shadow that is not essential (fails PS4 or PS5) forces the corresponding algebra to be wild.
Used in Section 2 to restrict the displayed lists to essential shadows; cited to [3, Lemma 5.2] and [6, Lemmas 3.2 and 3.3].
ad hoc to paper For every candidate left in S(n), the existence of a symmetric natural solution C of AC = 0 can be checked automatically; the Maple verification is omitted.
Section 2: 'Verifying this can be done automatically via standard computational environments like Maple, but we skip this for simplicity.' This is an unverified computational assumption that the shadow lists depend on.
standard math Odd-dimensional skew-symmetric matrices are singular and N(A^T)=N(A) for skew-symmetric A.
Used in the algorithm proof and in the PS3 nullspace filtering; standard linear algebra.
standard math The lexicographic relation ≼ is a total order on the finite set of shades, and every permutation orbit contains a unique minimal representative.
The algorithm uses A≺ and (−A)≺ to remove permutations and opposites; the totality and uniqueness are asserted in Section 2.

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Cite this review

Pith. "Pith review of Periodicity shadows II. Computational aspects." pith.science (2026). https://pith.science/paper/RIRMBLUM

@misc{pith2026241119682,
  author       = {Pith},
  title        = {Pith review of: Periodicity shadows II. Computational aspects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RIRMBLUM}},
  note         = {Machine review of arXiv:2411.19682}
}

read the original abstract

This article provides the second part of the research initiated in arXiv:2411.17381, where we introduced and investigated so called periodicity shadows, which are special skew-symmetric matrices related to symmetric algebras with periodic simple modules. In arXiv:2411.17381 we focused on theoretical aspects, whereas here we present complementary cosiderations concerning computational issues. Namely, we discuss an algorithm, which computes all tame periodicity shadows of given size (see Section 2), and then present lists of all tame periodicity shadows of small sizes, that is at most 6.

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Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[3]

Erdmann, A

K. Erdmann, A. Skowro\' n ski, Algebras of generalized quaternion type , Adv. Math. 349 (2019), 1036--1116

work page 2019
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Erdmann, A

K. Erdmann, A. Skowro\' n ski, Higher tetrahedral algebras , Algebr. Represent. Theory 22 (2019), no. 2, 387--406

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Erdmann, A

K. Erdmann, A. Skowro\' n ski, Higher spherical algebras , Archiv Math. 114 (2020), 25--39

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Skowro\' n ski, K

A. Skowro\' n ski, K. Yamagata, Frobenius algebras. I. Basic representation theory , EMS Textbooks in Mathematics, European Mathematical Society (EMS), Z\"urich, 2011. xii+650 pp

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Skowyrski, Two tilts of higher spherical algebras , Algebr

A. Skowyrski, Two tilts of higher spherical algebras , Algebr. Represent. Theory 25 (2021) 237--254

work page 2021
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Periodicity shadows I: A new approach to combinatorics of periodic algebras

A. Skowyrski, Periodicity shadows I. A new approach to combinatorics of periodic algebras , arXiv:2411.17381

work page Pith review arXiv

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [3]

Erdmann, A

K. Erdmann, A. Skowro\' n ski, Algebras of generalized quaternion type , Adv. Math. 349 (2019), 1036--1116

work page 2019

[2] [1]

Erdmann, A

K. Erdmann, A. Skowro\' n ski, Higher tetrahedral algebras , Algebr. Represent. Theory 22 (2019), no. 2, 387--406

work page 2019

[3] [2]

Erdmann, A

K. Erdmann, A. Skowro\' n ski, Higher spherical algebras , Archiv Math. 114 (2020), 25--39

work page 2020

[4] [4]

Skowro\' n ski, K

A. Skowro\' n ski, K. Yamagata, Frobenius algebras. I. Basic representation theory , EMS Textbooks in Mathematics, European Mathematical Society (EMS), Z\"urich, 2011. xii+650 pp

work page 2011

[5] [5]

Skowyrski, Two tilts of higher spherical algebras , Algebr

A. Skowyrski, Two tilts of higher spherical algebras , Algebr. Represent. Theory 25 (2021) 237--254

work page 2021

[6] [6]

Periodicity shadows I: A new approach to combinatorics of periodic algebras

A. Skowyrski, Periodicity shadows I. A new approach to combinatorics of periodic algebras , arXiv:2411.17381

work page Pith review arXiv