{"id":"45c08a13-6363-4379-8b6d-31fc36cf2989","arxiv_id":"2505.21977","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"With epsilon invertible and delta arbitrary, Rook-Brauer homology is isomorphic to symmetric group homology in all degrees and Motzkin homology vanishes in positive degrees.","lead":"This paper proves that the homology of the Rook-Brauer diagram algebra matches the homology of the symmetric group in every degree, and that the Motzkin algebra has no positive-degree homology, assuming only one of the two algebra parameters is invertible. It fixes a gap in an earlier attempt by Boyde and weakens the assumptions of a recent theorem by Fisher and Graves.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.6's asserted idempotent e=ε^{-1}T_a V_ab satisfies e^2=(δ/ε)e, not e; the direct-summand splitting of M_{X,{a,b}} fails for general δ.","rationale":"The reader correctly identified the splitting assertions in Lemma 4.6 as the load-bearing point, but classified them as compressed arguments that a fuller proof would resolve. The stress-test finds a concrete algebraic error in one of those assertions: the element claimed to be an idempotent is not idempotent unless δ=ε, and the splitting map fails even in a two-node example with δ=0. Because Theorem 4.3's inductive resolution depends on M_{X,{a,b}} being a direct summand, and Theorem 4.3 is then used for Rook-Brauer and Motzkin homology, the main theorems are not supported by the present argument. The proof would need either a genuinely different retraction that does not require δ-invertibility, or an additional hypothesis on δ, which would remove the claimed improvement over Fisher–Graves. This is a mathematical error in a central lemma, not merely an omitted verification, so the appropriate verdict changes from CONDITIONAL to REJECT, pending a corrected proof.","tokens_in":12282,"tokens_out":45484,"duration_ms":483694,"concrete_test":"In RBr_3 with ε=1 and δ=0, compute the product V_{12}T_1V_{12} explicitly in the diagram basis: if the multiplication rule of §3.1 is applied faithfully, the sole middle component is a loop, so the product is 0V_{12}=0. This contradicts Lemma 4.6, which requires the same product to equal εV_{12}=V_{12}. An independent check is to verify that (T_1V_{12})^2=0 and therefore ε^{-1}T_1V_{12} is not an idempotent.","verdict_should_be":"REJECT","load_bearing_attack":"Section 4.1, Lemma 4.6, third bullet: to split the inclusion M_{a,b}→RBr_n, the proof uses right multiplication by e=ε^{-1}T_aV_ab and asserts that e is idempotent and that αe=α for α∈M_{a,b}. Direct diagram composition with the multiplication rule in §3.1 gives the opposite scalar. For n=3, a=1, b=2, ε=1, δ=0, take α=V_{12} (with node 3 horizontal). Stacking V_{12}T_1 yields the diagram D with blocks {−1,−2}, {1}, {2}, {−3,3}; multiplying D on the right by V_{12} leaves a middle component {−1,−2} consisting of two contracted nodes. Under the paper's rule this component is a loop and contributes δ, so V_{12}T_1V_{12}=δV_{12}=0. Hence (T_1V_{12})^2=0, so e is not idempotent and the proposed retraction sends the generator V_{12} to 0 rather than to itself. The failure occurs exactly in the lemma used to prove Tor-vanishing for B_{X,x} via Lemma 4.7 and hence to drive the inductive resolution in Theorem 4.3. This is not a missing detail: the claimed splitting map is not a retraction for δ≠ε, and for δ=0 it is nilpotent on the summand it is supposed to split.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims that, for a unital commutative ring R with ε invertible and δ arbitrary, the homology of the Rook-Brauer algebra RBr_n(R,δ,ε), defined as Tor^{RBr_n}_*(1,1), is isomorphic to the homology of the symmetric group in all degrees, and that the homology of the Motzkin algebra M_n(R,δ,ε) vanishes in positive degrees. The proof follows the inductive-resolution framework of Boyd–Hepworth–Patzt: it constructs resolutions of the modules RBr_n/J_X and M_n/J_X using auxiliary modules A_{X,x}, B_{X,x}, M_{X,{a,b}}, and Y_{P,{x,y}}, and proves vanishing of positive Tor groups by induction. The main theorems are then derived from an analogue of Shapiro's Lemma for Rook-Brauer algebras.","tokens_in":12543,"tokens_out":13156,"duration_ms":148895,"significance":"If true, the result would be a substantial strengthening of the Fisher–Graves theorem, removing the invertibility assumption on δ. The inductive-resolution strategy is appropriate and the paper correctly identifies the relevant modules. Some parts of the argument, such as the splitting of A_{X,x} via right multiplication by ε^{-1}T_x, are sound. However, the central splitting lemma for the modules M_{X,{a,b}} is false under the paper's own multiplication rule, and the proof of exactness of the key resolutions is not adequately justified. Since these points are load-bearing for the inductive vanishing theorem, the main results are not established by the manuscript.","major_comments":[{"comment":"The element e = ε^{-1}T_aV_ab is not idempotent and does not split the inclusion M_{a,b} → RBr_n. Using the multiplication rule in §3.1, one computes V_ab T_a V_ab = δ V_ab. Hence e^2 = (δ/ε)e, and for α = V_ab one has αe = (δ/ε)α, not α. In the concrete case n=3, a=1, b=2, ε=1, δ=0, e^2=0 and the proposed retraction sends V_ab to 0. Therefore M_{X,{a,b}} is not shown to be a direct summand of RBr_n/J_{X−{a,b}}, and the inductive step of Theorem 4.3—which needs this summand to conclude Tor_{>0}(1,M_{X,{a,b}})=0—collapses. Since Theorem 5.1 and Theorem 2.1 depend on Theorem 4.3, the main result is not established.","section":"Section 4.1, Lemma 4.6 (third bullet)"},{"comment":"The proof of exactness at degree 1 is not valid as written. The assertion that the map A_{X,x} ⊕ B_{X,x} → RBr_n/J_{X−{x}} is injective is justified only by the sentence that A_x and B_X,x have no basis elements in common. For submodules spanned by disjoint basis subsets, an element of J_{X−{x}} can be a sum of a vector from A_x and a vector from B_X,x without either summand lying in J_{X−{x}}. One needs a proof that J_{X−{x}} ∩ (A_x + B_X,x) = (J_{X−{x}}∩A_x) ⊕ (J_{X−{x}}∩B_X,x), or an explicit basis argument for the quotient. The same gap appears in the Motzkin analogue, Proposition 4.15, which is used in Theorem 4.17 and hence in Theorem 2.2.","section":"Section 4.1, Proposition 4.8 and Section 4.2, Proposition 4.15"}],"minor_comments":[{"comment":"There is a typo: 'satisify' should be 'satisfy'.","section":"Corollary 2.3"},{"comment":"In the proof, 'the map in injective' should be 'the map is injective'.","section":"Lemma 4.16"},{"comment":"The proof is delegated entirely to the argument in [3] with the comment that it follows 'exactly as that of Theorem 4.1 of [3]'. Since the algebra and the hypotheses differ, the paper should at least state the spectral sequence or the explicit chain comparison maps that yield the mutually inverse isomorphisms.","section":"Theorem 5.1"},{"comment":"The notation J_m for J_{\\{n-m+1,\\ldots,n\\}} is potentially confusing because the subscript is a number rather than a subset; a brief clarification would help.","section":"Definition 4.2"}],"recommendation":"reject","confidential_remarks":"The central error occurs precisely in the lemma needed to make the inductive resolution work, and it is the same type of obstruction that the manuscript itself attributes to Boyde's earlier attempt. Because the claimed isomorphism for all δ relies on a false idempotent-splitting statement, the proof cannot be repaired by local editing; a different construction or additional hypotheses would be required. This places the paper below the standard for publication in its current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper aims for a genuine improvement over Fisher–Graves: proving the homology of Rook-Brauer and Motzkin algebras under only ε-invertibility, with δ arbitrary. The inductive-resolution strategy is appropriate, and the Y_P modules introduced for Motzkin algebras are a plausible new device. If the proof worked, this would be a real extension.\n\nIt does not work as written. Lemma 4.6 asserts that right multiplication by ε^{-1}T_aV_ab is idempotent and that αT_aV_ab = εα for α ∈ M_{a,b}. The second assertion is false. Take n=3, ε=1, δ=0, a=1, b=2, and α=V_{12}. Composing V_{12}T_1 gives the diagram D with blocks {−1,−2}, {1}, {2}, {−3,3}. Multiplying D on the right by V_{12} produces a middle-only loop and leaves the outer diagram V_{12}, so the product is δV_{12}=0. Thus αT_aV_ab = 0 ≠ εα. The same computation shows (T_aV_ab)^2 = δT_aV_ab, so e is idempotent only when δ=ε, not in general.\n\nThis is not a missing detail; it is a concrete error. The splitting of M_{X,{a,b}} as a direct summand of RBr_n/J_{X−{a,b}} fails, and with it Lemma 4.7 and the inductive resolution in Theorem 4.3 collapse. The main theorems 2.1 and 2.2 are therefore unproven. The paper correctly diagnoses Boyde's gap and organizes the material well, and the Y_P idea may be salvageable, but the central proof has a load-bearing false lemma.\n\nThe reader's conditional verdict is too generous. I would still send this to a serious referee because the question matters and the approach is sound in outline, but the referee should be asked to scrutinize Lemma 4.6 and the analogous Motzkin splitting claims. I would not cite it as a proof of the theorems in its current form.","headline":"A real attempt at a sharper homology theorem for Rook-Brauer and Motzkin algebras, but Lemma 4.6 contains an explicit false idempotent claim that brings the main proof down.","tokens_in":13108,"tokens_out":9816,"would_cite":false,"duration_ms":98570,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16E30","18G15","20J06"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that when $\\epsilon$ is invertible, the homology of the Rook-Brauer algebra is isomorphic to symmetric group homology in all degrees, and that Motzkin algebra homology vanishes in positive degrees.","keywords":["Rook-Brauer algebras","Motzkin algebras","homological stability","Tor groups","inductive resolution","diagram algebras","symmetric group homology"],"falsifier":"Compute $\\mathrm{Tor}^{RBr_2}_1(1,1)$ over $R=\\mathbb{Z}$ with $\\epsilon=1$ and, say, $\\delta=0$; Theorem 2.1 predicts $H_1(S_2)=\\mathbb{Z}/2$. A more targeted check is to test Lemma 4.6 directly: for $X=\\{1,2\\}$ in $RBr_2$, verify that right multiplication by $\\epsilon^{-1}T_1$ sends $J_{\\{2\\}}$ into itself and is idempotent on the quotient; a single diagram violating this would break the inductive step.","tokens_in":12012,"feed_emoji":"","tokens_out":10519,"duration_ms":103096,"temperature":0.7,"pith_summary":"Rook-Brauer algebras and Motzkin algebras are diagram algebras built from partitions into blocks of size at most two, and their homology is interpreted here as Tor with the trivial module. This paper proves that whenever $\\epsilon$ is invertible in the ground ring, with $\\delta$ arbitrary, the inclusion of the symmetric group algebra into the Rook-Brauer algebra induces an isomorphism in homology in every degree. It also proves that the Motzkin algebras have vanishing positive-degree homology. Together these results establish homological stability for both families, with a sharp stable range for the Rook-Brauer algebras. The earlier proofs of these statements required both parameters to be invertible; this paper removes the condition on $\\delta$.","feed_headline":"Rook-Brauer homology equals symmetric group homology","feed_subtitle":"With epsilon invertible, Motzkin homology vanishes; both families stabilize and the stable range is sharp.","key_machinery":"The central device is inductive resolution, a way of proving Tor vanishing by resolving a module by modules that already have vanishing positive Tor. The paper constructs such resolutions from left ideals and link-state modules: $A_x$ (diagrams in which a chosen node is isolated), $B_{X,x}$ (diagrams in which a chosen node connects to another node in $X$), $M_{\\{a,b\\}}$ (diagrams in which two nodes are connected), and, for the Motzkin algebras, $Y_P$ (modules built from a planar right-link state via splices and deletions). The load-bearing checks are the lemmas that identify the quotients $A_{X,x}$, $M_{X,\\{a,b\\}}$, and $Y_{P,\\{x,y\\}}$ as direct summands of simpler modules, using right multiplication by idempotents such as $\\epsilon^{-1}T_x$, $\\epsilon^{-1}T_aV_{ab}$, and $\\epsilon^{-k_0}\\gamma$.","core_discovery":"The central discovery is that the homological behaviour of the Rook-Brauer and Motzkin algebras is controlled entirely by their permutation diagrams. With $\\epsilon$ invertible in a unital commutative ring $R$ and $\\delta$ arbitrary, the inclusion $\\iota: RS_n \\to RBr_n(R,\\delta,\\epsilon)$ induces an isomorphism $$\\iota_*: H_*(S_n;1) \\to \\mathrm{Tor}^{RBr_n(R,\\delta,\\epsilon)}_*(1,1)$$ for all degrees, and $\\mathrm{Tor}^{M_n(R,\\delta,\\epsilon)}_*(1,1)$ is $1$ in degree zero and $0$ in positive degrees. The proof obtains this by building inductive resolutions of the modules $RBr_n/J_X$ and $M_n/J_X$ and then applying a Shapiro-type comparison to pass from the symmetric group algebra to the full Rook-Brauer algebra. Homological stability follows directly from the known sharp stable range for symmetric groups.","pith_inferences":["A natural extension is to check whether the vanishing theorem survives with nontrivial coefficients, such as the sign representation; if the direct-summand splitting is as robust as it appears, the Motzkin and Rook-Brauer homology should remain concentrated in the symmetric-group component.","Because $\\delta$ drops out of the homology, one might expect the higher Tor groups to be invariant under deforming $\\delta$, and the same inductive-resolution construction may provide explicit resolutions for other one-parameter diagram algebras.","The results suggest that the homological theory of these diagram algebras is inherited from the symmetric group, so representation-stability phenomena for $S_n$ should transfer to the Rook-Brauer algebras without extra parameter conditions.","A testable weakening would be to replace invertibility of $\\epsilon$ by the assumption that $\\epsilon$ is a non-zero-divisor; the splittings by $\\epsilon^{-1}T_x$ would then require more care and may reveal the exact boundary of the theorem."],"forward_implications":["Rook-Brauer homology equals symmetric group homology in every degree whenever $\\epsilon$ is invertible, so the higher Tor groups are independent of $\\delta$.","Motzkin algebras have no positive-degree homology with trivial coefficients; the only nonzero Tor group is $\\mathrm{Tor}^{M_n}_0(1,1)\\cong 1$.","Rook-Brauer algebras satisfy homological stability for $n\\ge 2i+1$, and this stable range is sharp.","Motzkin algebras also satisfy homological stability.","The inclusion and projection maps between $RS_n$ and $RBr_n$ give mutually inverse isomorphisms on homology, so the symmetric group part is a homology direct summand."],"supporting_citations":[{"why":"Introduces the inductive-resolution technique and the model resolution that Proposition 4.8 follows.","marker":"[1]"},{"why":"Provides the reason Shapiro's lemma fails for non-flat algebra inclusions and motivates the inductive method in this setting.","marker":"[2]"},{"why":"The Brauer-algebra homology paper whose Shapiro-type argument is adapted in the proof of Theorem 5.1.","marker":"[3]"},{"why":"Earlier attempt at Rook-Brauer homology assuming only $\\epsilon$ invertible; the argument was later found incorrect.","marker":"[4]"},{"why":"Proved the same main results under the additional assumption that $\\delta$ is invertible, and documented the flaw in the earlier attempt.","marker":"[7]"},{"why":"Nakaoka's sharp homological stability for symmetric groups, which turns Theorems 2.1 and 2.2 into stability statements.","marker":"[8]"},{"why":"Supplies the basis description (Proposition 3.4) used to identify the induced modules $RBr_n \\otimes_{RBr_m} 1$.","marker":"[9]"}],"fun_headline_variants":["Rook-Brauer homology equals symmetric group when epsilon invertible","Motzkin homology vanishes; Rook-Brauer homology stabilizes","Inductive resolution shows Rook-Brauer homology matches symmetric group","Homological stability proven for Rook-Brauer and Motzkin algebras","Rook-Brauer homology reduces to symmetric group; Motzkin homology trivial"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on the claim that certain modules built from diagrams split off as clean direct pieces of simpler modules, with the splitting given by multiplying on the right by specific idempotent elements such as $\\epsilon^{-1}T_x$; those splitting checks are stated briefly, and if any fails, the vanishing result and both main theorems collapse.","fun_headline_variants_meta":{"raw":{"variants":["Rook-Brauer homology equals symmetric group when epsilon invertible","Motzkin homology vanishes; Rook-Brauer homology stabilizes","Inductive resolution shows Rook-Brauer homology matches symmetric group","Homological stability proven for Rook-Brauer and Motzkin algebras","Rook-Brauer homology reduces to symmetric group; Motzkin homology trivial"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000569,"raw_usage":{"total_tokens":2627,"prompt_tokens":814,"completion_tokens":1813,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":430,"completion_tokens_details":{"reasoning_tokens":1716}},"tokens_in":430,"tokens_out":1813,"duration_ms":13123,"temperature":1.0,"reasoning_tokens":1716,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:20:08.554680+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $\\mathrm{Tor}^{RBr_2}_1(1,1)$ over $R=\\mathbb{Z}$ with $\\epsilon=1$ and, say, $\\delta=0$; Theorem 2.1 predicts $H_1(S_2)=\\mathbb{Z}/2$. A more targeted check is to test Lemma 4.6 directly: for $X=\\{1,2\\}$ in $RBr_2$, verify that right multiplication by $\\epsilon^{-1}T_1$ sends $J_{\\{2\\}}$ into itself and is idempotent on the quotient; a single diagram violating this would break the inductive step.","supporting_citations":[{"cited_title":"1, 1-27, available athttps://msp.org/pjm/2023/327-1/p01.xhtml","cited_arxiv_id":null,"evidence_quote":"Introduces the inductive-resolution technique and the model resolution that Proposition 4.8 follows."},{"cited_title":"3, 1437-1499, available athttps://msp.org/gt/2024/28-3/gt-v28-n3-p11-s.pdf","cited_arxiv_id":null,"evidence_quote":"Provides the reason Shapiro's lemma fails for non-flat algebra inclusions and motivates the inductive method in this setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The Brauer-algebra homology paper whose Shapiro-type argument is adapted in the proof of Theorem 5.1."},{"cited_title":"Ann.391(2025), 2173–2207, available at https://link.springer.com/article/10.1007/s00208-024-02960-3","cited_arxiv_id":null,"evidence_quote":"Earlier attempt at Rook-Brauer homology assuming only $\\epsilon$ invertible; the argument was later found incorrect."},{"cited_title":"Cohomology of rook-Brauer algebras and their subalgebras","cited_arxiv_id":"2412.14887","evidence_quote":"Proved the same main results under the additional assumption that $\\delta$ is invertible, and documented the flaw in the earlier attempt."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Nakaoka's sharp homological stability for symmetric groups, which turns Theorems 2.1 and 2.2 into stability statements."},{"cited_title":"Noetherianity of Diagram Algebras","cited_arxiv_id":"2409.10885","evidence_quote":"Supplies the basis description (Proposition 3.4) used to identify the induced modules $RBr_n \\otimes_{RBr_m} 1$."}],"review_version":1}