# Verification report — OWR-8415341-012 (Rossmann: characters of given degree of graphical groups over F_p)

Verification date: 2026-10-27 (UTC) for all three runs; the third run examined the text of the note.

**Verdict.** The note gives a **partial answer**: the polynomiality of the character counts is **refuted for an
explicit graph**; the general behaviour asked about **remains open**. Every statement of the note is proved, with
the two qualifications listed below.
- **Settled (negative answer to the precise core of the question, as the note reads it).** For the bipartite graph
  Γ_MK on the 8 points and 8 lines of the Möbius–Kantor configuration 8_3 (a point joined to the 5 lines not
  containing it; 16 vertices, 40 edges) and every odd prime p, the number of irreducible characters of degree p^3
  of G_{Γ_MK}(F_p) is p^10 (h(p) + (1 + (−3/p)) (p−1)^15) with an explicit integer polynomial h of degree 15
  (Theorem 1.1). It is a polynomial on each residue class mod 3 and agrees with no polynomial on any set of primes
  containing infinitely many primes ≡ 1 and infinitely many ≡ 2 mod 3. The non-polynomiality is proved without a
  computer; the coefficients of h are the output of an exact computation in integer arithmetic (not derived
  without a computer), recomputed by an independent implementation and confirmed by independent counts.
- **Also proved.** A short existence proof that some spanning subgraph of Γ_MK has a non-polynomial count
  (Theorem 1.2); polynomiality for i = 1 (all graphs) and i ≤ 2 (bipartite graphs), and non-polynomiality of some
  c_{Γ_MK,i}, 4 ≤ i ≤ 8 (Theorem 1.3); the matrix identity N(q) = h(q) + c(q)(q−1)^15 for every prime power q
  (Theorem 5.7), and the character statements for odd prime powers q with Rossmann's groups G_Γ(F_q)
  (Remark 2.2).
- **Conditional, existence only.** Assuming Belkale–Brosnan, Duke Math. J. 116 (2003), Theorem 11.1 (attributed
  there to Mnëv and Sturmfels; its proof is only sketched there), the counts c_{Γ,3} (Γ bipartite) span the point
  counts of all schemes of finite type over Z over R = S^{-1}Z[q], and some c_{Γ,3} is not PORC
  (Proposition 7.1, Corollary 7.2). No explicit non-PORC graph is known to us.
- **Not new.** The description of character degrees by ranks of alternating matrices (O'Brien–Voll, Trans. AMS 367
  (2015), Theorem B, of which Theorem 2.1 and Remark 2.2 are the case of graphical groups; the note gives a proof);
  the polynomiality of class numbers (Rossmann, Bull. LMS 2022, Theorem 1.2 = Rossmann–Voll, Mem. AMS 2024,
  Corollary 1.3); the realisability of 8_3 over C but not over R, with two realisations up to projectivities
  (classical; Brysiewicz–Gesmundo–Steiner, Beitr. Algebra Geom. 65 (2024), Lemma 3.22; Isham et al., Discrete
  Comput. Geom. 2026, §3.1.2), of which Proposition 3.2 is the same computation over an arbitrary field; the
  classical point count of y^2 = x^3 − x. The existence argument (inclusion and exclusion over supports, with a
  configuration that needs a quadratic irrationality) is of the same kind as Belkale–Brosnan's argument with the
  Fano matroid and may be regarded as known to specialists in this form. No priority is claimed.
- **Related results that do not answer the question.** For class-two groups of exponent p that are not graphical,
  O'Brien–Voll, Theorem 4.1 (generalising Boston–Isaacs), express the character counts through point counts of
  Pfaffian hypersurfaces; Rossmann, Proc. AMS 153 (2025), Theorem A: class numbers of Baer group schemes and
  numbers of linear orbits are wild modulo powers of q; Halasi–Pálfy 2011: class numbers of some class-two pattern
  groups are not polynomials. For graphical groups the class numbers are polynomials.
- **Open.** Which functions occur in general; which graphs have polynomial counts (for instance i = 2 for graphs
  that are not bipartite); which of c_{Γ_MK,4}, …, c_{Γ_MK,8} are not polynomials; an explicit or unconditional
  example that is not PORC; the ordinary representation theory of G_Γ(F_p) beyond the numbers of characters of
  each degree.

The note is unrefereed. Its status is "polynomiality refuted for an explicit graph; general behaviour open";
the problem is not solved.

## Statement checked
- **Primary source.** Oberwolfach Report 38/2021, "Computational Group Theory (hybrid meeting)", Oberwolfach Rep.
  18 (2021), no. 3, 2027–2087, doi:10.4171/OWR/2021/38; problem session, contribution of Tobias Rossmann,
  pp. 2081–2082 (read in the EMS Press file, PDF pages 55–56; fetched 2026-10-27). For a finite simple graph Γ
  with vertices v_1, …, v_n and an odd prime p, G_Γ(F_p) = ⟨x_1, …, x_n | [x_i, x_j] = 1 whenever v_i ≁ v_j, class
  ≤ 2, exponent dividing p⟩. Question: what does the ordinary representation theory of G_Γ(F_p) look like; in
  particular, how does #{χ ∈ Irr(G_Γ(F_p)) : χ(1) = p^i} behave as a function of p? Remarks there: the problem is
  equivalent to counting matrices of given rank in M_Γ(F_p) (antisymmetric, a_ij = 0 for non-adjacent v_i, v_j);
  the number of conjugacy classes of size p^i is a polynomial in p.
- **Same question.** T. Rossmann, Enumerating conjugacy classes of graphical groups over finite fields, Bull.
  Lond. Math. Soc. 54 (2022) 1923–1943, doi:10.1112/blms.12665, Question 1.10 (arXiv:2107.05564v3, Sec. 1.6), for
  prime powers q; known cases: edgeless graphs, paths (Marjoram), complete graphs (O'Brien–Voll, odd q); Rossmann
  suspects that the functions "might be rather wild" (quoting Belkale–Brosnan on symmetric matrices).
- **Rossmann's talk in the same report** (pp. 2064–2066): explicit unipotent groups with polynomial total
  orbit counts but non-PORC counts of orbits of a fixed size; wild examples from Belkale–Brosnan and Rossmann–Voll.
  These concern linear orbits, not characters of graphical groups.
- **Corpus record.** ulamai/UnsolvedMath, OWR-8415341-012 (release v1.8.0, commit d371ee9; status `open`). Its
  statement is a faithful paraphrase of the source's question ("determine the ordinary representation theory" for
  "what does it look like"; "nilpotency-class-two, exponent-p" for "class ≤ 2, exponent dividing p").

## Readings
| Reading | Answer | Where |
|---|---|---|
| Is c_{Γ,i}(p) always a polynomial in p (fixed Γ, i)? (the note's reading of the precise core) | **No**: Γ = Γ_MK, i = 3 | Theorem 1.1 |
| The same for all large p only | No (not even on any set of primes meeting both classes 1, 2 mod 3 infinitely often) | Theorem 1.1 |
| Existence of a non-polynomial pair (Γ, i) by a short argument | yes: a spanning subgraph of Γ_MK, i = 3 | Theorem 1.2 |
| Small degrees | polynomial for i = 1 (all Γ) and i ≤ 2 (Γ bipartite) | Theorem 1.3(a), (b) |
| Are the counts at least PORC? | for Γ_MK and i = 3: yes, period 3; in general: no, conditionally on [BB, Thm 11.1], existence only | Theorem 1.1; Corollary 7.2 |
| Prime powers q (Rossmann's Question 1.10) | the same statements for odd q; the matrix count for every q | Remark 2.2, Theorem 5.7 |
| "Ordinary representation theory" in general | not determined | — |

## Results in the paper
- **Theorem 2.1** (O'Brien–Voll, with a proof via the Baer group V × W): c_{Γ,i}(p) = p^{n−2i} #{A ∈ M_Γ(F_p) : rank A
  = 2i}. **Remark 2.2**: odd prime powers, via the Lazard correspondence (Rossmann 2022, §2.4); the case of
  graphical groups of O'Brien–Voll, Theorem B. **Lemma 2.3**: bipartite graphs, rank A = 2 rank B.
- **Lemma 3.1** (F1)–(F4) for MK; **Proposition 3.2**: exact realisations of MK over a field F correspond, after
  normalising a frame, to the roots of t^2 − t + 1 in F, with an explicit 8 × 8 matrix B(c) whose zeros are exactly
  the 24 incidences and whose other entries are units of Z[c]/(c^2 − c + 1); over F_q there are c(q)|PGL_3(q)|
  of them. **Corollary 3.3**: E(q) = c(q)(q−1)^15 rank-3 matrices with support exactly the non-incidences.
- **Theorem 1.2** (Section 4): Möbius inversion over the 2^40 subboards.
- **Section 5**: Lemma 5.1 (Möbius inversion in the subspace lattice of F_q^r; numbers of pairs of subspaces);
  Lemma 5.2 (Z_s, s ≤ 3; Z_3 through homomorphism counts into the incidence graph of PG(2,q)); Lemma 5.3 (rules
  R0–R3); Lemma 5.4 (every non-empty proper minor of the incidence graph H_MK with minimum degree ≥ 3 has a
  4-cycle; counting inequality and the 7 + 7 case analysis); Corollary 5.5 (homomorphism counts of proper minors
  are polynomials); Lemma 5.6 (Möbius inversion over point partitions, collinear count, closure lemma);
  Theorem 5.7 (N(q) = h(q) + c(q)(q−1)^15, every prime power q). The explicit h is computed by
  `rank3_formula.py` (exact integer arithmetic, 7 s; remainder 0).
- **Theorem 1.3** (Section 6): (a) SL_2-count of rank-2 alternating matrices; (b) Lemma 5.1 for r = 2;
  (c) from the class-number polynomial, Rossmann, Bull. LMS 2022, Theorem 1.2 (quoted there from Rossmann–Voll,
  Corollary 1.3).
- **Proposition 7.1, Corollary 7.2** (conditional on [BB, Theorem 11.1]): Vandermonde inversion over R; the
  numbers f_k(r) and a triangular system for maximal degrees; non-PORC via y^2 = x^3 − x.
- **Table 1**: N(p) and p^10 N(p) for p = 3, 5, 7, 11, 13; leading terms (1065 + (−3/p)) p^25 + (5033 −
  15(−3/p)) p^24.

## Computations (programs and outputs in reproducibility/)
- **Programs with which the results were obtained** (`scripts/`, `outputs/`): `rank3_formula.py` + `hompoly.py`
  (the computation of h, 7 s; test mode against brute force on four boards); `test_hompoly.py` (rules R0–R3
  against brute force in PG(2,2), PG(2,3): 81 comparisons, 0 mismatches); `mk_facts.py` (Lemma 3.1,
  Proposition 3.2, search over P^2(F_p), p ≤ 13); `zcount.c` (Z_3(p), p = 2, …, 13, by a different method);
  `rref3.c` (direct N(p), p = 2, 3); `rk3.c` (direct N(p), p = 2, 3, 5, 7); `mk_extra_checks.py` (rook
  congruence 4760; values p < 60); `gap/chardeg_check.g` (Theorem 2.1 against GAP for 11 groups); `check_fk.py`
  (27 cases of f_k(r)).
- **Written with the note**: `check_note.py` (8 s; ALL CHECKS PASSED): the printed statements, including h
  recomputed, N(2) by `rref3.c`, Table 1, and the matrix B(c).
- **Re-runs at the writing (2026-10-27)**: `mk_facts.py`, `rank3_formula.py test`, `test_hompoly.py`,
  `mk_extra_checks.py`, `check_fk.py`, `rk3.c` (p = 2, 3, 5), `gap/chardeg_check.g`, `rank3_formula.py mk` (inside
  `check_note.py`) and verification run A's `minors.c`: outputs identical to the recorded ones.
- **Programs of the three verification runs**: `verification_run_A/`, `verification_run_B/`,
  `independent_run_2/` (the third run), each with its own programs and recorded outputs.

## Independent verification runs
Three independent verification runs, all AI-assisted, examined the results; each re-derived the proofs and wrote
its own programs. None found a wrong statement or a gap in a proof. Runs A and B examined the first written
version of the results; the third run examined the text of the note, first the parts
written after runs A and B.

| Item | Run A | Run B | Third run |
|---|---|---|---|
| Statement and reading of the question against the source | CONFIRMED | CONFIRMED | CONFIRMED_WITH_FIXES (reading marked as such; "might be wild") |
| Theorem 2.1, Lemma 2.3 | CONFIRMED (re-derived) | CONFIRMED (re-derived; GAP on 12 further graphs, pc presentations; cross-check with the p-quotient algorithm) | CONFIRMED; O'Brien–Voll Theorem B read (arXiv v2); GAP on 10 further groups built from Heisenberg groups, three over F_9 |
| Remark 2.2 (prime powers) | — | — | CONFIRMED (Lazard correspondence as in Rossmann 2022, §2.4; wording fixed) |
| Proposition 3.2, Corollary 3.3 | CONFIRMED (sympy from generic unknowns; all 120 determinants; search p ≤ 13) | CONFIRMED (exact realisations for all primes p ≤ 31, assumption-free for p ≤ 7; E(2) = 0, E(3) = 2^15, E(5) = 0 directly) | CONFIRMED (B(c) and y_j symbolically; exact realisations for all 18 prime powers q ≤ 32, using only that the points are distinct) |
| Theorem 1.2 | CONFIRMED | CONFIRMED | CONFIRMED |
| Lemmas 5.1–5.3 | CONFIRMED (pair counts by enumeration over F_2^3, F_3^3) | CONFIRMED (pair counts for q ≤ 9) | CONFIRMED (re-derived; pair counts by enumeration over F_2^3, F_3^3, F_5^3) |
| Lemma 5.4 | CONFIRMED; exhaustive enumeration of all 447,153,316 minors with at least one point and one line: 9,193,080 with minimum degree ≥ 3, only H_MK itself without a 4-cycle | CONFIRMED_WITH_FIXES (statement for non-empty minors; own exhaustive enumeration) | CONFIRMED (rewritten proof re-derived; run A's enumeration re-run, identical) |
| Lemma 5.6 | CONFIRMED; brute-force hom(H_MK) = c(q)\|PGL_3(q)\| + P_0(q) at q = 2, 3, 5, 7 | CONFIRMED | CONFIRMED; brute force at q = 2, 3, 4, 5 |
| Corollary 5.5 | CONFIRMED; brute-force homomorphism counts for 20 actual proper minors at q = 2, 3, 5 (60 comparisons) and 6 at q = 7 | used | CONFIRMED; 183 brute-force comparisons in all |
| Theorem 5.7, qualitative part | CONFIRMED | CONFIRMED | CONFIRMED |
| Explicit h | CHECKED: re-run identical; direct N(p), p = 2, 3, 5, 7 (own program); Z_3 at p = 2..13 | CHECKED: own direct counts (row-space DP p = 2, 3; partition function p = 2, 3, 5, 7 and q = 4, 8, 9) | RECOMPUTED by an independent implementation of Section 5: identical h and P_0; direct N(2) (two ways), N(3) |
| Remark 5.8 | — | rook number 4760 | CONFIRMED (own rook count; (q−1)^4 ∤ h) |
| Theorem 1.3 | CONFIRMED_WITH_FIXES (cite the class-number input exactly) | CONFIRMED | CONFIRMED (formula of (a) by brute force, six graphs) |
| Proposition 7.1, Corollary 7.2 | CONFIRMED given [BB, Thm 11.1]; f_k(r) in 16 more cases | CONFIRMED given [BB, Thm 11.1]; f_k(r) in 57 cases | CONFIRMED given [BB, Thm 11.1]; f_k(r) in 50 cases; the point count of y^2 = x^3 − x for all primes < 20,000 |
| Package | — | — | re-run from source.zip: identical outputs |
| Literature and novelty | not searched (local extracts only) | searched; no prior answer found; fixes for attribution and related work | searched again; no prior answer found; fixes for attribution and credit |

**Corrections required by runs A and B**, all applied:
1. (A1) Header comments of `hompoly.py` and `rank3_formula.py` in `reproducibility/`: lemma numbers of the note;
   the Heawood remark replaced. Code unchanged.
2. (A2) "The formula also holds at p = 2" now refers to the matrix count N(2) only (Section 1.2, Theorem 5.7);
   the character statement needs p odd.
3. (A3) The note states that h is the result of an exact computation in integer arithmetic, not derived without a
   computer, confirmed independently, and that Proposition 7.1 is conditional on a theorem whose source gives
   only a sketch (abstract, Sections 1.2, 5, 7, 8).
4. (A4) Theorem 1.3(c) cites Rossmann, Bull. LMS 2022, Theorem 1.2 (quoted there from Rossmann–Voll,
   Corollary 1.3), and Corollary B for an explicit formula.
5. (A5) The exhaustive enumeration of minors and the brute-force homomorphism counts on actual minors are reported
   in the Verification paragraph and in `reproducibility/README.md`; the programs and outputs of run A are in
   `reproducibility/verification_run_A/`.
6. (B1) Lemma 5.4 is stated for non-empty proper minors; part (a) of the proof says "non-empty".
7. (B2) The realisability of 8_3 is attributed as classical (see fix 2 of the third run for the final wording).
8. (B3) Halasi–Pálfy, J. Group Theory 14 (2011) 841–854, doi:10.1515/jgt.2010.081, in the related work, with the
   reason why it does not answer the question.
9. (B4) Status wording: polynomiality refuted for an explicit graph; general behaviour open; unrefereed.
10. (B5) Prime powers: proved (Remark 2.2 and Remark 5.8(b)); confirmed at q = 4, 8, 9 by run B
    (`reproducibility/verification_run_B/`).

**Corrections required by the third run**, all applied:
1. Corpus version: the bibliography gave "Version 1.6.0"; it now gives release v1.8.0 (commit d371ee9), the
   snapshot used by all runs.
2. Realisability of 8_3: the attribution "the criterion goes back to MacLane 1936, see also Coxeter 1950,
   Grünbaum 2009" (sources that had not been read) was replaced by sources read in the third run: the
   realisability over C but not over R, with two realisations up to projectivities, is classical
   (Brysiewicz–Gesmundo–Steiner 2024, Lemma 3.22, where the matroid is called the MacLane matroid; Isham et al.
   2026, §3.1.2, realisation space Spec Q[z]/(z^2 − z + 1) over Q); the criterion over an arbitrary field is the
   same computation (Proposition 3.2) and is not claimed as new. Abstract, Section 1.3, Section 8 adjusted.
3. O'Brien–Voll read (arXiv:1203.3050v2): Theorem B is cited directly (Section 1.3, Remark 2.2, with how graphical
   groups fit its hypotheses), Proposition 2.4 for complete graphs (Section 1.1).
4. Related work: O'Brien–Voll, Theorem 4.1 (after Boston–Isaacs, Example 4.3 there) and Rossmann, Proc. AMS 153
   (2025) 479–495, Theorem A, added with the reason why they do not answer the question; also in Section 8.
5. Corollary 7.2: "a classical result of Gauss [Ireland–Rosen]" became "a classical formula (see
   [Ireland–Rosen, Chapter 18])"; Ireland–Rosen is listed as not read; the formula was checked numerically.
6. Section 8: Rossmann "suspected that these functions might be wild" (not "are wild").
7. Section 1.1: the precise core is introduced as the note's reading.
8. Corollary 3.3: the sentence on rank X = 3 was rewritten.
9. Section 8, Verification: the third run is described; which direct counts used two programs is made precise;
   four web searches; dates of the searches.
10. Remark 2.2: "Sections 3 to 6 are proved for every prime power q" became "the arguments of Sections 3–6 apply
    verbatim to prime powers q".
Minor changes with these: Remark 5.8(a) notes that (q−1)^4 divides E; Proposition 7.1, Step 2, describes the
group action explicitly and says that the diagonal coefficients are units of R; Section 1.2 "c(q)(q−1)^15 such
matrices"; Klein–Lewis–Morales (2014) in the q-rook sentence; two specific open questions in Section 8.

**Changes made when the note was written.** The numbering was changed (first version → note: Theorem 1 → 2.1,
Theorem 2 → Lemma 2.3, Theorem 3 → Proposition 3.2 and Corollary 3.3, Theorem 4 → Theorem 1.2, Theorem 5 → 1.1 and
5.7, Theorem 6 → 1.3, Theorem 7 → Proposition 7.1 and Corollary 7.2, Lemmas 3.1–3.5 → 5.1–5.4 and 5.6); an
explicit matrix B(c) replaces the list of 40 determinants in Proposition 3.2; the elementary integrality fact (∗)
is stated once in Section 5; the prime-power statements were added. These changes were examined by the third
run and are covered by `check_note.py`.

## Relation to the literature, novelty and scope
- **Searches (2026-10-27).** arXiv (all papers of T. Rossmann; queries on graphical groups,
  characters of class-two groups, q-rook numbers, matrices with prescribed zeros, support constraints, the
  Kontsevich conjecture, the Möbius–Kantor configuration); works citing Rossmann 2022, Lewis et al. 2011,
  Stembridge 1998 and O'Brien–Voll 2015 in OpenAlex and Semantic Scholar; zbMATH Open; Crossref; four web
  searches. No answer to Rossmann's question was found, and no explicit count of matrices of given rank with
  prescribed zeros that fails to be a polynomial in odd characteristic. A search that finds nothing is not a
  proof of novelty.
- **Closest results.** q-rook numbers (Lewis–Liu–Morales–Panova–Sam–Zhang 2011, Question 5.3; Haglund 1998 for
  Ferrers boards; Klein–Lewis–Morales 2014 for further boards; Chen–Selover 2026, boards with ≤ 6 rows and columns
  polynomial); Stembridge's Fano board (count depending on the parity of q, polynomial on odd q); Belkale–Brosnan
  2003 (symmetric matrices, universality); Rossmann–Voll 2024 (uniformity of average kernel sizes; class numbers
  polynomial; their remark that the Belkale–Brosnan conclusion is unknown to them for arbitrary or antisymmetric
  matrices with constrained supports); O'Brien–Voll 2015, Theorem 4.1, and Boston–Isaacs (character counts of
  non-graphical class-two groups through Pfaffian hypersurfaces); Rossmann 2025 (class numbers of Baer group
  schemes wild modulo powers of q); Halasi–Pálfy 2011 (class numbers of some class-two pattern groups not
  polynomial); Rossmann's talk (non-PORC orbit counts of unipotent groups).
- **What was read.** The source (both contributions); Rossmann 2022 (arXiv v3), Rossmann–Voll (arXiv v2),
  Belkale–Brosnan (arXiv v1), O'Brien–Voll (arXiv v2; Sections 1, 2, 4), Rossmann 2025 (arXiv v2; Sections 1,
  4), Lewis et al. (arXiv v3), Lewis–Morales (arXiv v2), Chen–Selover (arXiv v1), Brysiewicz–Gesmundo–Steiner
  (arXiv v1, Section 3), Isham et al. (arXiv v1, Section 3), in the sections cited; the abstracts of Halasi–Pálfy
  and Klein–Lewis–Morales. Not read: Stembridge 1998 (known through Lewis et al., Lewis–Morales, Chen–Selover),
  Haglund 1998 (through Lewis et al.), Marjoram 1999 (cited after Rossmann), Ireland–Rosen (the point count of
  y^2 = x^3 − x, checked numerically for all primes below 20,000). MacLane 1936, Coxeter 1950 and Grünbaum 2009
  were not read and are no longer cited.
- **Scope.** The note refutes the polynomiality for one explicit graph and describes its count; it does not
  determine the possible behaviour of the counts in general. No priority is claimed.

## 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/