{"id":"5a2c4d78-4976-4232-9cec-3dd217cc96aa","arxiv_id":"2502.06017","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Rational SU(3)-equivariant spectra are Quillen equivalent to differential graded objects of an explicitly constructed abelian category A(SU(3)).","lead":"This paper gives a complete algebraic description of rational SU(3)-equivariant cohomology theories by splitting the group's subgroup data into 18 building blocks. The result is a concrete step in a broad program that aims to replace topological equivariant spectra with ordinary algebra.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"§4.C and Theorem 4.3 give inconsistent lists of the seven exceptional finite subgroups (A5×C3 vs A6); since A6 is not even realizable in SU(3), the block decomposition and hence A(SU(3)) are ambiguous as printed.","rationale":"The reader flagged the same inconsistency as minor; I am elevating it because it is the one place where the paper's internal text contradicts the theorem that defines the model category. I am not making the external-reference issue the primary complaint: although the main Quillen equivalence depends on [8]–[10] and [5,2], those dependencies are explicitly declared and could be discharged by reading the companion papers. The exceptional-list mismatch, by contrast, is a defect in this paper's own statement of its central input. A computational check against the classical classification will settle it; until then the paper cannot be accepted as it stands. The conditional verdict remains appropriate.","tokens_in":12864,"tokens_out":34835,"duration_ms":337144,"concrete_test":"Use GAP or Magma together with the Miller–Blichfeldt–Dickson/Ludl tables: for each candidate finite group in the two conflicting lists, construct a faithful unitary 3-dimensional representation and compute its normalizer and W_G in SU(3). In particular, test whether A6 occurs and whether there is a center-free exceptional group with W_G=C3. If A6 is absent, correct Theorem 4.3 row 15 to the center-free partner of the group named in §4.C, fix the W_G=1 row accordingly, and re-count the blocks. If the count changes from 18, the model category and the Quillen equivalence change.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is an assembly over the 18 blocks of Theorem 4.3. That theorem is internally inconsistent with the classification that precedes it. Section 4.C lists the seven Type E finite subgroups as PSL2(7), A5×C3, PSL2(7)×C3, A6·3, G36·3, G72·3, G216·3. Theorem 4.3 replaces A5×C3 by a row (1,A6) with W_G=C3, while retaining (1,A6·3) with W_G=1. These are not notational variants: A5×C3 has order 180, whereas A6 has order 360 and has no faithful 3-dimensional complex representation, so it cannot be a subgroup of SU(3) at all. The 0-dimensional singleton blocks contribute Q[W_G(H)]-modules as factors of A(SU(3)); replacing one finite subgroup by a non-realizable group changes the abelian category, and the statement 'the category of rational SU(3)-spectra is Quillen equivalent to DG-A(SU(3))' has no determinate target until the list is corrected. This is load-bearing because Theorem 4.3 is the input that Section 5 assembles; it is not a harmless typo in an example.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper describes the space of conjugacy classes of subgroups of SU(3) as a disjoint union of 18 Zariski clopen blocks, each dominated by a pair (H_e, F). For every block it identifies the algebraic input (sheaf of rings, component structure, and Weyl-group data) needed for a rational algebraic model, and then assembles results from companion papers to assert a Quillen equivalence between rational SU(3)-spectra and the category of differential graded objects of the abelian category A(SU(3)). Sections 2 and 3 recall the block decompositions for proper subgroups and analyze fusion from U(2) to SU(3); Section 4 performs the new subgroup classification and states the 18-block partition in Theorem 4.3; Section 5 indicates the model structures for the blocks.","tokens_in":13129,"tokens_out":5779,"duration_ms":48462,"significance":"If correct, the paper would give a complete algebraic model for rational SU(3)-equivariant cohomology, a substantial non-toral example of the general Greenlees conjecture. The block decomposition and the normalizer computations in Sections 3 and 5 are concrete and useful, and the explicit 18-block structure is a meaningful contribution. However, the central Quillen equivalence is not proved here; it is imported from the same-author preprints [8,9,10] and [5,2], so the result is conditional on those works. Moreover, the internal inconsistency in the list of exceptional finite subgroups between §4.C and Theorem 4.3 makes the definition of A(SU(3)) ambiguous as printed, and must be corrected before the main claim is well posed.","major_comments":[{"comment":"The list of Type E finite subgroups in §4.C is inconsistent with the table in Theorem 4.3. §4.C states the seven exceptional groups are PSL2(7), A5×C3, PSL2(7)×C3, A6·3, G36·3, G72·3, G216·3, whereas Theorem 4.3 replaces A5×C3 by a row (1,A6) with W_G(H)=C3 while retaining (1,A6·3). These are not notational variants: A5×C3 has order 180, while A6 has order 360 and admits no faithful 3-dimensional complex representation, so A6 cannot be a subgroup of SU(3). Since Theorem 4.3 is the input that Section 5 assembles into the model A(SU(3)), the definition of the target category is ambiguous and the central Quillen equivalence is not well posed until the list is corrected.","section":"§4.C and Theorem 4.3"},{"comment":"The central claim, that rational SU(3)-spectra are Quillen equivalent to DG-A(SU(3)), is not proved in this paper. Section 5 assembles models block by block, but the actual equivalences for the 1-dimensional blocks are delegated to the same-author preprints [8], [9], [10], and the toral block to [5, 2]. These are not peer-reviewed and are not included, so a reader cannot verify the main theorem from this manuscript alone. The authors should either include the necessary proofs or explicitly state that the main result is conditional on these companion papers and make them publicly available.","section":"§1.E, §5"}],"minor_comments":[{"comment":"Typo: 'matices' should be 'matrices'.","section":"§1.G"},{"comment":"In the table, the row (T^2, C1) uses 'C1' for the trivial group, whereas elsewhere the trivial group is denoted '1'; please unify the notation.","section":"Theorem 4.3"},{"comment":"The reference [15] is incomplete; it gives volume and page range but no journal or book title.","section":"References"},{"comment":"The phrase 'The one dominated by the maximal torus A(G|toral) is treated in [5, 2]' is grammatically awkward and should be rephrased for clarity.","section":"§5.C"},{"comment":"The sentence 'Those with a two dimensional faithful representations' contains an agreement error; it should be 'a two-dimensional faithful representation'.","section":"§4.C"}],"recommendation":"major_revision","confidential_remarks":"The paper is the fifth in a series and relies heavily on the author's own preprints. The editor may wish to verify that the companion papers [8], [9], [10] are in suitable shape, since the main theorem of this paper rests on them. Also, the inconsistent list of exceptional subgroups in §4.C versus Theorem 4.3 is a blocking issue that must be corrected before the central claim can be assessed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this is a genuine milestone: the first explicit 18-block decomposition of X_{SU(3)} and a proposed abelian category A(SU(3)) that should model rational SU(3)-spectra. The subgroup analysis is careful, the block count is arithmetically consistent, and the normalizer lemmas are convincing. Second, the paper as printed has a load-bearing inconsistency: Section 4.C lists the exceptional finite subgroup A5×C3, while Theorem 4.3 replaces it with (1,A6). A6 has order 360 and no faithful 3-dimensional complex representation, so it cannot be a subgroup of SU(3). Since Theorem 4.3 defines the blocks that Section 5 assembles into A(SU(3)), the target category is ambiguous until this is fixed. This is not a cosmetic typo.\n\nThe genuinely new content is the fusion analysis specific to SU(3) and the identification of the 11 blocks not inherited from U(2). The paper is also honest about what it does not prove: the Quillen equivalence is assembled from four same-author preprints, all on arXiv. That is standard for a capstone paper, but it means the central theorem is not self-contained and a referee will need to inspect those preprints.\n\nSoft spots in proportion: the A5×C3/A6 issue is the main one; it should be corrected and the Weyl group for the affected block re-derived. There are also minor typos, e.g., a malformed sentence in the proof of Lemma 3.6. The reliance on companion papers is a structural feature rather than a flaw, though it does cap the paper's independence.\n\nWho this is for: equivariant homotopy theorists working on algebraic models for rational G-spectra. They will want to read this after (or with) the companion papers. It deserves a serious referee: the framework is right and the result is important, but the printed version needs correction before the claimed equivalence is determinate. I would send it to review, with the expectation of a substantive revision fixing the subgroup list and clarifying the status of the imported results.","headline":"A real milestone for rational SU(3)-spectra, but the printed central theorem has an inconsistent exceptional subgroup list (A5×C3 vs A6) that must be fixed before the claimed equivalence is well-defined.","tokens_in":13683,"tokens_out":4304,"would_cite":false,"duration_ms":39462,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55P42","55P91","55N91"],"pacs":[],"model":"deepseek-v4-flash","headline":"Rational SU(3)-equivariant cohomology theories are Quillen equivalent to differential graded objects of the abelian category A(SU(3)), built from an 18-block partition of subgroup conjugacy classes.","keywords":["rational equivariant cohomology","SU(3)-spectra","Quillen equivalence","algebraic model","subgroup conjugacy classes","sheaves of rings","cotoral order","18-block decomposition"],"falsifier":"Check the claimed 18-block partition directly: if any closed subgroup of $\\mathrm{SU}(3)$ is not conjugate to a subgroup of $\\mathrm{U}(2)$ and not conjugate to one of the seven listed exceptional finite groups, then Theorem 4.3 is incomplete. A second check is to compute a known rational $\\mathrm{SU}(3)$-equivariant cohomology group, for example Borel cohomology of $\\mathrm{SU}(3)/T$ with constant coefficients, from the sheaf-ring data and compare it with the standard answer.","tokens_in":12645,"feed_emoji":"🧩","tokens_out":9085,"duration_ms":78620,"temperature":0.7,"pith_summary":"This paper claims that rational SU(3)-equivariant cohomology theories are completely described by an explicit abelian category $\\mathcal{A}(\\mathrm{SU}(3))$, so that the homotopy theory of rational $\\mathrm{SU}(3)$-spectra becomes a piece of ordinary homological algebra. The category is assembled from the space of conjugacy classes of closed subgroups of $\\mathrm{SU}(3)$, which is partitioned into 18 Zariski-clopen blocks, each dominated by a subgroup of the form $(H_e, F)$. For every block the paper specifies a sheaf of rings and a component structure, and it assembles results from companion papers to conclude that rational $\\mathrm{SU}(3)$-spectra are Quillen equivalent to differential graded objects of $\\mathcal{A}(\\mathrm{SU}(3))$. If correct, this settles the proposed algebraic-model classification for $\\mathrm{SU}(3)$ and all of its subgroups.","feed_headline":"Rational SU(3) cohomology collapses to an 18-block algebraic model","feed_subtitle":"All rational SU(3)-equivariant cohomology theories are governed by sheaves over 18 subgroup blocks.","key_machinery":"The load-bearing object is the partition of $X_G = \\mathrm{Sub}(G)/G$ into blocks dominated by pairs $(H_e,F)$ with finite Weyl group, stated as Theorem 4.3. A block $V^G_{(H_e,F)}$ is the closure under cotoral specialization of a neighbourhood of the dominating subgroup; a subgroup $K$ is cotoral in $H$ when $K$ is normal in $H$ with torus quotient. The algebraic model over each block is a sheaf of modules over a sheaf of rings with stalk $H^*(B\\,W^e_G(K))[W^d_G(K)]$, together with a component structure recording the action of the discrete part of the Weyl group. These pieces multiply together, $\\mathcal{A}(\\mathrm{SU}(3)) = \\prod_i \\mathcal{A}(\\mathrm{SU}(3)|V_i)$, and the sheaf data over cotoral inclusions encodes the Localization Theorem for equivariant cohomology.","core_discovery":"The central claim is a complete structural description of rational $\\mathrm{SU}(3)$-equivariant cohomology. The space $X_{\\mathrm{SU}(3)} = \\mathrm{Sub}(\\mathrm{SU}(3))/\\mathrm{SU}(3)$ of conjugacy classes of closed subgroups is shown to split as a disjoint union of 18 Zariski-clopen blocks $V^G_{(H_e,F)}$, each dominated by a connected subgroup $H_e$ together with a finite subgroup $F$ of its Weyl group. Over each block the model is a sheaf of modules over a sheaf of rings whose stalk at a subgroup $K$ is the twisted group ring $H^*(B\\,W^e_G(K))[W^d_G(K)]$, equipped with a component structure coming from the finite group $\\pi_0(W_G(K))$. Joining these blocks yields an abelian category $\\mathcal{A}(\\mathrm{SU}(3))$, and the paper assembles the supporting models to conclude that the category of rational $\\mathrm{SU}(3)$-spectra is Quillen equivalent to the category of differential graded objects of $\\mathcal{A}(\\mathrm{SU}(3))$.","pith_inferences":["An implicit consequence is that maps between rational $\\mathrm{SU}(3)$-equivariant cohomology theories, and the spectral sequences that compute them, should become Ext computations inside $\\mathcal{A}(\\mathrm{SU}(3))$; the paper does not spell this out.","The same fusion-and-block method suggests an explicit route to algebraic models for nearby rank-2 groups such as $\\mathrm{PSU}(3)$ or $\\mathrm{U}(3)$, whose subgroup spaces differ mainly through normalizers and fusion of conjugacy classes.","A testable extension is to compute a concrete rational $\\mathrm{SU}(3)$-equivariant cohomology theory, such as Borel cohomology of a homogeneous space, directly from the sheaf stalks and compare with classical calculations; a mismatch would pinpoint which block's sheaf data needs adjustment."],"forward_implications":["Rational $\\mathrm{SU}(3)$-equivariant cohomology theories are classified by algebraic objects, so computations can be carried out as sheaf cohomology and Ext over the 18-block space.","The model factorizes as a product of 18 blocks, so every rational $\\mathrm{SU}(3)$-spectrum is the product of its pieces of geometric isotropy supported on single blocks.","Blocks inherited from $\\mathrm{U}(2)$ survive in $\\mathrm{SU}(3)$ with only a small amount of fusion; restriction from $\\mathrm{SU}(3)$ to $\\mathrm{U}(2)$ induces category isomorphisms on those blocks.","Nine of the blocks are 0-dimensional singletons, where the model reduces to modules over the group ring of a finite Weyl group, i.e. ordinary finite-group cohomology."],"supporting_citations":[{"why":"Constructs algebraic models for all relevant one-dimensional blocks, applied here to five of the one-dimensional blocks.","marker":"[8]"},{"why":"Constructs models for rank-2 toral blocks of mixed type, used for the block of the maximal torus normalizer.","marker":"[9]"},{"why":"Constructs the seven-block algebraic model for rational U(2)-spectra, from which seven SU(3) blocks are inherited with small fusion.","marker":"[10]"},{"why":"Supplies the algebraic model for the toral block dominated by the maximal torus T^2.","marker":"[5, 2]"},{"why":"Shows that modules over the rational group ring of a finite Weyl group model pieces with finite Weyl group, covering the nine 0-dimensional blocks.","marker":"[14]"},{"why":"Identifies the space of conjugacy classes of subgroups as the spectrum of finite rational G-spectra, providing the Zariski topology and block structure.","marker":"[6]"},{"why":"Classifies finite subgroups of SU(3), which fixes the seven exceptional 0-dimensional blocks in the partition.","marker":"[16]"}],"fun_headline_variants":["Rational SU(3) cohomology: 18-block algebraic model","18 blocks govern all rational SU(3) cohomology","Rational SU(3) cohomology classified by 18 sheaf blocks","SU(3) equivariant cohomology: 18 blocks, one model"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the algebraic models for the one-dimensional and toral blocks, taken from companion papers, are correct, together with the classical classification of finite subgroups of $\\mathrm{SU}(3)$ that fixes the seven exceptional 0-dimensional blocks; the present paper assembles these ingredients rather than reproving them.","fun_headline_variants_meta":{"raw":{"variants":["Rational SU(3) cohomology: 18-block algebraic model","18 blocks govern all rational SU(3) cohomology","Rational SU(3) cohomology classified by 18 sheaf blocks","SU(3) equivariant cohomology: 18 blocks, one model"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000811,"raw_usage":{"total_tokens":3533,"prompt_tokens":899,"completion_tokens":2634,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":515,"completion_tokens_details":{"reasoning_tokens":2554}},"tokens_in":515,"tokens_out":2634,"duration_ms":18551,"temperature":1.0,"reasoning_tokens":2554,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T17:00:09.013696+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the claimed 18-block partition directly: if any closed subgroup of $\\mathrm{SU}(3)$ is not conjugate to a subgroup of $\\mathrm{U}(2)$ and not conjugate to one of the seven listed exceptional finite groups, then Theorem 4.3 is incomplete. A second check is to compute a known rational $\\mathrm{SU}(3)$-equivariant cohomology group, for example Borel cohomology of $\\mathrm{SU}(3)/T$ with constant coefficients, from the sheaf-ring data and compare it with the standard answer.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows that modules over the rational group ring of a finite Weyl group model pieces with finite Weyl group, covering the nine 0-dimensional blocks."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Identifies the space of conjugacy classes of subgroups as the spectrum of finite rational G-spectra, providing the Zariski topology and block structure."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Classifies finite subgroups of SU(3), which fixes the seven exceptional 0-dimensional blocks in the partition."}],"review_version":1}