# Verification report — OWR-14299292-011 (M. Lee, Oberwolfach Report 27/2025, Problem 1: fixed-point-free representations of A_p for p and p − 2 prime)

Verification date: 2026-10-21.

**Verdict.** The note gives a **partial answer: yes for p = 5, no for p = 7, 13, 19 (13 and 19 conditional on
published decomposition matrices); open for p >= 31.** Every statement it makes is proved, except that the
negative answers for p = 13 (r = 2, 3, 5) and p = 19 (r = 2, 3, 5, 7) are stated, and proved, only conditionally
on the published tables.
- **Settled without tables.** p = 5: yes, exactly in characteristic 2 (A_5 = SL_2(4) on F_4^2; over F_2 a
  4-dimensional irreducible module with endomorphism ring F_4). p = 7: no, in every characteristic (explicit
  identities 1 = sum c_j z_j s_{K_j} in F_2A_7 and F_3A_7). For every twin prime p >= 7: no in every
  characteristic r >= p/2 (r = p, r = p − 2, r > p, p/2 < r < p − 2); in particular p = 13 with r = 7 and p = 19
  with r = 11, 13. Characteristic 0: no.
- **Settled conditionally.** p = 13 with r = 2, 3, 5 and p = 19 with r = 2, 3, 5, 7: no, assuming the
  decomposition matrices of S_13 (mod 2, 3, 5), S_19 (mod 2) and S_17 (mod 3, 5, 7) published by the Modular
  Atlas project (generated from GAP's character table library) are correct.
- **Reduction.** For odd r: if the simple F S_{p−2}-modules on which a (p−2)-cycle is fixed-point-free are only
  the natural module and its sign twist, the answer for (p, r) is no (Lemma 4.1); this classification is a case
  of Zalesski's open Problem 4 (arXiv:2509.24018, p. 6). No reduction for r = 2 is known.
- **Not new.** The case r > p is Lee's own remark (Lemma 3.1 gives its short proof). The exclusion of
  p/2 < r < p − 2 is a standard Brauer-tree argument (Lemma 3.2); for a p-cycle alone, Zalesski attributes the
  case l > n/2 to Thompson (J. Algebra 319 (2008), Lemma 3.2).
- **Open.** Every twin prime p >= 31 (31, 43, 61, 73, 103, ...) with a prime r < p/2.

The note is unrefereed.

## Statement checked
- **Primary source.** Oberwolfach Report 27/2025, "Computational Group Theory" (workshop 1–6 June 2025; organisers
  H. Dietrich, B. Eick, G. Nebe, C. Roney-Dougal), Oberwolfach Rep. 22 (2025), no. 2, 1389–1450,
  doi:10.4171/owr/2025/27; "Summary of the Problem Session", report page 1442, "Melissa Lee. Problem 1". Read in the
  open publisher file (https://ems.press/content/serial-article-files/51860, 62 pages, sha256
  `818c47a49ce2fd05b13dd30e2bd9826f869fedf5e8b1c3a9d6d77bf0c1e356b0`), page index 53.
  - The source assumes that p and p − 2 are prime, asks for a faithful irreducible representation of A_p "over a
    field in any characteristic r > 0" in which all elements of orders p and p − 2 act fixed-point-freely, and
    remarks that it is easy to show this is not the case if r > p.
- **Corpus record.** ulamai/UnsolvedMath, record OWR-14299292-011, title "Fixed-Point-Free Representations of
  Alternating Groups", status `open` (v1.8.0 snapshot d371ee9, tag `literature-status:open`; its literature review of
  2026-08-22 records only the r > p obstruction). The record's clean statement ("Let p and r be primes. Does the
  alternating group A_p admit ...") omits the hypothesis that p − 2 is prime; its `original_statement` starts at the
  question sentence (so the hypothesis and the r > p remark are missing there too) and also carries the text of
  Problem 2 of the session (the Ellis–Harper derangement conjecture), which is not part of this question. The note
  answers the source's question (twin primes p) and claims nothing for other primes p.

## Readings
| Reading | Answer | Where |
|---|---|---|
| For which pairs (p, r) does such a representation exist? (per characteristic, as the source's remark on r > p indicates) | p = 5: exactly r = 2; p = 7: none; p = 13, 19: none (conditional for r < p/2, r != p − 2); p >= 31: only r < p/2 possible, open | Theorem 1.1, Problem 1.2 |
| "in any characteristic" = for some r | yes for p = 5; no for p = 7, 13, 19 (13 and 19 conditional); open for p >= 31 | same |
| "in any characteristic" = for every r | no for every p (already r > p fails, Lee's remark) | Lemma 3.1 |
| "irreducible" over the given field vs absolutely irreducible | equivalent questions (field change) | Lemma 2.1 |
| faithful | automatic (A_p simple; the trivial module has fixed vectors) | Section 1.1 |
| characteristic 0 | no (not the question; covered in passing) | Lemma 3.1 |
| projective representations (2.A_p), only one of the two classes, Problem 2 of the session | not the question; not treated | Section 1.1 |

## Results in the paper
- **Theorem 1.1** (a) reductions (Lemmas 2.1–2.4): the field does not matter; "all elements of orders p and p − 2"
  is equivalent to "g and h"; r = p, p − 2 impossible; criterion φ(g) = −φ(1)/(p−1), φ(h) = −φ(1)/(p−3), so
  (p−1)(p−3)/2 divides dim D (lcm(p−3, (p−1)/2) on A_p, via Zolotarev's lemma); A_p equivalent to S_p (Clifford
  theory, Frobenius reciprocity). (b) r > p and characteristic 0 (Lemma 3.1, Lee's remark). (c) p/2 < r < p − 2
  (Lemma 3.2: blocks of weight <= 1, Brauer-tree alternating sum for an edge, at most two hooks per weight-one
  block, so |φ(g)| <= 2 and dim D <= 2(p−1) < (p−1)(p−3)/2 for p > 7). (d) p = 5 (Proposition 5.1). (e) p = 7
  (Proposition 6.2). (f) p = 13, 19 (Proposition 7.1, conditional part (b)). (g) transposition twist (Lemma 4.1:
  Y = S_q x <t>, (y, t) conjugate to z in S_p, four elements y, two 2 x 2 eliminations with constant 8).
- **Problem 1.2** (open cases) and **Remark 8.1**: Zalesski's expected answer to Problem 4 of the survey (dimension < p)
  together with Lemma 2.4(ii) would give no for all p >= 7 (conditional; suggested by the second verification run).
- **Appendix A**: the 18 Modular Atlas files with SHA-256, seven marked as used in the proof.

## Computations (programs and outputs in reproducibility/)
- `check_note.py` (standard library, about 13 s; TOTAL failures: 0): p = 5 isomorphism and fixed points; both p = 7
  certificate files (1251 and 1675 terms) with a negative control; A_7 class and coset counts; Lemma 2.4 arithmetic
  and Zolotarev's lemma for the 15 twin primes p <= 200; Murnaghan–Nakayama values at a p-cycle for p = 5, 7, 13, 19;
  the hook count of Lemma 3.2 for all 124 pairs (p, r) with p <= 200 (and a control for r < p/2); the constants 8 and
  the conjugacy (y, t) ~ z of Lemma 4.1 for q = 11, 17.
- `original/`: the programs with which the results were first obtained and their 26 recorded outputs (p = 5
  certificate; ideal ranks dim I = 2520 for p = 7, r = 2, 3; decomposition-matrix programs for S_7, S_13, S_17,
  S_18, S_19; transposition-twist test).
- `writing_stage/identify_fpf.py` (fix A3): the modules on which an n-cycle (or the given cycle) is fixed-point-free,
  identified by their Brauer characters on all r-regular classes, for all 18 tables. For S_13 mod 2, 3, 5 and S_19
  mod 2 the (n−2)-cycle is fixed-point-free on no simple module; for S_17 mod 3, 5, 7 (and 11, 13) the 17-cycle is
  fixed-point-free exactly on N and N x sgn.
- `tables/fetch_tables.py`: fetch-and-verify of the 18 tables (not redistributed).
- `verification_run_B_meataxe/`: the second run's MeatAxe-type program (simple modules of A_5, A_7 over F_2, F_3).
- `independent_run_2/`: the programs and outputs of the third verification run (on the final text; see its README):
  certificate checker, own parser and Brauer-character solver for all 18 tables with the block check, the exhaustive
  single-entry alteration test, lemma checks, the p = 5 check in SL_2(4), and the log of the package re-run.
- All 48 recorded outputs were reproduced by `rerun_all.sh` on 2026-10-21 (about 45 s), again by the third run from an
  extracted copy of `source.zip` (43 s; 48 of 48 identical up to the ignored timing lines).
- **Limits.** The table steps check complete parsing, row order, James's unitriangularity, consistency of the
  overdetermined system M B = X with the ordinary characters on all r-regular classes, integrality and positive
  degrees. A changed entry in the row of an r-core (a block of defect zero, whose column has no other nonzero entry)
  is invisible to these checks: among all single +1 changes of the seven tables used in the proof, 233 of 7,676
  (S_13 mod 5), 172 of 32,076, 760 of 60,885 and 4,830 of 74,844 (S_17 mod 3, 5, 7) go unnoticed, none for S_13 mod
  2, 3 and S_19 mod 2. The third run's block check (nonzero decomposition numbers only between partitions with the
  same r-core; no violation in the 18 tables) detects every one of them. No such check can exclude a consistent but
  wrong table, so the decomposition numbers are not proved.

## Independent verification runs
The results were first obtained with proofs and the programs in `original/`. Three independent verification runs,
all AI-assisted, followed (2026-10-21), each with its own programs. Runs A and B examined the first write-up; the
third run examined the final text of the note (below the table).

| Item | Run A | Run B |
|---|---|---|
| Statement and reading against the source | CONFIRMED (source read) | CONFIRMED (source read) |
| Lemmas 2.1–2.4 (field, classes, unipotent elements, criterion, S_p vs A_p) | CONFIRMED (re-derived); fix A1 (integrality) | CONFIRMED (re-derived) |
| Lemma 3.1 (r > p, characteristic 0) | CONFIRMED | CONFIRMED |
| Lemma 3.2 (p/2 < r < p − 2) | CONFIRMED; fix A5 (size of the core) | CONFIRMED modulo standard Brauer-tree theory; hook count checked to p = 200; fix B1 (attribution) |
| Lemma 4.1 (transposition twist) | CONFIRMED (constants 8 checked) | CONFIRMED (re-derived; determinants 8 with sympy) |
| Lemma 6.1 (ideal criterion) | CONFIRMED | CONFIRMED |
| p = 5 | CONFIRMED (proof; original program re-run) | CONFIRMED (own search for the isomorphism, negative control) |
| p = 7 | CONFIRMED (own certificate files, table-free) | CONFIRMED (own MeatAxe; own ideal ranks with all rows) |
| p = 13, 19 | CONFIRMED relative to the tables (own parser and solver; Brauer-character identification; 3 PDFs re-downloaded) | CONFIRMED relative to the tables (own parser and solver; all 18 PDFs re-downloaded, same SHA-256; negative control: 41 inconsistent pairs); fix B2 (provenance) |
| Re-run of the original programs | p = 5 program | all 26 outputs reproduced |
| Literature and bibliography | Zalesski's Problem 4 and the Thompson reference checked | searches (arXiv, zbMATH, Semantic Scholar, OpenAlex, Crossref, 2 web searches); Crossref check of the references; Thompson's paper refused (403) |
| Scope PARTIAL, open set p >= 31 with r < p/2 | CONFIRMED | CONFIRMED |
| Overall | CONFIRMED | CONFIRMED_WITH_FIXES (wording and citations only) |

Required fixes, all made in the note and the package: A1 integrality argument in Lemma 2.4(i); A2 the conditional in
the abstract, the theorem, the introduction's summary and the conclusion; A3 identification of the fixed-point-free
modules by Brauer characters (`identify_fpf.py`); A4 description of two stray error-output files of the first
write-up, which are kept out of the package; A5 size p − r >= 2 of the hook core; B1 attribution of the case
p/2 < r < p − 2; B2 provenance of the tables (URL, access date, CTblLib, SHA-256 appendix, load-bearing files);
B3 clarification of the case p = 7 in Section 8; B4 the conditional Remark 8.1.

**Third run (on the final text, 2026-10-21).** It examined the note line by line before any change, read again the
problem session and the first page of the OWR report (workshop "Computational Group Theory", 1–6 June 2025; problem
session of 5 June 2025; report pp. 1389–1450, problem on p. 1442), the corpus record, pp. 4–6 and the reference list of
arXiv:2509.24018 and the Modular Atlas and CTblLib pages cited, re-derived every proof (verdict: no gap), and checked the
bibliographic data with Crossref. Its own programs (`independent_run_2/`, written from the text):
- both p = 7 certificate files: every generator a 7-cycle or a 5-cycle, every representative even, the expanded sum is
  the identity of F_rA_7 (r = 2: 1251 terms, 240 distinct subgroups; r = 3: 1675 terms, 245 distinct subgroups);
  negative controls; consistent with `ideal_rank.py` and with the second run's MeatAxe lists;
- all 18 tables with its own pypdf extraction and parser (rows by position; label strings such as "11121" are
  ambiguous), Murnaghan–Nakayama, unitriangularity, M B = X (0 inconsistent pairs), identification by Brauer
  characters: S_13 mod 2, 3, 5: the 13-cycle exactly on N (and N ⊗ sgn), the 11-cycle on none; S_19 mod 2: the
  19-cycle only on D^(18,1), the 17-cycle on none; S_17 mod 3, 5, 7 (and 11, 13): exactly N and N ⊗ sgn; maxima of
  |φ| at the n-cycle 1, 2, 4 as stated; block check: no violation;
- the exhaustive single-entry alteration test of the seven load-bearing tables (numbers under **Limits** above);
- the finite statements of Lemmas 2.4, 3.2 (124 pairs), 4.1 (206 cycle types) and (2); Proposition 5.1 directly in
  SL_2(4) (no invariant F_2-subspace, commutant F_4);
- SHA-256 of the seven load-bearing files equal to Appendix A; A13.2partmod3, A19.2partmod2, A17.2partmod7 downloaded
  again with the same values; `tables/fetch_tables.py` run on local copies (all verified) with a negative control
  (one altered byte rejected); the whole package re-run from the archive (48 of 48 outputs reproduced);
- literature: arXiv API (7 queries), OpenAlex (0 works citing the OWR report), Crossref, zbMATH, one web search; no
  answer found; Thompson's paper refused again (HTTP 403), not worked around.
It found no mathematical error and required seven corrections, all made: (1) no gendered pronoun for the proposer
(abstract, Section 1.1; also for Zalesski in Section 1.3, Remark 8.1 and this report; Zenodo description); (2) the
date of the problem session; (3) the counts of the certificate files (terms, not subgroups); (4) the account of what
the table checks detect (Section 7); (5) the description of this run (Section 9); (6) the account of the searches and
sources (Scope and priority: four web searches, a Semantic Scholar keyword search refused with HTTP 429, the cited
textbooks not re-read); (7) this report and the package README.

## Relation to the literature, novelty and scope
- **Searches (2026-10-21, repeated in part by the third run).** arXiv API (alternating group and eigenvalue 1;
  fixed-point-free representations of alternating groups; twin primes; author queries Zalesski and Lee; unisingular
  groups; Brauer characters at cycles), zbMATH API, Semantic Scholar (including the 5 works citing arXiv:2509.24018;
  one keyword search of run B was refused with HTTP 429 twice), Crossref and OpenAlex (0 works citing the OWR
  report), four web searches in all. No printed answer to the question was found in these sources. A search that
  finds nothing is not a proof of novelty.
- **Closest work.** A. E. Zalesski, arXiv:2509.24018 (v1, 2025-09-28), Problem 4 (p. 6): irreducible
  representations of A_p, characteristic l != 0, p, in which an element of order p has minimal polynomial of degree
  < p; open; expected dimension < p; the case l > n/2 attributed to J. G. Thompson, J. Algebra 319 (2008) 558–594,
  Lemma 3.2 (doi:10.1016/j.jalgebra.2006.04.040; the lemma itself was not read: the publisher's site refused
  automated access with HTTP 403, to run B and again to the third run, not worked around). Context only: Chen–Lee–Mitrović–O'Brien–Xia arXiv:2508.12652,
  Freedman–Lee arXiv:2607.23423 (full texts searched by run B; no A_p), Zalesski arXiv:2509.05770, Velmurugan
  arXiv:2412.20894 (other conditions or characteristic 0; abstracts read).
- **What was read.** The problem session and the first page of the OWR report; pp. 4–6 and the reference list entry
  [64] of arXiv:2509.24018; the index, directory (A13) and references pages of the Modular Atlas; the CTblLib home page
  and its information page for A13. Textbook facts (Brauer characters, Clifford theory, blocks of defect one and Brauer
  trees, Murnaghan–Nakayama rule, Nakayama's conjecture, James's unitriangularity; Navarro, Brauer, Dade, Feit, James,
  Parker, Holt–Rees) are cited, not re-read; Crossref confirms Feit's Chapter VII (pp. 269–339) but not its title.
- **Data provenance.** Modular Atlas decomposition matrices, https://www.math.rwth-aachen.de/~MOC/decomposition/,
  files tex/A<n>/A<n>.2partmod<r>.pdf, accessed 2026-10-21; created automatically from GAP's character table library
  (CTblLib, T. Breuer; home page lists version 1.3.11; generating version not stated). Primary references for the
  original computations of the individual tables were not traced.
- **Scope and priority.** The note's contribution is the explicit answers for p = 5, 7, 13, 19 (13 and 19 only
  modulo the published tables), the reduction of Lemma 4.1 (not found in print in this form; no priority claimed),
  and the exclusion of all r >= p/2 assembled for this question. The general question stays open.

## Suggested dataset label
`open`, unchanged: an unrefereed partial answer (evidence: unrefereed_claim); at most "partially solved" at the
maintainers' discretion; never "solved" — partial answer: yes for p = 5, no for p = 7, 13, 19 (13 and 19
conditional on published decomposition matrices); open for p >= 31.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/