{"id":"d37dcaae-2386-4a18-85cf-94f5b20f7466","arxiv_id":"2606.21881","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Presents a necessary and sufficient condition for an orthogonal system to be a simple-minded system over domestic Brauer graph algebras, with constructions for 2-domestic ones.","lead":"This paper examines the relationship between simple-minded systems and weakly simple-minded systems in the stable module category of self-injective algebras. It gives a necessary and sufficient condition for an orthogonal system to be simple-minded over domestic Brauer graph algebras and constructs examples for the 2-domestic case.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly flags the class restriction, but that restriction is not a flaw in the argument—it is the scope of the result. With no further technical detail available to expose a gap in the necessity or sufficiency proof, the central claim stands as stated.","tokens_in":1543,"tokens_out":273,"duration_ms":11209,"concrete_test":"Take the smallest 2-domestic Brauer graph algebra (e.g., the one corresponding to a cycle of length 3 with appropriate multiplicities) and enumerate all orthogonal systems of size equal to the number of simples; check whether each satisfies the stated condition if and only if it is simple-minded by direct computation of Hom and Ext groups in the stable category.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a necessary and sufficient condition characterizing when an orthogonal system in A-stmod is simple-minded, but only for the restricted class of domestic Brauer graph algebras (a tame subclass of self-injective algebras). The paper also constructs examples for the 2-domestic case as a byproduct. No internal inconsistency, hidden assumption in a derivation, or unsupported step is identifiable from the abstract and description; the restriction to this class is explicit and appropriate for obtaining a clean characterization.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper studies the relationship between simple-minded systems and weakly simple-minded systems in the stable module category A-stmod of a self-injective algebra A over an algebraically closed field. It presents a necessary and sufficient condition for an orthogonal system to be a simple-minded system when A is a domestic Brauer graph algebra, and constructs a class of simple-minded systems over 2-domestic Brauer graph algebras as a byproduct.","tokens_in":1608,"tokens_out":189,"duration_ms":8530,"significance":"If the stated condition is correct, the result supplies an explicit characterization of simple-minded systems within a concrete tame subclass of self-injective algebras, which may facilitate further classification work in the representation theory of tame algebras. The explicit constructions for the 2-domestic case provide verifiable examples that can be checked against the definitions.","major_comments":[],"minor_comments":[],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive report and recommendation to accept the manuscript.","responses":[],"tokens_in":1043,"tokens_out":34,"duration_ms":5104,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is a clean characterization: for domestic Brauer graph algebras, an orthogonal system in the stable module category is simple-minded exactly when it meets the stated condition. The paper also produces a family of examples for the 2-domestic subclass as a side result.\n\nWhat is actually new is the restriction of the earlier notions of simple-minded and weakly simple-minded systems to this specific tame family, together with the explicit condition and constructions that follow from the domestic structure. The work stays inside the existing framework rather than inventing new general tools.\n\nThe paper does what it sets out to do. The domestic restriction lets the authors reach a precise if-and-only-if statement that would be harder in the general self-injective case. The examples appear to be direct and checkable.\n\nThe obvious limitation is scope. Everything is stated only for domestic Brauer graph algebras; there is no claim or discussion about what happens for other self-injective algebras or even other tame classes. That is explicit in the abstract, so it is not a hidden flaw, but it keeps the result narrow.\n\nThe argument pattern looks standard for the area and does not show circularity or free parameters. The full text would need to be checked for the actual derivations, but nothing in the setup raises an immediate red flag.\n\nThis is for specialists already working on representation theory of self-injective algebras and Brauer graph algebras. Readers outside that corner will not find much to use.\n\nIt deserves a serious referee. The characterization and examples are concrete enough to be worth checking in detail.","headline":"This note gives a necessary and sufficient condition for orthogonal systems to be simple-minded over domestic Brauer graph algebras, plus some explicit examples in the 2-domestic case.","tokens_in":2080,"tokens_out":398,"would_cite":false,"duration_ms":17710,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Over domestic Brauer graph algebras an orthogonal system is a simple-minded system precisely when it satisfies the given condition.","keywords":["simple-minded systems","weakly simple-minded systems","self-injective algebras","Brauer graph algebras","orthogonal systems","stable module category","domestic algebras"],"falsifier":"An explicit orthogonal system over a domestic Brauer graph algebra that meets the condition but fails to be simple-minded, or vice versa.","tokens_in":2424,"feed_emoji":"","tokens_out":495,"duration_ms":23828,"temperature":0.7,"pith_summary":"The paper examines the relationship between simple-minded systems and weakly simple-minded systems inside the stable module category of self-injective algebras. For domestic Brauer graph algebras it supplies a necessary and sufficient condition that turns an orthogonal system into a simple-minded system. A reader would care because these systems organize indecomposable objects and support classification in the representation theory of the algebra. The work also produces explicit examples of simple-minded systems for the 2-domestic subclass.","feed_headline":"Condition turns orthogonal systems into simple-minded ones","feed_subtitle":"The necessary and sufficient test applies to domestic Brauer graph algebras and yields examples for the 2-domestic case.","key_machinery":"The necessary and sufficient condition that distinguishes when an orthogonal system qualifies as a simple-minded system over domestic Brauer graph algebras.","core_discovery":"Over domestic Brauer graph algebras, an orthogonal system in A-stmod is a simple-minded system if and only if it satisfies the necessary and sufficient condition presented for this class.","pith_inferences":["The condition might extend to other tame self-injective algebras.","It could support a full classification of simple-minded systems over domestic Brauer graph algebras.","Similar tests may connect to the study of silting or tilting objects in related categories."],"forward_implications":["Simple-minded systems can be identified for domestic Brauer graph algebras by checking the condition.","A class of simple-minded systems exists over 2-domestic Brauer graph algebras.","The distinction between simple-minded systems and weakly simple-minded systems is resolved for this algebra class."],"fun_headline_variants":["Condition makes orthogonal systems simple-minded over domestic Brauer graph algebras","Orthogonal system qualifies as simple-minded iff condition met in Brauer algebras","Condition links orthogonal to simple-minded systems over domestic Brauer algebras","Necessary condition identifies simple-minded systems among orthogonals in Brauer algebras"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The algebras are domestic Brauer graph algebras.","fun_headline_variants_meta":{"raw":{"variants":["Condition makes orthogonal systems simple-minded over domestic Brauer graph algebras","Orthogonal system qualifies as simple-minded iff condition met in Brauer algebras","Condition links orthogonal to simple-minded systems over domestic Brauer algebras","Necessary condition identifies simple-minded systems among orthogonals in Brauer algebras"]},"model":"grok-4.3","cost_usd":0.01072,"raw_usage":{"total_tokens":4627,"prompt_tokens":463,"num_sources_used":0,"completion_tokens":72,"cost_in_usd_ticks":107199500,"prompt_tokens_details":{"text_tokens":463,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4092,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":463,"tokens_out":72,"duration_ms":26935,"temperature":1.0,"reasoning_tokens":4092,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T11:27:31.041336+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit orthogonal system over a domestic Brauer graph algebra that meets the condition but fails to be simple-minded, or vice versa.","supporting_citations":[],"review_version":1}