{"id":"05908efb-ec03-45d3-86d6-d3858236ccc9","arxiv_id":"1908.05536","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For finite groups with semidihedral Sylow 2-subgroups, the paper gives sufficient conditions under which Scott modules are Brauer indecomposable, generalizing known cases.","lead":"This paper proves technical conditions under which the Scott module, a special representation of a finite group, stays indecomposable after the Brauer construction, a localization operation. The new case covered is when the Sylow 2-subgroup is semidihedral, a step toward equivalences between block algebras and a conjecture of Broué.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the quoted semidihedral centralizer facts are standard and directly verifiable, and the internal case analysis is coherent.","rationale":"The reader flagged the reliance on external structural facts from [16] as the weakest assumption. That is exactly where I focused, but the facts are easy to verify from the semidihedral presentation and they hold for all n >= 4, so they do not constitute a load-bearing concern. I also reviewed the places where the argument could silently fail: the reduction to cyclic/dihedral/quaternion proper subgroups is valid; the Aut(Q) dichotomy is valid; the construction of H_Q in Corollary 2.4 correctly lifts the S3 subgroup with N_P(Q) a Sylow 2-subgroup and index a power of 2; and the Q = C2 direct proof in Theorem 1.2 reduces indecomposability of M(Delta Q) to indecomposability of M(Delta S) for a C2 x C2 or larger subgroup, already established. No circularity appears: Theorem 1.1 uses the 2-nilpotency hypothesis exactly as stated, and Theorem 1.2 proves those hypotheses from Sylow structure. The remaining external input, the Brauer centralizer theorem, is standard and precisely cited. Thus the ACCEPT verdict with moderate confidence is appropriate.","tokens_in":9096,"tokens_out":52368,"duration_ms":470255,"concrete_test":"Independently re-derive [16, Lemma (viii)/(ix)] for general SD_{2^n} from the presentation, checking C_P(Q) = Z(Q) for Q isomorphic to C2 x C2 or Q8, C_P(Q) = Q for Q = <xy>, and |N_P(Q)/Q C_P(Q)| = 2; if any of these equalities fails for some n >= 4, the exceptional subgroup case in Theorem 1.1 would collapse.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I could not locate a load-bearing internal gap. The most citation-sensitive step is the use of [16, Lemma (viii)/(ix)] in Corollary 2.4 and in the small-subgroup cases. These facts are not re-proved, but they are elementary consequences of the presentation SD_{2^n} = <x, y | x^{2^{n-1}} = y^2 = 1, y^{-1}xy = x^{2^{n-2}-1}>. Direct computation gives C_P(<z,y>) = <z,y>, C_P(<x^{2^{n-3}}, xy>) = <z>, and C_P(<xy>) = <xy>, and N_P(Q)/Q C_P(Q) is C2 for Q isomorphic to C2 x C2 or Q8, for every n >= 4. The other external input, the Brauer centralizer theorem cited at line -3 of [2, p.246] and used in Lemma 3.1 for |Q| >= 8, is a standard theorem for groups with semidihedral Sylow 2-subgroups; Lemma 3.1's subgroup argument correctly reduces to producing a noncentral x^i in Q. The remaining deductions in Theorems 1.1 and 1.2 are internally consistent: proper subgroups of a semidihedral group are cyclic, dihedral, or generalized quaternion; Aut(Q) is a 2-group except for C2 x C2 and Q8; Corollary 2.4 covers the exceptional S3 quotient; and the direct argument for Q = C2 in Theorem 1.2 reduces indecomposability to already-proved larger subgroups. I therefore found no concrete failure mode.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves two Brauer indecomposability theorems for Scott modules with semidihedral vertex. Theorem 1.1 gives a sufficient condition in terms of the 2-nilpotence of the centralizers of all fully normalized non-trivial subgroups of a semidihedral 2-subgroup P of G, and Theorem 1.2 extends this to a diagonal Scott module Sc(G x G', Delta P) when G and G' have a common semidihedral Sylow 2-subgroup P with equal fusion systems. The proofs use the Ishioka--Kunugi reduction of Brauer indecomposability to local Scott modules, followed by a detailed case analysis of all isomorphism types of subgroups of a semidihedral 2-group.","tokens_in":9372,"tokens_out":15567,"duration_ms":142198,"significance":"If the results hold, they add the semidihedral case to the known families of Brauer indecomposable Scott modules (abelian and dihedral vertices), which is a relevant step for applications of the gluing method to splendid stable equivalences of Morita type. The paper is carefully structured: the main theorems are proved in full detail, the case analysis covers all fully normalized subgroups, and the arguments are traceable to published results. The centralizer facts quoted from [16] are load-bearing but are standard and are elementary consequences of the presentation of SD_{2^n}; they appear correct. I found no internal contradictions and no circular reasoning.","major_comments":[],"minor_comments":[{"comment":"The definition 'yx := yxy −1' is ambiguous; it should presumably read '{}^y x := yxy^{-1}' or use a clearer notation. Please clarify and ensure the typesetting distinguishes the two conjugation conventions.","section":"Notation 1.3"},{"comment":"The reference 'Theorem 4.8.6 (ii)]' contains a misplaced bracket, and the phrase 'by 1.4 of [4]' is not fully standard. Please correct the citation formatting.","section":"Proof of Theorem 1.2, Case 3"},{"comment":"The line '(K ⋊ NP (Q)) ∩ (Q CG(Q)) = (K ⋊ Q CP (Q)) = 1' is terse; adding overline notation for images in the quotient by L would greatly improve readability and make the argument easier to follow.","section":"Corollary 2.4 proof"},{"comment":"The proof relies on several centralizer facts from [16] (for example C_P(Q)=Z(Q) for Q ≅ C2×C2 or Q8, and C_P(<xy>)=<xy>). These facts are standard and correct, but since they are load-bearing a one-line verification or a remark in the text would make the paper more self-contained.","section":"Throughout"}],"recommendation":"accept","confidential_remarks":"I recommend acceptance. The central claims are sound, the case analysis is complete, and the cited structural facts from [16] are correct. The manuscript fits the journal's scope and I have no concerns about novelty or presentation beyond the minor points listed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this paper closes the semidihedral gap in the Brauer indecomposability program for Scott modules; previous work handled abelian and dihedral vertices, and the semidihedral case needs genuinely new case analysis around C2×C2 and Q8 subgroups. Second, the main theorems are provably correct as far as I can tell. I checked the structural facts about semidihedral 2-groups quoted from [16] directly from the presentation, and they are exactly as stated.\n\nWhat's new: Theorem 1.1 gives a sufficient condition—saturated fusion system plus 2-nilpotence of centralizers of fully normalized subgroups—for Sc(G,P) to be Brauer indecomposable. Theorem 1.2 shows that if G and G′ have a common semidihedral Sylow 2-subgroup P and equal fusion systems, then the diagonal Scott module Sc(G×G′, ΔP) is Brauer indecomposable. The proof of Theorem 1.2 is the heart of the paper: Lemma 3.1 shows that any subgroup of P of order at least 8 contains an element x^i whose centralizer is 2-nilpotent by Brauer's classical theorem, and the remaining small subgroups C2, C4, and C2×C2 are handled separately. The C2 case is the most delicate, using Burry–Carlson–Puig and a Mackey argument to rule out decompositions; I found this part coherent and complete.\n\nSoft spots, in proportion: the paper leans on unproved centralizer facts from [16]—notably that C_P(Q)=Z(Q) for Q≅C2×C2 or Q8, and C_P(Q)=Q for the order-4 subgroup <xy>. These are elementary and correct, but a referee should ask the authors to state them explicitly or include a short derivation, since the case analysis for small subgroups rests entirely on them. That is a minor presentation issue, not a substantive flaw. The paper also assumes fluency with fusion systems and Scott modules; the notation is dense but standard for the intended audience. I saw no circularity and no fitted parameters; the self-citations are to established tools rather than load-bearing prior claims.\n\nBottom line: this is a solid technical extension for block theory specialists. It does not reinvent the wheel, but it fills a known gap and supplies a useful tool for constructing stable equivalences of Morita type for blocks with semidihedral defect groups—relevant to Broué's conjecture. I would cite it if I worked in this area, and I would definitely send it to a specialist referee. Recommendation: accept, with minor requests for self-contained statements of the quoted centralizer facts.","headline":"Solid, correct closing of the semidihedral case; worth a careful referee, with minor requests for self-contained centralizer facts.","tokens_in":9936,"tokens_out":3700,"would_cite":true,"duration_ms":35662,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20C20","20C05","20C15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that, in characteristic 2, Scott modules with semidihedral vertex are Brauer indecomposable when all fully normalized centralizers are 2-nilpotent.","keywords":["Brauer indecomposability","Scott modules","semidihedral 2-groups","fusion systems","Brauer construction","2-nilpotent centralizers","Morita equivalence","modular representation theory"],"falsifier":"Compute $C_P(Q)$ directly in $SD_{2^n}$ for $Q\\cong C_2\\times C_2$, $Q\\cong Q_8$, and $Q=\\langle xy\\rangle$: if it is not respectively $Z(Q)$, $Z(Q)$, and $Q$, the proof's reduction fails. Alternatively, search for a finite group $G$ with semidihedral Sylow $2$-subgroup, saturated fusion system, and $2$-nilpotent $C_G(Q)$ for all fully normalized $Q$, for which $\\mathrm{Sc}(G,P)$ is not Brauer indecomposable; even one such example would disprove Theorem 1.1.","tokens_in":8868,"feed_emoji":"","tokens_out":21459,"duration_ms":174693,"temperature":0.7,"pith_summary":"The paper establishes a sufficient condition for the Scott module of a finite group over a semidihedral $2$-subgroup to be Brauer indecomposable: if the fusion system $\\mathcal{F}_P(G)$ is saturated and $C_G(Q)$ is $2$-nilpotent for every fully $\\mathcal{F}_P(G)$-normalized nontrivial subgroup $Q$ of $P$, then the Brauer quotient $\\mathrm{Sc}(G,P)(Q)$ is indecomposable as a $k[Q\\,C_G(Q)]$-module for every $Q\\le P$. This matters because Brauer indecomposability of $p$-permutation bimodules is a standard precondition for the gluing method that constructs splendid stable equivalences of Morita type between blocks, a route toward the abelian defect group conjecture. The result extends earlier Brauer-indecomposability theorems for abelian and dihedral vertices to the semidihedral case. A companion theorem treats the diagonal Scott module $\\mathrm{Sc}(G\\times G',\\Delta P)$ when two groups share a semidihedral Sylow $2$-subgroup $P$ and the same fusion system, which is the bimodule form needed for such equivalences.","feed_headline":"Proved: semidihedral Scott modules stay Brauer-indecomposable","feed_subtitle":"A 2-nilpotent centralizer condition keeps Scott-module Brauer quotients indecomposable, aiding block equivalences.","key_machinery":"The load-bearing mechanism is the reduction supplied by [9]: by Theorem 2.1, for saturated $\\mathcal{F}_P(G)$ the Brauer indecomposability of $\\mathrm{Sc}(G,P)$ is equivalent to indecomposability of the local Scott modules $\\mathrm{Res}_{Q C_G(Q)}^{N_G(Q)}\\,\\mathrm{Sc}(N_G(Q),N_P(Q))$ for fully normalized $Q$, and Theorem 2.2 gives a sufficient condition via a subgroup $H_Q\\le N_G(Q)$ for which $N_P(Q)$ is a Sylow $p$-subgroup and the index $|N_G(Q):H_Q|$ is a $p$-power. The proof feeds this reduction with a case analysis of the semidihedral group $SD_{2^n}=\\langle x,y\\mid x^{2^{n-1}}=y^2=1,\\ y^{-1}xy=x^{2^{n-2}-1}\\rangle$: the classification of its three maximal subgroups from [16], the centralizer identities $C_P(Q)=Z(Q)$ for $Q\\cong C_2\\times C_2$ or $Q_8$ and $C_P(Q)=Q$ for $Q=\\langle xy\\rangle$, the $2$-nilpotency result from [2] for $C_G(x^i)$ with $x^i\\notin\\{1,z\\}$, and an $S_3$-subgroup construction (Lemma 2.3) used to build $H_Q$ in the exceptional cases.","core_discovery":"The paper's central claim is Theorem 1.1: for an algebraically closed field $k$ of characteristic $2$, if $G$ has a semidihedral $2$-subgroup $P$, the fusion system $\\mathcal{F}_P(G)$ is saturated, and $C_G(Q)$ is $2$-nilpotent for every fully $\\mathcal{F}_P(G)$-normalized nontrivial subgroup $Q$ of $P$, then the Scott module $\\mathrm{Sc}(G,P)$ is Brauer indecomposable. Theorem 1.2 extends this to $\\mathrm{Sc}(G\\times G',\\Delta P)$, where $G$ and $G'$ have the same semidihedral Sylow $2$-subgroup $P$ and equal fusion systems $\\mathcal{F}_P(G)=\\mathcal{F}_P(G')$. The proof handles fully normalized $Q$ by dividing into cases: large subgroups are covered by known $2$-nilpotency of centralizers of noncentral powers in the maximal cyclic subgroup, while the exceptional subgroups $C_2\\times C_2$, $C_4$, and $C_2$ are treated with explicit centralizer computations inside the semidihedral group and an $S_3$-subgroup argument.","pith_inferences":["The proof's case split suggests that for a Sylow semidihedral $2$-subgroup, Lemma 3.1 already forces $C_G(Q)$ to be $2$-nilpotent once $|Q|\\ge 8$, so the actual restriction imposed by Theorem 1.1 concerns only the small subgroups $C_2$, $C_4$, and $C_2\\times C_2$; proving the centralizer condition for those from fusion-theoretic axioms alone would remove the main hypothesis.","The same reduction strategy—classify subgroups, verify $2$-nilpotency except for a short list, then handle the exceptions through automorphism groups and $S_3$-subgroup constructions—looks transferable to other defect groups with known subgroup structure, such as generalized quaternion or elementary abelian $p$-groups.","A direct, self-contained proof of the quoted semidihedral centralizer identities would make Theorem 1.1 independent of structural classifications and would let the exceptional-case verification be checked mechanically for small $n$.","The diagonal result suggests a testable criterion: equality of fusion systems plus Brauer indecomposability may be enough for $\\mathrm{Sc}(G\\times H,\\Delta P)$ to induce a stable equivalence of Morita type for semidihedral defect groups; constructing such an equivalence for an explicit pair of groups would probe the limits of the gluing method."],"forward_implications":["For every group satisfying the hypotheses, each Brauer quotient $\\mathrm{Sc}(G,P)(Q)$ is indecomposable as a $k[Q\\,C_G(Q)]$-module, so the Scott module can act as a building block in gluing constructions for stable equivalences of Morita type.","When two groups share a semidihedral Sylow $2$-subgroup and have the same fusion system, the diagonal Scott module $\\mathrm{Sc}(G\\times G',\\Delta P)$ is Brauer indecomposable, providing the bimodule datum needed for a splendid stable equivalence between their principal blocks.","For each fully normalized $Q$, the Brauer quotient is isomorphic to the local Scott module $\\mathrm{Sc}(N_G(Q),N_P(Q))$, so global indecomposability is controlled by local indecomposability.","The result places semidihedral vertices on the same footing as abelian and dihedral vertices in the sequence of Brauer-indecomposability results for Scott modules."],"supporting_citations":[{"why":"Supplies the reduction theorem and the $H_Q$ criterion that turn Brauer indecomposability of $\\mathrm{Sc}(G,P)$ into indecomposability of local Scott modules for fully normalized subgroups.","marker":"[9]"},{"why":"Provides the structural facts about semidihedral $2$-groups: the three maximal subgroups, the centralizer identities $C_P(Q)=Z(Q)$ for $C_2\\times C_2$ or $Q_8$, $C_P(Q)=Q$ for $\\langle xy\\rangle$, and the conjugacy classes of involutions and elements of order $4$.","marker":"[16]"},{"why":"Gives the $2$-nilpotency of $C_G(x^i)$ for $x^i$ outside $\\{1,z\\}$, which handles subgroups of order at least $8$ and cyclic subgroups.","marker":"[2]"},{"why":"Supplies the $S_3$-subgroup lemma (Lemma 2.3) and the proof template from the dihedral vertex case that this paper adapts.","marker":"[13]"},{"why":"Defines Brauer indecomposability, proves the base case that the Brauer quotient at $P$ is indecomposable as $P C_G(P)$-module, and provides Lemma 4.3(ii) used at the start of the proof.","marker":"[12]"},{"why":"Provides fusion-system facts: fully normalized subgroups are fully centralized, and $C_P(Q)$ is a Sylow $2$-subgroup of $C_G(Q)$ in the order-$4$ and $C_2\\times C_2$ cases.","marker":"[17]"},{"why":"Used to show $\\mathrm{Aut}(Q)$ is a $2$-group for nonexceptional subgroups $Q$, reducing the generic case to $2$-nilpotency of $N_G(Q)$.","marker":"[7]"}],"fun_headline_variants":["Semidihedral Scott modules: a new Brauer-indecomposability condition","Brauer-indecomposable Scott modules for semidihedral vertices","When semidihedral Scott modules stay Brauer-indecomposable","New criterion for Brauer-indecomposable Scott modules","Generalizing Brauer indecomposability to semidihedral Scott modules"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The theorem rests on the hypothesis that for every nontrivial fully fusion-normalized subgroup $Q$ of $P$, the centralizer $C_G(Q)$ has a normal $2$-complement; the proof also depends on the quoted identities $C_P(Q)=Z(Q)$ for $Q\\cong C_2\\times C_2$ or $Q_8$ and $C_P(Q)=Q$ for $Q=\\langle xy\\rangle$ inside semidihedral groups, and the argument collapses if those identities are wrong.","fun_headline_variants_meta":{"raw":{"variants":["Semidihedral Scott modules: a new Brauer-indecomposability condition","Brauer-indecomposable Scott modules for semidihedral vertices","When semidihedral Scott modules stay Brauer-indecomposable","New criterion for Brauer-indecomposable Scott modules","Generalizing Brauer indecomposability to semidihedral Scott modules"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000749,"raw_usage":{"total_tokens":3371,"prompt_tokens":1018,"completion_tokens":2353,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":634,"completion_tokens_details":{"reasoning_tokens":2260}},"tokens_in":634,"tokens_out":2353,"duration_ms":17500,"temperature":1.0,"reasoning_tokens":2260,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:11:12.118173+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $C_P(Q)$ directly in $SD_{2^n}$ for $Q\\cong C_2\\times C_2$, $Q\\cong Q_8$, and $Q=\\langle xy\\rangle$: if it is not respectively $Z(Q)$, $Z(Q)$, and $Q$, the proof's reduction fails. Alternatively, search for a finite group $G$ with semidihedral Sylow $2$-subgroup, saturated fusion system, and $2$-nilpotent $C_G(Q)$ for all fully normalized $Q$, for which $\\mathrm{Sc}(G,P)$ is not Brauer indecomposable; even one such example would disprove Theorem 1.1.","supporting_citations":[{"cited_title":"Craven, A","cited_arxiv_id":null,"evidence_quote":"Used to show $\\mathrm{Aut}(Q)$ is a $2$-group for nonexceptional subgroups $Q$, reducing the generic case to $2$-nilpotency of $N_G(Q)$."},{"cited_title":"Ishioka, N","cited_arxiv_id":null,"evidence_quote":"Supplies the reduction theorem and the $H_Q$ criterion that turn Brauer indecomposability of $\\mathrm{Sc}(G,P)$ into indecomposability of local Scott modules for fully normalized subgroups."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the structural facts about semidihedral $2$-groups: the three maximal subgroups, the centralizer identities $C_P(Q)=Z(Q)$ for $C_2\\times C_2$ or $Q_8$, $C_P(Q)=Q$ for $\\langle xy\\rangle$, and the conjugacy classes of involutions and elements of order $4$."},{"cited_title":"Brauer, Some applications of the theory of blocks of charact ers of finite groups III, J","cited_arxiv_id":null,"evidence_quote":"Gives the $2$-nilpotency of $C_G(x^i)$ for $x^i$ outside $\\{1,z\\}$, which handles subgroups of order at least $8$ and cyclic subgroups."},{"cited_title":"Koshitani, C","cited_arxiv_id":null,"evidence_quote":"Supplies the $S_3$-subgroup lemma (Lemma 2.3) and the proof template from the dihedral vertex case that this paper adapts."},{"cited_title":"Kessar, N","cited_arxiv_id":null,"evidence_quote":"Defines Brauer indecomposability, proves the base case that the Brauer quotient at $P$ is indecomposable as $P C_G(P)$-module, and provides Lemma 4.3(ii) used at the start of the proof."},{"cited_title":"Linckelmann, Introduction to fusion systems, In: Group Representation Theory, Edited by EPFL Press, 79–113, Lausanne, 2007","cited_arxiv_id":null,"evidence_quote":"Provides fusion-system facts: fully normalized subgroups are fully centralized, and $C_P(Q)$ is a Sylow $2$-subgroup of $C_G(Q)$ in the order-$4$ and $C_2\\times C_2$ cases."}],"review_version":1}