REVIEW 1 major objections 6 minor 8 references
Arbitrarily large $\mathcal{O}$-Morita Frobenius numbers
T0 review · 1 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read For every prime $l$ and every $n$, some finite group block has $O$-Morita Frobenius number $n$.
desk verdict Livesey constructs blocks with O-Morita Frobenius number exactly n for any n, a genuinely new result that looks correct; the main soft spot is an unproved generalization of an Eaton–Livesey lemma that a referee should ask the author to spell out. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is a family of blocks $B_\varphi = O G_{l'} e_\varphi$, where $G_{l'}$ is a semidirect product of a normal defect group $D = D_{t_1}\times D_{t_2}$ (a product of two homocyclic $l$-groups, each a direct product of cyclic groups of the same $l$-power order) by an $l'$-group $E_{l'}$ acting faithfully on $D$, and $e_\varphi$ is the block idempotent attached to a character $\varphi$ of a central $l'$-subgroup $Z_{l'}$. The key identity is the classification of all Morita equivalences among these blocks: $B_\varphi$ is Morita equivalent to $B_\theta$ exactly when $\varphi=\theta$, or when $t_1=t_2$ and $\varphi=\theta^{-1}$. That identity reduces the Frobenius number to the multiplicative order of $\varphi$ under the $l$-power map. The proof is carried by two tools: a proposition that passes Morita equivalences down to quotients by normal $l$-subgroups and lifts trivial-source bimodules back up, and a structure theorem for trivial-source Morita bimodules as direct summands induced from graph subgroups $\Delta_\gamma=\{(d,\gamma(d))\}$.
What would settle it
Take $l=2$ and $n=2$; choose a prime $p\equiv 1\pmod 3$ (for example $p=7$), pick $t_1\neq t_2$, and let $\vartheta$ be a character of $Z_{l'}$ of order $3$. The theorem predicts $mf_O(B_\vartheta)=2$; computing this number by an independent method and obtaining any value other than $2$ would refute the construction.
Extended reading notes
Core claim
The central result is Theorem 3.6: for every prime $l$ and every $n\in\mathbb{N}$, there exists an $O$-block $b$ with $mf_O(b)=n$. The blocks are built as $B_\varphi = O G_{l'} e_\varphi$, where $G_{l'}$ is a semidirect product of a normal defect group $D=D_{t_1}\times D_{t_2}$ (a product of two homocyclic $l$-groups) by an $l'$-group $E_{l'}$ acting faithfully on $D$, and $e_\varphi$ is the block idempotent attached to a character $\varphi$ of a central $l'$-subgroup $Z_{l'}$. The proof classifies all Morita equivalences among these blocks (Proposition 3.5): $B_\varphi$ is Morita equivalent to $B_\theta$ precisely when $\varphi=\theta$, or when $t_1=t_2$ and $\varphi=\theta^{-1}$. Since the $l$-power Frobenius twist sends $B_\varphi$ to $B_{\varphi^{l^m}}$, the Morita Frobenius number of $B_\varphi$ is the smallest $m$ with $\varphi^{l^m}=\varphi$. Choosing $\varphi$ of order $l^n-1$ makes that smallest $m$ equal to $n$, and a standard theorem on primes in arithmetic progressions supplies a prime $p$ with $p-1$ divisible by $l^n-1$ to realize the construction.
Load-bearing premise
The proof assumes that a cited proposition about Morita equivalences, quotients by normal $l$-subgroups, and trivial-source bimodules, proved in the literature for one group, remains true without change when the two sides are different groups and different blocks; the author states the proof is identical but does not write it out.
Editorial extensions
If this is right
- For any fixed prime $l$, the $O$-Morita Frobenius numbers of blocks are unbounded as the defect group varies; there is no universal constant that bounds them.
- The blocks realizing $n$ have defect groups of order $l^{t_1+t_2}$ with $t_1\neq t_2$, so the defect groups grow with $n$; the theorem therefore does not contradict the conjecture that the numbers are bounded for blocks with a fixed defect group.
- Because no example is known of two blocks that are Morita equivalent over $k$ but not over $O$, the paper observes that the same construction strongly suggests the Morita Frobenius numbers over $k$ are also unbounded, which would settle the open question motivating the paper.
- For the block $B_\vartheta$ attached to a character $\vartheta$ of order $l^n-1$, the $m$-th Frobenius twist is Morita equivalent to the original block precisely when $n$ divides $m$; in particular the block returns to itself only after $n$ twists.
Reading between the lines
- Because the classification of equivalences is obtained through trivial-source bimodules, the same argument likely gives $mf_k(B_\vartheta)=mf_O(B_\vartheta)$ for these blocks; if so, the constructed examples would already show unbounded Morita Frobenius numbers over $k$, without needing a new example that distinguishes $k$-equivalence from $O$-equivalence.
- The only step the author asserts without proof is the extension of a cited proposition to two distinct groups; a direct verification of Proposition 2.2 in that setting would make the proof entirely self-contained.
- The template of a faithful $l'$-action on a product of two homocyclic $l$-groups could be varied to control further invariants of the block, such as its inertial quotient, while keeping the Frobenius number arbitrarily large.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the O-Morita Frobenius number mf_O(b), the least n for which a block b of a finite group algebra over a complete discrete valuation ring O of residue characteristic l is Morita equivalent to its l^n-th Frobenius twist. The main theorem (Theorem 3.6) constructs, for every prime l and every natural number n, a finite group G_l' and a block B_ϑ of OG_l' with mf_O(B_ϑ)=n. The group is a semidirect product (D_{t1}×D_{t2})⋊E_l', where D_t is the kernel of the product map in a product of p copies of C_{l^t}; the block is selected by a character ϑ of a central l'-subgroup Z_l'. The proof pinpoints the Morita equivalence classes of the blocks B_ϕ (Proposition 3.5) by using a character-bijection argument to reduce to the quotient by the normal l-subgroup D, then a trivial-source/Puig induction classification, and finally a central-character argument involving commutators of lifts. Theorem 3.6 then follows by taking ϑ of order l^n−1, so that the smallest twist returning to ϑ is the order of l modulo l^n−1.
Significance. The theorem, if correct, establishes that O-Morita Frobenius numbers are unbounded, and the authors correctly note that this is strong evidence for the unboundedness of ordinary Morita Frobenius numbers, a question raised by Benson and Kessar. The construction is explicit and the heart of the argument — the classification of Morita equivalences among the B_ϕ — is checkable and elegant, reducing a block-theoretic statement to the multiplicative order of l modulo l^n−1. I was able to verify Lemmas 3.1–3.4 and the numerical part of Theorem 3.6 without finding errors. The main weakness is not the strategy but a missing proof of a quoted generalization that the argument uses in an essential way.
major comments (1)
- [§2, Proposition 2.2] Proposition 2.2 is stated for two possibly distinct groups H1,H2, normal l-subgroups Q1,Q2 and distinct blocks b1,b2, but the proof is only justified by the sentence 'This is proved in [4, Propositions 4.3,4.4], with the added assumption that H1=H2, Q1=Q2 and b1=b2. However, the proof in this more general setting is identical.' This is a load-bearing point: Proposition 3.5 applies the statement to H1=H2=G_l', Q1=Q2=D, with b1=B_ϕ and b2=B_ϑ distinct, and part (2) is what allows the conclusion that the Morita-equivalence bimodule M has trivial source, which in turn is needed for the Puig inductive-structure step [8, 7.6]. Since the fixed-point functor for a normal l-subgroup is not exact, the passage from trivial source of D M = M D to trivial source of M is not formal, and the claimed 'identical' proof is not supplied. Please give a complete proof of Proposition 2.2 in the stated generality, or an explicit statement in [4] that covers it.
minor comments (6)
- [§2, Proposition 2.2] The notation Q1M and MQ2 is used without definition. Please define Q1M = {m ∈ M | q m = m for all q ∈ Q1} and MQ2 = {m ∈ M | m q = m for all q ∈ Q2}, and make explicit that 'Q1M = MQ2' denotes the condition that these submodules coincide.
- [§3, proof of Proposition 3.5] In the displayed formula involving eϕ(OΔγ↑(D×Zl′)×(D×Zl′)↑Gl′×Gl′)eϕ, the idempotents should be e_ϕ on the left and e_ϑ on the right, since M is a B_ϕ-B_ϑ-bimodule. The subsequent notation γ(OD) ⊗O ϕ Oϑ confirms the intended form, but the displayed formula as written is inconsistent.
- [§3, Theorem 3.6] The statement 'Let p be a prime different from l such that p ≡ 1 mod (l^n − 1), the existence of which is guaranteed by the Dirichlet prime number theorem' needs one extra sentence for the case l=2, n=1, where the congruence is vacuous; one should additionally choose p (for instance p=7) with p−1 not a power of 2.
- [§3, definition of a] The phrase 'a := v_l(p−1), the largest power of l dividing p−1' should say 'a := v_l(p−1), the exponent of l in p−1'; the subsequent use of l^a shows that the exponent is intended.
- [§3, proof of Proposition 3.5, Eq. (4)] The equality rk_O(B_ϕ) = [G_l' : Z_l'] is asserted without proof. It follows because e_ϕ is a primitive central idempotent of Z_l' and KG_l'e_ϕ has dimension |G_l'|/|Z_l'|, but this deserves a sentence for the reader.
- [§3, proof of Proposition 3.5] The normalization 'By Lemmas 3.2 and 3.3, we may assume that γ = Id_D' is terse. A short explanation of how the automorphisms from Lemma 3.3 transform an arbitrary Morita-equivalence bimodule, and how the t1=t2 case replaces ϕ by ϕ^{-1}, would make the argument easier to verify.
Circularity Check
No circularity: the main theorem is an explicit construction whose final step is a direct character-order computation; the only self-citations are non-circular technical lemmas.
full rationale
The derivation is self-contained in the relevant sense. The paper defines mf_O(b) in Definition 1.1 and then constructs explicit blocks B_theta whose O-Morita Frobenius number is computed by the order of theta via Proposition 3.5 and Theorem 3.6. The key classification Proposition 3.5 is not an input to the construction; it is proved using Lemmas 3.1-3.4 and an external result from Eaton-Livesey [4, Propositions 4.3,4.4], quoted as Proposition 2.2. That cited result has stated assumptions (normal l-subgroups and Morita equivalent blocks) that do not include the target conclusion, namely the existence of blocks with mf_O(b)=n, and it is parameter-free rather than fitted to the present examples. The only possible concern is that Proposition 2.2 extends [4] from one group and one block to two possibly distinct groups and blocks with the sentence 'the proof in this more general setting is identical'; this is a mathematical gap or correctness risk, not a circularity, because no equation or fitted parameter in the present paper is being reused as a prediction. The final step of Theorem 3.6 merely compares theta^{l^m} with theta for a character of order l^n-1; this is a direct order computation, not a restatement of the definition or of a fitted quantity. Self-citations to [3] and [4] are background or technical tools and are not used to assert the main theorem by authority. Therefore no circular step is exhibited, and the score is 0.
Assumptions & free parameters
assumptions (4)
- standard math Dirichlet's theorem on primes in arithmetic progressions
- domain assumption Puig's classification of trivial-source Morita equivalences ([8, 7.6])
- domain assumption Eaton-Livesey [4, Propositions 4.3,4.4] for Morita equivalences and normal l-subgroups
- standard math Schur-Zassenhaus theorem
Cite this review
Pith. "Pith review of Arbitrarily large $\mathcal{O}$-Morita Frobenius numbers." pith.science (2026). https://pith.science/paper/MXI5NWC7
@misc{pith2026190806680,
author = {Pith},
title = {Pith review of: Arbitrarily large $\mathcalO$-Morita Frobenius numbers},
year = {2026},
howpublished = {\url{https://pith.science/paper/MXI5NWC7}},
note = {Machine review of arXiv:1908.06680}
}
abstract
We construct blocks of finite groups with arbitrarily large $\mathcal{O}$-Morita Frobenius numbers. There are no known examples of two blocks defined over $\mathcal{O}$ that are not Morita equivalent but the corresponding blocks defined over $k$ are. Therefore, the above strongly suggests that Morita Frobenius numbers are also unbounded, which would answer a question of Benson and Kessar.
Reference graph
Works this paper leans on
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work page 2020
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D. Benson and R. Kessar, Blocks inequivalent to their Frobenius twists , J. Algebra, 315(2) (2007), 588–599. 9
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C. W. Eaton, F. Eisele and M. Livesey, Donovans conjecture, blocks with abelian defect groups and discrete valuation rings , Math. Z. 295 (2020), 249–264
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N. Farrell and R. Kessar, Rationality of blocks of quasi-simple finite groups , Represent. Theory 23 (2019), 325–349
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work page 2005
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[8]
L. Puig, On the local structure of Morita and Rickard equivalences be tween Brauer blocks, Progress in Math. 178, Birkh¨ auser Verlag, Basel (1999). 10
work page 1999
Reviewed August 14, 2026 · model on record in the stance chip above.
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