{"id":"5d26cad0-c297-47b6-873e-3699298770ec","arxiv_id":"1908.06680","paper_version":5,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For every prime l and every positive integer n, there exists a finite group block whose O-Morita Frobenius number is exactly n.","lead":"This paper constructs finite group blocks whose O-Morita Frobenius number, a measure of how many Frobenius twists are needed to return to the original block, can be any chosen integer. The construction suggests that the related Morita Frobenius numbers over the residue field are also unbounded, a question raised by Benson and Kessar.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2.2 is quoted for distinct blocks/groups but only proved for a single block; Prop 3.5's trivial-source conclusion and hence Theorem 3.6 depend on this unproved generalization.","rationale":"The stress-test pass confirms the reader's weakest assumption. The main theorem is a corollary of Prop 3.5, and the only unproved input in the chain is Prop 2.2 in the generality needed. The author explicitly notes the generalization and says the proof is identical, but does not provide it; because the step is used to conclude that an arbitrary Morita-equivalence bimodule has trivial source, it is load-bearing rather than a cosmetic omission. A formal verification could remove the concern, but none is present. The construction itself is explicit, the rank arguments are checkable, and no circularity or obvious internal inconsistency appears. However, the correctness risk is higher than 'low' until Prop 2.2 is either proved in the stated generality or replaced by a reference covering H1=H2, b1 not equal to b2. Thus the conditional verdict is appropriate; no change from the reader's verdict.","tokens_in":7885,"tokens_out":19612,"duration_ms":226045,"concrete_test":"Obtain [4, Propositions 4.3 and 4.4] and re-derive them with H1, H2, Q1, Q2, b1, b2 possibly distinct. If any displayed equality or module identification in the proof requires b1=b2 (for example, identifying the dominated quotient blocks on the two sides, or using a self-equivalence of a single block algebra), then Prop 2.2 is not established in the needed generality. A secondary check in the exact setting of Prop 3.5: verify that if D M = M D has trivial source, then M has trivial source without assuming phi=theta; if the only available proof identifies B_phi^D with B_theta^D, the lifting step is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main theorem is derived from Prop 3.5, whose converse requires proving that any Morita-equivalence bimodule M between B_phi and B_theta has trivial source. This is obtained by applying Prop 2.2(2) to H1=H2=G_l', Q1=Q2=D, with distinct blocks b1=B_phi and b2=B_theta. But Prop 2.2 is stated for two possibly distinct groups and blocks, while the quoted result [4, Propositions 4.3,4.4] is proved only for H1=H2, Q1=Q2, b1=b2. The text says 'the proof in this more general setting is identical' without giving it. This is not a cosmetic gap: the fixed-point submodule D M = M D being a trivial-source Morita equivalence between the dominated blocks only yields trivial source of M if a lifting statement for splendid equivalences through a normal l-subgroup is valid for two distinct blocks. Since D is an l-group and O is l-modular, the fixed-point functor is not an exact retract, so the assertion is nontrivial. The subsequent use of [8, 7.6] to write M as a direct summand of an induced trivial-source module depends entirely on this conclusion; without it, the stabilizer and rank argument in Prop 3.5 has no basis, and the proof of mf_O(B_theta)=n collapses.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":8123,"tokens_out":28276,"duration_ms":286324,"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":[{"comment":"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.","section":"§2, Proposition 2.2"}],"minor_comments":[{"comment":"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.","section":"§2, Proposition 2.2"},{"comment":"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.","section":"§3, proof of Proposition 3.5"},{"comment":"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.","section":"§3, Theorem 3.6"},{"comment":"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.","section":"§3, definition of a"},{"comment":"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.","section":"§3, proof of Proposition 3.5, Eq. (4)"},{"comment":"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.","section":"§3, proof of Proposition 3.5"}],"recommendation":"major_revision","confidential_remarks":"The only substantive obstacle is the unproved generalization in Proposition 2.2. If the authors can provide a complete proof of that proposition in the generality stated, or point to a reference that literally covers the distinct-block case, I would be willing to accept the paper. The reliance on the author's own previous paper is not itself a problem, but this particular step is load-bearing and should be self-contained."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Mike,\n\nThe headline is real: for every prime l and every n, the paper constructs an O-block whose O-Morita Frobenius number is exactly n. That is the first unboundedness result for the O-version, and the previous literature only had examples with mfO equal to 1, 2, or 4. The construction is explicit and the proof is mostly concrete: a carefully chosen group E, a normal defect group D, blocks B_theta with cyclic l'-inertial quotient, and then a reduction of mfO(B_theta) to the multiplicative order of l modulo l^n-1. That reduction is clean and is the heart of the paper.\n\nWhat is good: the paper is serious, checkable work. The character-theoretic Lemma 3.4 and the stabilizer/rank argument in Proposition 3.5 are clearly written. The use of Dirichlet to pick the prime p is fine. The paper does not overclaim: the abstract correctly says the unboundedness of the k-version is only suggested, not proved.\n\nThe soft spot is exactly what the stress-test flags. Proposition 2.2 quotes Eaton–Livesey [4, Propositions 4.3,4.4] for the case H1=H2, Q1=Q2, b1=b2, then says the proof in the more general setting of two distinct groups and blocks is identical. That is not obviously harmless. The fixed-point submodule D M = M D being a trivial-source Morita equivalence between dominated blocks is not by itself enough to conclude M has trivial source unless you know the right lifting statement through the normal l-subgroup D. The rest of Proposition 3.5, including the use of Puig [8, 7.6], depends on that. My own reading is that the generalization is probably true and the proof probably does go through verbatim, but the author should be asked to write it out. It is a moderate gap, not a fatal one, and the central argument has no other visible weaknesses.\n\nThere is also a small presentational issue: the paper says \"strongly suggests\" the k-version is unbounded, which is fair but could be misread as more than a suggestion.\n\nWho is this for? People working on Donovan's conjecture, Morita Frobenius numbers, and block theory with normal defect groups. It deserves a serious referee. The unproved generalization in Proposition 2.2 should be fixed before publication, but the main theorem is novel and likely correct.\n\nRecommendation: send it to peer review; request a proof of Proposition 2.2 in the stated generality.\n\n— [Your name]","headline":"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.","tokens_in":8681,"tokens_out":1765,"would_cite":true,"duration_ms":21290,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20C20"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every prime $l$ and every $n$, some finite group block has $O$-Morita Frobenius number $n$.","keywords":["Morita Frobenius number","O-Morita Frobenius number","Frobenius twist","block of a finite group","Morita equivalence","defect group","trivial-source module","modular representation theory"],"falsifier":"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.","tokens_in":7621,"feed_emoji":"🔢","tokens_out":14432,"duration_ms":132907,"temperature":0.7,"pith_summary":"The paper proves that $O$-Morita Frobenius numbers of blocks of finite groups are unbounded: for every prime $l$ and every positive integer $n$, it constructs a block $b$ defined over the coefficient ring $O$ whose $O$-Morita Frobenius number $mf_O(b)$ is exactly $n$. This number is the smallest power $l^m$ of the Frobenius twist after which $b$ and its twisted block $b^{(l^m)}$ become Morita equivalent as algebras over $O$; a block with value $1$ is already equivalent to its twist. Earlier constructions produced only small values, and it was open whether these numbers could be arbitrarily large. Since no example is known where two blocks are Morita equivalent over the residue field $k$ but not over $O$, the paper argues the same construction strongly suggests the ordinary Morita Frobenius numbers over $k$ are also unbounded, which would settle an open question.","feed_headline":"For each n, some group block returns to itself after n twists","feed_subtitle":"These numbers measure when a block returns to itself under Frobenius twists; new examples show they are unbounded.","key_machinery":"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 $\\Delta_\\gamma=\\{(d,\\gamma(d))\\}$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the technical propositions on Morita equivalences, quotients by normal $l$-subgroups, and trivial-source bimodules that Proposition 2.2 extends to two distinct groups.","marker":"[4]"},{"why":"Gives the structure theorem for trivial-source Morita bimodules (cited as [8, 7.6]) used to reduce equivalence to automorphisms of the defect group.","marker":"[8]"},{"why":"Defines the Morita Frobenius number and connects its boundedness to a finiteness conjecture for blocks with a fixed defect group, setting up the invariant whose unboundedness is proved.","marker":"[7]"},{"why":"Provides the first examples of nontrivial Morita Frobenius numbers and the questions about unboundedness that the paper addresses.","marker":"[1]"}],"fun_headline_variants":["Arbitrarily large Morita Frobenius numbers exist","For every n, a block that needs n Frobenius twists","Morita Frobenius: no bound on twist-return","Blocks with prescribed twist counts: unbounded","Each n: a block returning after n Frobenius twists"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Arbitrarily large Morita Frobenius numbers exist","For every n, a block that needs n Frobenius twists","Morita Frobenius: no bound on twist-return","Blocks with prescribed twist counts: unbounded","Each n: a block returning after n Frobenius twists"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000282,"raw_usage":{"total_tokens":1661,"prompt_tokens":932,"completion_tokens":729,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":548,"completion_tokens_details":{"reasoning_tokens":646}},"tokens_in":548,"tokens_out":729,"duration_ms":6863,"temperature":1.0,"reasoning_tokens":646,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:38:30.841939+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the technical propositions on Morita equivalences, quotients by normal $l$-subgroups, and trivial-source bimodules that Proposition 2.2 extends to two distinct groups."},{"cited_title":"Puig, On the local structure of Morita and Rickard equivalences be tween Brauer blocks, Progress in Math","cited_arxiv_id":null,"evidence_quote":"Gives the structure theorem for trivial-source Morita bimodules (cited as [8, 7.6]) used to reduce equivalence to automorphisms of the defect group."},{"cited_title":"Kessar, A remark on Donovan’s conjecture , Arch","cited_arxiv_id":null,"evidence_quote":"Defines the Morita Frobenius number and connects its boundedness to a finiteness conjecture for blocks with a fixed defect group, setting up the invariant whose unboundedness is proved."},{"cited_title":"Benson and R","cited_arxiv_id":null,"evidence_quote":"Provides the first examples of nontrivial Morita Frobenius numbers and the questions about unboundedness that the paper addresses."}],"review_version":1}