# Verification report — OWR-12978-004 (Montgomery, Oberwolfach Report 20/2014, Question 9: Frobenius–Schur indicators and gauge equivalence; the group-algebra case)

Verification date: 2026-10-20.

**Verdict.** The note gives a **complete negative answer to Question 9 in the case of group algebras** (the case
the source calls open), in the strongest reading of the question and also with R. Ng's extra condition of equal
fusion rules. The corpus record bundles two further questions, which the note does not settle, so the record as a
whole is **partial**.
- **Settled by the note.** If two groups of odd order form a Brauer pair, their group algebras have equal
  exponents and a bijection of simple modules that preserves all higher indicators ν_n (n ≥ 1) and the fusion
  rules, but their representation categories are not tensor equivalent (Theorem 1.3). An explicit pair of
  Camina groups of order 3^6 is constructed, with complete proofs of all its properties (Theorem 1.4).
- **Cited, not reproved.** The non-equivalence uses exactly one external theorem: Etingof–Gelaki, Isocategorical
  groups, IMRN 2001, Corollary 1.4 (groups of odd order are categorically rigid), with Definition 1.1 and
  Lemma 3.1 of that paper.
- **Known before.** Brauer pairs of odd order (Dade 1964; Eick–Müller 2006; Lewis 2009; Nenciu 2008, 2009); the
  pair of order 3^6 is of a kind described by Lewis and is not claimed to be new as a Brauer pair.
- **Elementary example (not found in print).** For general semisimple Hopf algebras without Ng's condition, an
  elementary negative answer (C^Heis(3) against C[(Z/3)^3], Remark 5.1) is given in the note without any novelty
  claim; Mignard–Schauenburg (arXiv 1708.06538, Remark 4.5)
  observe that indicators do not classify pointed fusion categories (rank 8, quasi-Hopf case; that paper, read in
  full, contains no such pair with trivial cocycle, i.e. no pair of Hopf algebras).
- **Not settled.** Lentner's Question 6(c) (interpretation of diagram folding) and Lu's Question 7 (a Hopf ideal
  complement of the maximal pointed Hopf subalgebra H'; relations between H and H'). For 7(a) the note has only a
  remark that is conditional on a reading of H' (H' was introduced in Lu's talk, joint work with Fan and Yu; the
  source cites no paper for it, and the paper of these authors that presumably contains the definition was not
  read).
- **Novelty.** No printed answer to Question 9 was found in the sources searched; both ingredients are in print
  and the combination is short, so it may be folklore. No priority is claimed.

The note is unrefereed.

## Statement checked
- **Problem source.** K. A. Brown, K. R. Goodearl, T. H. Lenagan, J. J. Zhang (organisers), Mini-Workshop:
  Infinite Dimensional Hopf Algebras, Oberwolfach Report 20/2014, Oberwolfach Rep. 11 (2014), no. 2, 1111–1137,
  doi:10.4171/OWR/2014/20. Read in the publisher's open-access PDF: Montgomery's abstract (pp. 1126–1127) and
  the Open Questions (pp. 1131–1134).
- **Definitions (p. 1126).** H semisimple Hopf algebra over C, Λ its integral with ε(Λ) = 1,
  x^[n] = Σ x_1 ⋯ x_n the n-th Hopf power, ν_n(V) = χ(Λ^[n]); for groups the classical indicator. The abstract
  states that the indicators are gauge invariants (citing Mason–Ng 2005 and Ng–Schauenburg 2008) and that it is
  open whether all indicator values determine the category "up to tensor equivalence" (p. 1127).
- **Question 9 (pp. 1133–1134).** With simple modules V_1, …, V_n, exponent m and
  S = {ν_j(V_i) : 1 ≤ i ≤ n, 1 ≤ j ≤ m}: if H and H' have the same set S, are their module categories gauge
  equivalent? Remarks of the source: ν_m(V_i) = dim V_i, so equal sets give H ≅ H' as algebras; trivially true
  if H is the group algebra of an abelian group; raised in 2006; "still seems to be open for group algebras";
  R. Ng suggested also requiring the same fusion rules.
- **Bundled questions.** Question 6(c) (S. Lentner, p. 1133) and Question 7(a),(b) (D.-M. Lu, p. 1133), as
  described in the verdict.
- **Corpus record.** UnsolvedMath record OWR-12978-004 (snapshot d371ee9), label: open.

## Readings
- **The datum S.** Literally a set of numbers, S cannot determine the algebra: C[Z/2] and C[Z/3] both have
  S = {0, 1}, but C^2 ≇ C^3. This contradicts the source's own remark that equal sets force H ≅ H' and would make
  the abelian case false. The note therefore reads S as the indexed family of indicator sequences and answers
  the strongest form (Question 1.2 of the note): equal exponents, a bijection σ of simple modules with
  ν_n(σV) = ν_n(V) for all n ≥ 1, and σ preserving the fusion rules. A counterexample to this form is a
  counterexample to the literal-set, the multiset and Ng's readings at once.
- **Gauge equivalence** is read as tensor (monoidal) equivalence, the weakest reading (for group algebras:
  Etingof–Gelaki, Lemma 3.1, isocategorical = gauge equivalent group algebras). No braided or symmetric
  structure is involved.
- **Not covered.** Invariants not mentioned in the source (generalized indicators ν_{n,k}, the S-matrix of the
  Drinfeld double, Kuperberg-type invariants of Chang–Ng–Wang) are not examined for the pair.

## Results in the paper
- **Lemma 2.1.** For C[G]: Λ = |G|^{-1} Σ g, ν_n(χ) = |G|^{-1} Σ_C |C| χ(C^[n]); exp C[G] = exp G (with the
  definition of the exponent of Etingof–Gelaki, MRL 1999, Definition 2.1; their Proposition 2.2(2)); fusion
  coefficients from the table and the class sizes.
- **Corollary 2.3** (Etingof–Gelaki, Definition 1.1 and Corollary 1.4): if |G1| is odd and Rep(G1), Rep(G2)
  are tensor equivalent, then G1 ≅ G2. The exact last sentence of Corollary 1.4 (arXiv v1, p. 3) is quoted;
  the journal version was not compared with arXiv v1; the zbMATH review of the journal version (Zbl 0988.20003)
  states the same structure result.
- **Lemmas 3.1, 3.2 and Theorem 1.3.** In a Brauer pair (definition with power maps for all n ∈ Z, as in the
  zbMATH review of Eick–Müller and in Lewis), n = 0 gives τ({1}) = {1}; group orders, class sizes, dimensions,
  exponents, all ν_n and all fusion coefficients agree; Corollary 2.3 excludes a tensor equivalence for odd
  order.
- **Propositions 4.1, 4.2 and Theorem 1.4.** P(φ) = V × Z, V = F_9^2 = F_3^4, Z = F_9, cocycle
  c_φ(v, w) = v1 w2 + Σ_k carry(v_k, w_k) φ(e_k); φ1(x1, x2) = x2, φ2(a + bi, c + di) = b + di. Proved in the
  text: group law; [g, h] = (0, v1 w2 − v2 w1); Z(P) = P' = 0 × Z; Camina property; 89 classes (9 central, 80
  of size 9); g^3 = (0, φ(πg)); exponent 9; 81 linear characters and 8 of degree 9 (9ω^{λ(z)} on the centre, 0
  elsewhere); explicit fusion rules; power maps; the bijection τ induced by the F_3-linear involution
  α(a, b, c, d) = (a, c, b, d) with φ2 ∘ α = φ1; indicator sequences 1 × (1,…,1), 80 × (0,0,1,0,0,1,0,0,1),
  8 × (0,…,0,9); literal set {0, 1, 9}; Ω = {g : g^3 = 1} of order 81, abelian in P1 and not in P2, so P1 ≇ P2.
- **Remark 4.3.** Lewis's Main Theorem (odd p, Camina class 2: Brauer pair iff equal |P:P'|, |P'|, |℧_1|) gives a
  second proof that P1, P2 form a Brauer pair; printed odd-order pairs (Dade, Eick–Müller) give Theorem 1.3
  independently of Section 4.
- **Remark 5.1.** C^Heis(3) and C[(Z/3)^3]: same indexed indicator data 1 × (1,1,1), 26 × (0,0,1), exponent 3,
  different Grothendieck rings; so the general Hopf case without Ng's condition has an elementary negative answer,
  and the source's remark on abelian group algebras needs both algebras to be abelian group algebras.
- **Remark 5.2.** With Nenciu's Brauer t-tuples (zbMATH review only): for every t ≥ 2, t group algebras of odd
  order with pairwise matching data and fusion rules, pairwise not tensor equivalent.
- **Remark 5.3.** (a) Meir–Szymik's Adams-operation criterion (zbMATH review) is satisfied by the pair.
  (b) Computation only: the Drinfeld doubles D(P1), D(P2) have equal multisets of (dim, twist, ν_1..ν_9) over
  their 7281 simple modules (14 types); the D(G) indicator formula is derived in the note and was not compared
  with printed formulas. (c) Stronger invariants not examined.
- **Remark 5.4.** Questions 6(c), 7(b) not treated; Question 7(a): conditional remark (C^{Q8}), not part of the
  result. H' is attributed, as in the source (p. 1125), to Lu's talk (joint work with Fan and Yu); the paper
  Fan–Lu–Yu 2016 presumably contains the definition and was not read.

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation. Theorem 1.4 is a finite statement; the programs confirm it.
- `check_note.py` (standard library, exact arithmetic in Z and Z[ω], about 10 s): the cocycle identity on all
  81^3 triples, identity, inverses, commutators, centre, classes, cubes, element orders, |℧_1| = 9, induced
  characters Ind_Z^P(λ) = 9χ_λ, exact orthonormality, fusion rules, τ and σ (table, class sizes, power maps
  n = 0..18), ν_1..ν_18 by summation over all elements and by the class formula, Ω; Remarks 3.3(c), 5.1, 5.4.
  Output: TOTAL failures: 0.
- `original/brauer_pair_729.py` (numpy, exact, about 25 s): in addition associativity on all 729^3 triples.
  Output ends with "ALL CHECKS PASSED".
- `original/double_indicators.py` (numpy, exact in Q(ζ_9), about 7 s): Remark 5.3(b).
- `original/q7_q8_check.py`: the finite part of Remark 5.4.
- `independent_run_2/` (third verification run, see below): `run2_presentation.py` (numpy, sympy; 133 checks) and
  `run2_misc.py` (standard library; 12 checks), 0 failures.
- All programs, including those of the verification runs, were re-run on 2026-10-20 from a copy of the folder;
  every recorded output was reproduced (`reproducibility/RERUN_LOG.txt`). A second complete re-run from a fresh
  extraction of the source archive (`independent_run_2/rerun_package_log.txt`) reproduced every output; the three
  "K of order 4" lines of `q7_q8_check.py` come in an order that depends on Python's hash seed
  (`PYTHONHASHSEED=10` reproduces the recorded file byte for byte).

## Independent verification runs
Three independent verification runs, all AI-assisted, each with its own programs (in
`reproducibility/verification_run_A/`, `reproducibility/verification_run_B/` and
`reproducibility/independent_run_2/`). The first two examined the first written version of the result; the third
examined the text of the note.
- **First run.** Read the source PDF (Questions 6–9 and Montgomery's abstract) and checked the statement and the
  reading of S, including the two supporting facts (C[Z/2], C[Z/3]; the abelian case under the indexed reading).
  Re-derived Lemmas 2.1, 3.1, 3.2 and all parts of Propositions 4.1 and 4.2 line by line, including the boundary
  cases v = 0, n = 0, negative n and 9 | n. Checked Etingof–Gelaki (arXiv math/0007196 v1) Definition 1.1,
  Lemma 3.1 (with its proof), Theorem 1.3, Corollary 1.4, Remark 1.5 and the mechanism of Lemma 3.6; Lewis's Main
  Theorem and MAGMA remark (arXiv 0809.4695); the zbMATH review of Eick–Müller. Its own program (exact arithmetic
  in Z[ζ_9]) verified the group axioms on all 729^3 triples, the class structure, the eight characters of degree 9
  as induced characters from the abelian normal subgroup π^{-1}(F_9 × 0) of order 81, orthonormality, τ and σ
  (table, class sizes, power maps n = 0..18, indicators n = 1..18) and all fusion coefficients; it re-ran the
  original programs with identical outputs. Verdict: confirmed, no mathematical gap. Required corrections (all
  editorial): the proof that exp C[G] = exp G; the exact wording of Etingof–Gelaki Corollary 1.4 and the
  statement that the journal version was not compared; the two-line justifications in the Question 7(a) remark;
  Kashina–Montgomery–Ng and Keilberg's papers in the list of sources not read; the labelling of the D(G) formula as
  derived and not compared with print; τ described through α.
- **Second run.** Built the groups with its own code and computed the character tables from scratch by Burnside's
  algorithm (rounded into Z[ω], then checked exactly); all 89^3 class structure constants; power maps n = 0..18;
  σ found by matching rows; fusion rules; indicators; Ω (commuting pairs 6561 and 2673). Five negative controls
  (broken cocycle; D8 against Q8; φ of rank one; wrong class bijection; perturbed table) are detected. It
  re-derived the D(G) formula, validated it against a brute-force Hopf power on explicitly built induced modules
  for S3, Q8, D8 (n = 1..4, deviation < 1e-14) and recomputed the doubles (floating point, rounded). It re-ran the
  original programs (byte-identical outputs), searched the literature (arXiv, zbMATH, Crossref, web), checked the
  bibliography with Crossref and zbMATH, and found the elementary example of Remark 5.1 and the related remark of
  Mignard–Schauenburg. Verdict: confirmed. Required corrections: Remark 5.1 with the statement that the new
  content is the group-algebra case; the journal reference of Mignard–Schauenburg arXiv 1708.02796 (Lett. Math.
  Phys. 111 (2021), paper no. 60, verified with Crossref).
- **Further additions when the note was written.** Nenciu's Brauer t-tuples (Remark 5.2), the sentence on stronger
  invariants (Remark 5.3(c)), the exact program `check_note.py`, and the re-run of the whole package.
- **Third run (verification of the written note).** Read the note line by line and the sources again: the report
  (pp. 1125–1127, 1131–1134), Etingof–Gelaki 2001 (Definition 1.1, Theorems 1.2, 1.3, Corollary 1.4, Remark 1.5,
  Lemmas 3.1, 3.6), Etingof–Gelaki 1999 (Definition 2.1, Proposition 2.2, Section 4), Lewis, the zbMATH reviews
  used, the abstract of Ng–Schauenburg 2008, and Mignard–Schauenburg arXiv 1708.06538 in full. Re-derived all
  lemmas, propositions and remarks (including the constants 6561 and 2673 for the commuting pairs in Ω, the
  centralizers π^{-1}(F_9 v) of order 81, and 7281 simple modules of D(P) with Σ dim^2 = 729^2). Its own programs
  (`independent_run_2/`): the groups defined by the presentation read off from Proposition 4.1(b),(c) and
  enumerated by Todd–Coxeter coset enumeration (order 729 as output), with a bijective homomorphism onto P(φ)
  checked on all pairs; the characters of degree 9 as traces of explicit monomial representations induced from
  linear characters of the abelian subgroup π^{-1}(F_9·(1,1)), checked to be representations on all pairs; exact
  checks in Z[ζ_9] of orthonormality, power maps n = 0..18, ν_1..ν_18, all 89^3 fusion coefficients, τ, σ and Ω,
  with two negative controls; a second program for Remarks 3.3(c), 5.1 and 5.4 and for the rank-8 statement. It
  re-ran all programs from an extracted copy of the source archive (all outputs reproduced, see above), checked
  all 18 DOIs with Crossref and searched again (arXiv API, zbMATH Open, OpenAlex, Semantic Scholar, one web search).
  Verdict: confirmed; no mathematical error; not found in print. Required corrections (all made): the attribution
  of H' in Remark 5.4 (to Lu's talk, as in the source, instead of to the unread paper of Fan–Lu–Yu); in Remark 5.1,
  that Mignard–Schauenburg contains no pair with trivial cocycle and that the five categories Vec_G of rank 8 have
  pairwise different indicator data (with multiplicity), with credit to their Corollary 3.3 for the formula
  ν_n(S_g) = [g^n = 1]; the exact description of Etingof–Gelaki Theorem 1.2; the twist statement for general
  finite-dimensional Hopf algebras marked as standard and unused; the precise form of the D8/Q8 sentence in
  Remark 3.3(c); the search record; the hash-seed remark for `q7_q8_check.py`.
- **Lead wording fix of 2026-10-20:** credit sentence aligned with Remark 5.1 (abstract, Sections 1.3 and 6).

## Relation to the literature, novelty and scope
- **Credit.** The question is Montgomery's (2006; Oberwolfach Report 20/2014, Question 9); the fusion-rule variant
  is R. Ng's. Higher indicators: Kashina–Sommerhäuser–Zhu (Mem. AMS 181 (2006), no. 855); gauge invariance:
  Mason–Ng (Adv. Math. 190 (2005)) and Ng–Schauenburg (Trans. AMS 360 (2008); abstract read). Exponent of a Hopf
  algebra: Etingof–Gelaki (MRL 6 (1999)). Non-equivalence: Etingof–Gelaki (IMRN 2001). Brauer pairs: Dade
  (J. Algebra 1 (1964)), Eick–Müller (J. Algebra 304 (2006)), Lewis (Arch. Math. 92 (2009)), Nenciu (J. Algebra
  Appl. 7 (2008); J. Algebra 322 (2009)).
- **Closest printed statements.** Mignard–Schauenburg, arXiv 1708.06538, Remark 4.5 (indicators do not classify
  pointed fusion categories; rank 8; 47 categories, 34 sets of indicators; for non-trivial ω these are
  representation categories of quasi-Hopf, not Hopf, algebras). The paper, read in full, contains no pair of
  non-equivalent pointed categories with trivial cocycle and equal indicators (its named coincidences are between
  two cocycles on one group), and in rank 8 the five categories Vec_G with trivial cocycle have pairwise different
  indicator data counted with multiplicity; the Heis(3) pair is not in it. Its Corollary 3.3 contains, for ω = 1,
  the formula ν_n(S_g) = [g^n = 1] used in Remark 5.1.
  Mignard–Schauenburg, Lett. Math. Phys. 111 (2021): modular data do not determine modular categories (braided
  analogue). Meir–Szymik, Indiana Univ. Math. J. 70 (2021): monoidal equivalences preserve Adams operations Ψ^p for
  odd p (consistent with the pair).
- **What was read.** The source PDF (pp. 1125–1127 and 1131–1134); Etingof–Gelaki 2001 (arXiv v1) and the zbMATH
  review of the journal version; Etingof–Gelaki 1999 (arXiv v2: Definition 2.1, Proposition 2.2, Section 4); Lewis (arXiv v1); Mignard–Schauenburg
  1708.06538 (in full) and 1708.02796; abstracts of Ng–Schauenburg 2008, Kashina–Montgomery–Ng 2012,
  Chang–Ng–Wang 2025; abstracts and text searches of Keilberg 1708.06583, Iovanov–Mason–Montgomery 2014 and
  Schauenburg 1604.02378. Only through zbMATH reviews: Dade, Eick–Müller, Nenciu (both), Meir–Szymik. Not read:
  Kashina–Sommerhäuser–Zhu, Mason–Ng (cited as in the source), Keilberg 2012 and 2014, Fan–Lu–Yu (so the
  definition of H' in Question 7 is unknown). A statement on the four gauge classes of 8-dimensional quasi-Hopf
  algebras was checked in the abstract of Ng–Schauenburg 2008 and is not used in the note.
- **Searches (20 October 2026).** arXiv interface (several queries; some refused with HTTP 429 and replaced by
  single abstract pages), zbMATH Open (isocategorical groups, Brauer pairs, indicators as gauge or complete
  invariants, the proposer's later work, the authors named above), Crossref (bibliography; all 18 DOIs checked
  again by the third run), nine web searches, and the manuscripts of a public release of AI-generated
  mathematical results (github.com/openai/math, October 2026). The third run added 13 arXiv API queries (FS
  indicators with gauge invariants, "distinguish", "complete invariant", pointed fusion categories, Camina groups,
  Brauer pairs, isocategorical groups, and others) and 5 zbMATH queries. OpenAlex returned the records of the
  report and of Etingof–Gelaki 2001 but refused the lists of citing works (HTTP 429, also after a wait and a
  retry). Semantic Scholar listed the five works citing the report (2015–2020, all on Nichols algebras and
  Gelfand–Kirillov dimension) and 53 works citing Etingof–Gelaki 2001 (titles screened, six abstracts read); none
  concerns Question 9. Works citing Montgomery's 2006 formulation could not be traced (the source gives no
  reference for it). No printed answer to Question 9 was found in these sources; this is not a proof of
  novelty.
- **Scope for the corpus.** Question 9: answered (negatively) for group algebras, in every reading, also with Ng's
  condition; the general Hopf case without Ng's condition by an elementary example. Record OWR-12978-004 as a
  whole: partial, since Questions 6(c) and 7 are not settled.

## 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/