{"id":"c7b1b75d-d711-4cc2-b9b0-c354f9a1c0ed","arxiv_id":"2501.03198","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper gives an A-infinity functor strategy to compute Namikawa-Weyl groups for quiver varieties with symplectic resolutions, and uses it to reproduce Yaochen Wu's classification.","lead":"This paper develops a categorical method for computing Namikawa-Weyl groups, the symmetry groups that govern deformations of symplectic singularities, and applies it to quiver varieties. The main output is a case-by-case list of these groups, which reproduces a result previously obtained by Yaochen Wu.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 6.1's proof asserts F is a strict A-infinity equivalence, but Section 3 only proves a faithful embedding; fiberwise surjectivity onto pi^{-1}(L) is unproved, so the monodromy classification may be incomplete.","rationale":"The central claim is the classification of Namikawa-Weyl groups in Theorems 6.2 and 7.2, and the mechanism is Proposition 6.1, which identifies the bundle pi^{-1}(L) -> L with (M x E)/Gamma -> M/Gamma using the functor F. The proof of Proposition 6.1 depends on F being a strict A-infinity equivalence on the relevant twisted completions. This is not what is proved: Proposition 3.1 gives only a strict faithful embedding, and Appendix B establishes preservation of honest modules and stability but not essential surjectivity. The sentence 'Since any theta-polystable representation ... lies in Tw{S1,...,Sk}' only places such a representation in the target category, not in the image of F. A faithful embedding can miss objects; geometrically, missing a theta-stable module over a leaf would mean the corresponding projective line or singular point in the exceptional fiber is not accounted for, so the monodromy group computed in Theorem 6.2 could be too small. This is an internal gap, not a disagreement with prior consensus. It is likely fillable: the local-to-global comparison is plausible and the final classification recovers Yaochen Wu's theorem. Appendix C.3 explicitly skips non-generic stability cases for Proposition 5.1, which is a further incompleteness, but the F-equivalence gap is the more fundamental obstruction. For these reasons the conditional verdict is retained without change.","tokens_in":27497,"tokens_out":6144,"duration_ms":60446,"concrete_test":"In the Appendix A example (Q with alpha=(1,2,1), theta=(-1,-1,3), local quiver A3 with theta'=(-3,+1,+1,+1)), enumerate all theta-stable representations in pi^{-1}(L), including the middle one-parameter family corresponding to the middle node of A3, and list all points e in the local exceptional fiber E. Compute F(m,e) for each pair and check whether every theta-stable representation in pi^{-1}(L) is isomorphic to F(m,e) for some (m,e). If any of the three families is missed, Proposition 6.1(2) is false; if all are hit, repeat the test on a D4 leaf, and in any case supply a general essential-surjectivity argument before relying on Theorem 6.2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 6.1(2) is not supported by the established properties of F. In the proof, the author states that the functor F_{S1,...,Sk}: Tw{tilde S1,...,tilde Sk} -> Tw{S1,...,Sk} is a strict A-infinity equivalence; however Section 3 and Proposition 3.1 only establish a strict faithful unital embedding, and Lemma B.3 only shows that subrepresentations of F(X) lie in the essential image. The subsequent step -- \"Since any theta-polystable representation in pi^{-1}([S1,...,Sk]) lies in Tw{S1,...,Sk}, the map F: E -> pi^{-1}([S1,...,Sk]) is also surjective\" -- is invalid: lying in the target category does not imply lying in the essential image of a faithful embedding. If some theta-polystable module over the leaf is not F(e) for any e in the local exceptional fiber E, then the bundle identification in Proposition 6.1 fails and the monodromy action computed in Theorem 6.2 may omit contributions from those fibers. No separate proof of essential surjectivity onto pi^{-1}(L) is supplied. Appendix C.3 likewise announces that non-generic stability cases are skipped for Proposition 5.1, but the fundamental gap is the unproved equivalence in Proposition 6.1.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a categorical strategy for computing Namikawa-Weyl groups of quiver varieties, aiming to recover and systematize Yaochen Wu's results. The strategy is to use a local-to-global A-infinity functor F from twisted complexes over the Kleinian local quiver to twisted complexes over the original quiver, to identify the exceptional fiber bundle over each codimension-2 leaf, and to read off monodromy as the action of the labeled local quiver symmetry group on the Kleinian exceptional fiber. The main results are Theorem 6.2, which lists the Weyl group for each local type (A_n with possible identifications, D_n with possible identifications, E_6, E_7, E_8), and Theorem 7.2, which reduces the general case to the canonical decomposition via products and symmetric products. The paper also states a generalization to moduli spaces of representations of arbitrary Calabi-Yau-2 algebras.","tokens_in":27747,"tokens_out":3598,"duration_ms":34099,"significance":"If the proof gaps are repaired, the paper would provide a genuinely useful computational framework: it gives an explicit, parameter-free recipe that reduces Namikawa-Weyl groups of quiver varieties to a finite list of local symmetry computations, and it connects the categorical Calabi-Yau-2 structure to deformation-theoretic Weyl groups. The paper is honest in flagging several unsupported assumptions and omitted cases, but those admissions currently block the central theorem. The claimed recovery of Wu's theorem is a falsifiable benchmark, and the case-by-case tables in Theorem 6.2 and Appendix C are concrete and testable. The absence of fitted parameters and the presence of explicit representation-theoretic descriptions of exceptional fibers are strengths.","major_comments":[{"comment":"The proof of fiberwise bijectivity of F is not supported by the established properties of F. The text states that F_{S1,...,Sk}: Tw{tilde S1,...,tilde Sk} -> Tw{S1,...,Sk} is a strict A-infinity equivalence, but Section 3 and Proposition 3.1 establish only a strict faithful unital embedding. Lemma B.3(3) shows only that subrepresentations of F(X) lie in the essential image of F, not that every object in Tw{S1,...,Sk}, or every theta-polystable module in the target fiber, is isomorphic to F(Y) for some Y. The subsequent sentence, \"Since any theta-polystable representation in pi^{-1}([S1,...,Sk]) lies in Tw{S1,...,Sk}, the map F:E -> pi^{-1}([S1,...,Sk]) is also surjective,\" is therefore invalid: lying in the target category does not imply belonging to the essential image of a faithful embedding. This gap is load-bearing because Proposition 6.1(2) is the identification of the fiber bundle, and Proposition 6.1(3) and Theorem 6.2 inherit it. The author needs either to prove essential surjectivity onto pi^{-1}(L), or to construct the inverse bundle map by a separate argument, or to restrict the claims to the essential image.","section":"Section 6, proof of Proposition 6.1(2)"},{"comment":"The proof of Proposition 5.1 is incomplete for non-generic localized stability parameters. Appendix C.3 explicitly says \"we decide to treat only two specific cases\" and \"we therefore skip all the other cases,\" yet Proposition 5.1 concludes \"This exhausts all options.\" No argument is supplied that the two treated partial resolutions (A3 with theta'=(+1,-1,+1,-1) and D4 with theta'=(-1,-1,-1,-1,+2)) cover all non-generic theta' that can arise from a pseudo-generic global stability parameter and a primitive symmetry in Figure 4.1. Since Proposition 5.1 is used in Proposition 6.1(3) to identify the monodromy action, the omitted cases could change the claimed Weyl groups in Theorem 6.2. The paper needs either a case analysis showing that all other combinations are impossible, or an explicit treatment of each remaining combination, or a precise statement restricting Theorem 6.2 to the cases actually proved.","section":"Section 5 and Appendix C.3, Proposition 5.1"},{"comment":"The cyclicity of the A-infinity model of Mod Pi_Q is assumed without proof. The text states \"We are not aware of a specific proof of the cyclicity in the literature, but it appears to be well-known among experts and we shall simply assume it.\" This assumption is used in the construction of F in the proof of Proposition 3.1, where the A-infinity structures on the two categories are identified by matching bases of Hom spaces, and in the formality statement of Theorem 2.13. Because the local-to-global functor is the central technical tool, an unproved structural property of the ambient A-infinity category is load-bearing. The author should either supply a proof or a precise citation for cyclicity of the minimal model of Mod Pi_Q, or state explicitly that the main results depend on this assumption.","section":"Section 2.5, Theorem 2.12"}],"minor_comments":[{"comment":"The phrase \"Goreinstein singularities\" should read \"Gorenstein singularities.\"","section":"Introduction, paragraph 2"},{"comment":"The displayed isomorphism writes the symmetric products as S^{n1} M(Q, alpha_k) x ... x S^{nk} M(Q, alpha_k); this should presumably be S^{n1} M(Q, alpha_1) x ... x S^{nk} M(Q, alpha_k).","section":"Section 2.3, Theorem 2.5"},{"comment":"The phrase \"explicit explicit description\" contains a duplicated word.","section":"Before Theorem 6.2"},{"comment":"The displayed stability parameter in the D4 case is written both as theta'=(-1,-1,+5,-1,-1) in the text and as theta'=(+1,+1,-5,+1,+1) in Figure C.4; the sign convention should be reconciled.","section":"Appendix C.1, D4 title"},{"comment":"In the final sentence of the D4 discussion, \"the two singular points correspond to the outer three nodes\" appears to be a typo; the preceding sentence says there are three singular points, so this should likely read \"the three singular points correspond to the outer three nodes.\"","section":"Appendix C.3, D4 case"}],"recommendation":"major_revision","confidential_remarks":"The paper overlaps considerably with Yaochen Wu's prior theorem, and the novel contribution is the categorical method rather than the final classification. The main gap in Proposition 6.1 is a genuine essential surjectivity problem that the current text does not resolve. If the author can supply the missing equivalence or a substitute argument, the paper would be a worthwhile contribution; in its present state, the central theorem is not fully supported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this paper's advertised classification is not new — it recovers Yaochen Wu's theorem — and the proof as written has one load-bearing gap plus a few unproved structural assumptions. But the method is genuinely different from Wu's and, if repaired, could give a template for other Calabi-Yau-2 moduli spaces. Worth a serious referee, not a desk reject.\n\nWhat's actually new: the local-to-global A-infinity functor F, the bundle identification (M × E)/Γ → π^{-1}(L), and the symmetry analysis of labeled local quivers. The paper does a lot of honest, explicit work in the appendices: the exceptional fiber computations for A3, D4, D_n, and the proof that the naive symmetry group equals the automorphism group of the labeled local quiver (Proposition 4.1) look correct and are useful. The stability-preservation lemma (Lemma 3.2) is also careful given its assumptions. The writing is clear and the organization is logical.\n\nSoft spots. The big one: the proof of Proposition 6.1(2) asserts that F is a 'strict A-infinity equivalence' on twisted completions, but Section 3 and Proposition 3.1 only establish a faithful embedding. The step 'since any θ-polystable representation in π^{-1}(...) lies in Tw{S1,...,Sk}, the map F is surjective' does not follow. Lying in the target category does not imply being in the essential image of a faithful embedding. Without fiberwise surjectivity of F onto π^{-1}(L), the bundle identification and hence the monodromy computation are not established. This is exactly the stress-test concern, and it lands.\n\nThere are also two explicitly admitted assumptions: Theorem 2.12 assumes cyclicity of the A∞-structure on Mod ΠQ ('we shall simply assume it'), and after Theorem 2.13 the paper assumes that Davison's formal model agrees with the cyclic model. These might be fillable, but they are structural assumptions, not minor details. And Appendix C.3 only treats two non-generic stability cases while Proposition 5.1 claims all cases; the paper says 'we skip all the other cases.' That is another gap, though perhaps less load-bearing because the two cases are the 'most singular' ones.\n\nThe classification itself matches Wu's theorem, so the final answer is almost certainly correct. The issue is that the proof's central mechanism isn't yet rigorous. That said, the gaps look identifiable and potentially repairable; the strategy is coherent and the paper is honestly written.\n\nBottom line: send it to a competent referee. The referee should demand a proof of essential surjectivity of F (or a different argument for fiberwise bijectivity) and a resolution of the cyclicity/formality compatibility. If those get fixed, the paper would be a solid contribution as a method paper. For my own work, I would not cite it yet; I would cite Wu for the classification and wait for the revised version.","headline":"A new method for Namikawa-Weyl groups of quiver varieties, but the proof has a load-bearing gap in the local-to-global functor and the classification itself is not new.","tokens_in":28263,"tokens_out":3490,"would_cite":false,"duration_ms":30382,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G20","14E15","16E45"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every quiver variety with a symplectic resolution has its Namikawa-Weyl group determined by local quiver symmetries.","keywords":["Namikawa-Weyl group","quiver varieties","symplectic singularities","Calabi-Yau-2 algebras","A-infinity categories","monodromy","Kleinian singularities","preprojective algebras"],"falsifier":"Compute the fiber $\\pi^{-1}(x)$ over a point $x$ of an $A_3$-type codimension-2 leaf and compare its dimension with the exceptional fiber of the local quiver; Proposition 6.1 predicts equality, so any mismatch — or any $\\theta$-polystable representation in $\\pi^{-1}(x)$ not isomorphic to $F(X)$ for honest $X$ — falsifies the bundle identification. Independently, writing down the cyclic $A_\\infty$-structure asserted in Theorem 2.12 and checking compatibility with the formal model would settle whether the hom-space computations used to define $F$ are valid.","tokens_in":27239,"feed_emoji":"📐","tokens_out":20004,"duration_ms":160985,"temperature":0.7,"pith_summary":"This paper aims to compute the Namikawa-Weyl group — the finite group that governs how the universal Poisson deformations of a symplectic singularity and of its symplectic resolution are related — for every quiver variety $M(Q,\\alpha)$ that admits a symplectic resolution. The proposed strategy is categorical: encode the local model of each codimension-2 leaf by a Kleinian quiver, then use an $A_\\infty$-functor $F$ from the local vertex simples to the global simples to identify the entire exceptional fiber bundle over the leaf. Once the bundle is identified, the monodromy around loops in the leaf is exactly the action of the symmetry group of the labeled local quiver on the Kleinian exceptional fiber, and the Weyl group is read off as the invariant part of the classical Weyl group under that action. The main theorem lists the resulting Weyl groups case by case ($A_n$, $C_2$, $B_{n-1}$, $D_n$, $G_2$, $E_6$, $E_7$, $E_8$) and gives a product formula for canonical decompositions, matching the answer previously obtained by direct deformation-theoretic comparison.","feed_headline":"Namikawa-Weyl groups computed from local quiver symmetries","feed_subtitle":"A local-to-global functor reads the monodromy of each codimension-2 leaf off the symmetry group of a Kleinian quiver.","key_machinery":"The load-bearing object is the local-to-global $A_\\infty$-functor $F$ — a functor between categories with higher composition products that preserves those products — from $\\operatorname{Tw}\\{\\widetilde S_1,\\dots,\\widetilde S_k\\}\\to\\operatorname{Tw}\\{S_1,\\dots,S_k\\}$, built by matching the vertex simples of the local Kleinian quiver $(Q',\\alpha')$ with a chosen collection of global simples of dimensions $\\beta_1,\\dots,\\beta_k$. The $A_\\infty$-structure is the Calabi-Yau-2 structure on the module category of the preprojective algebra $\\Pi_Q$, with higher products vanishing by formality for $\\Sigma$-sequences, so the hom-spaces between simples are determined by the combinatorial dimension formula $\\dim\\operatorname{Ext}^1(M,N)=\\dim\\operatorname{Hom}(M,N)+\\dim\\operatorname{Hom}(N,M)-(\\dim M,\\dim N)$. The functor preserves honesty of twisted complexes and stability with respect to localized stability parameters. Bundling $F$ over the parameter space $M=\\prod_i M_s(Q,\\beta_i)\\setminus\\Delta$ and taking the quotient by the label-symmetry group $\\Gamma$ identifies the global exceptional bundle with the local one, reducing monodromy to the action of $\\Gamma$ on the Kleinian exceptional fiber $E$.","core_discovery":"The central claim is that for any quiver setting $(Q,\\alpha)$ with $\\alpha\\in\\Sigma_{0,0}$, indivisible or of type $(2,2)$, the Namikawa-Weyl group of $M(Q,\\alpha)$ is determined by the codimension-2 leaves as follows. For a leaf $L$ with isotropic decomposition $\\alpha=\\sum_i m_i\\beta_i$, the reduced local quiver $(Q',\\alpha')$ is a Kleinian quiver, and the group $\\Gamma=\\operatorname{Aut}(Q',\\alpha',(\\beta_1,\\dots,\\beta_k))$ of label-preserving quiver automorphisms acts on the exceptional fiber $E$ of the local resolution by Dynkin automorphisms. Proposition 6.1 identifies the global exceptional bundle $\\pi^{-1}(L)\\to L$ with the quotient bundle $(M\\times E)/\\Gamma\\to M/\\Gamma$ via the local-to-global functor $F$, so the monodromy homomorphism $\\pi_1(L)\\to\\operatorname{Aut}(E)$ factors through $\\Gamma$. Consequently the Weyl group associated to the leaf is the $\\Gamma$-invariant part of the classical Weyl group of the local Kleinian type, and Theorem 6.2 lists the possibilities: $A_n$ (with $C_2$ in the $A_3$ cases where opposite roots coincide), $D_n$ (with $B_{n-1}$ or $G_2$ in the $D_4$ and $D_{\\ge5}$ identification cases), and $E_6,E_7,E_8$ when all roots are distinct. Theorem 7.2 extends this to arbitrary $\\alpha$ via the canonical decomposition: $W(M(Q,\\alpha)) = \\prod_i W(M(Q,\\alpha_i))$, with symmetric powers contributing no new factor. The monodromy computation agrees with the earlier deformation-theoretic computation, so the paper presents an independent route to the same Weyl groups.","pith_inferences":["Extension: if the functor $F$ is later shown to be full, the same bundle identification would prove the analogous Weyl-group formula for arbitrary Calabi-Yau-2 moduli spaces; the paper sketches the setup but does not complete that proof.","Extension: the classification suggests a converse pattern — non-simply-laced factors $C_2$, $B_{n-1}$, $G_2$ arise exactly when root identifications collapse a simply-laced local Dynkin diagram; testing this against deformation-theoretic computations on more examples would clarify the scope.","Extension: comparing the partial-resolution monodromy of Appendix E with the full-resolution Dynkin automorphism on a leaf outside the two model cases would test whether the 'naive monodromy' shortcut is a general phenomenon or an artifact of those examples."],"forward_implications":["For a leaf of type $A_3$ with one pair of opposite roots identified, the Weyl group is $C_2$; if both pairs are identified it remains $C_2$.","For a $D_4$ leaf, one identified pair gives $B_3$, one identified triple gives $G_2$, and two identified pairs give $B_3$; for $D_n$ with $n\\ge5$, any identification of one or two outer pairs gives $B_{n-1}$.","The monodromy of the resolution over a codimension-2 leaf depends only on which simples are permuted by the loop, not on the loop's path inside the leaf, because the map $\\pi_1(L)\\to\\Gamma$ forgets the trajectory.","For a quiver variety with canonical decomposition $\\alpha=\\sum_i n_i\\alpha_i$, the Namikawa-Weyl group is the direct product $\\prod_i W(M(Q,\\alpha_i))$; in particular, replacing a factor by its $n_i$-fold symmetric product does not change the Weyl group.","The same categorical local-to-global strategy applies to moduli spaces of representations of arbitrary Calabi-Yau-2 algebras whose codimension-2 strata have the analogous product form, giving a template for computing Namikawa-Weyl groups beyond quiver varieties."],"supporting_citations":[{"why":"supplies the symplectic stratification of quiver varieties, the local quiver construction, and the criterion for existence of symplectic resolutions that defines the class of varieties studied.","marker":"[3]"},{"why":"gives the theorem that universal Poisson deformation bases are linked by a Galois covering and the semi-explicit product formula for the Namikawa-Weyl group that the paper computes.","marker":"[12]"},{"why":"describes the exceptional fibers of Kleinian singularities through socles, which identifies the action of quiver automorphisms on the fiber in Proposition 5.1.","marker":"[6]"},{"why":"proves vanishing of higher $A_\\infty$ products for $\\Sigma$-sequences in Calabi-Yau-2 categories, letting the paper compute the $A_\\infty$-structure used to define $F$.","marker":"[8]"},{"why":"supplies the Calabi-Yau-2 property of preprojective algebras, the categorical framework in which the local-to-global functor is constructed.","marker":"[7]"},{"why":"is the earlier deformation-theoretic computation of the Namikawa-Weyl groups that the paper's monodromy calculation recovers.","marker":"[13]"},{"why":"provides the root-and-canonical-decomposition input used in the product theorem and in the enumeration of codimension-2 leaves.","marker":"[5]"}],"fun_headline_variants":["Weyl groups from leaf symmetries of quiver varieties","Local-to-global functor yields Weyl group for quiver settings","Namikawa-Weyl groups via Kleinian quiver automorphisms","Quiver Weyl groups read off codimension-2 leaf symmetries","Symplectic singularity Weyl groups from local quiver data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The bundle identification assumes that every module over a codimension-2 leaf is produced by the local-to-global functor $F$ from the local Kleinian model, and the paper also assumes without a written proof that the cyclic $A_\\infty$-model it computes with exists and agrees with the formal model; if either premise fails, the monodromy computation could be incomplete.","fun_headline_variants_meta":{"raw":{"variants":["Weyl groups from leaf symmetries of quiver varieties","Local-to-global functor yields Weyl group for quiver settings","Namikawa-Weyl groups via Kleinian quiver automorphisms","Quiver Weyl groups read off codimension-2 leaf symmetries","Symplectic singularity Weyl groups from local quiver data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000871,"raw_usage":{"total_tokens":3836,"prompt_tokens":1075,"completion_tokens":2761,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":691,"completion_tokens_details":{"reasoning_tokens":2669}},"tokens_in":691,"tokens_out":2761,"duration_ms":19511,"temperature":1.0,"reasoning_tokens":2669,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:52:47.229339+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the fiber $\\pi^{-1}(x)$ over a point $x$ of an $A_3$-type codimension-2 leaf and compare its dimension with the exceptional fiber of the local quiver; Proposition 6.1 predicts equality, so any mismatch — or any $\\theta$-polystable representation in $\\pi^{-1}(x)$ not isomorphic to $F(X)$ for honest $X$ — falsifies the bundle identification. Independently, writing down the cyclic $A_\\infty$-structure asserted in Theorem 2.12 and checking compatibility with the formal model would settle whether the hom-space computations used to define $F$ are valid.","supporting_citations":[{"cited_title":"Poisson deformations of affine symplectic varieties II","cited_arxiv_id":null,"evidence_quote":"gives the theorem that universal Poisson deformation bases are linked by a Galois covering and the semi-explicit product formula for the Namikawa-Weyl group that the paper computes."},{"cited_title":"On the exceptional fibres of Kleinian singularities","cited_arxiv_id":null,"evidence_quote":"describes the exceptional fibers of Kleinian singularities through socles, which identifies the action of quiver automorphisms on the fiber in Proposition 5.1."},{"cited_title":"Purity and 2-Calabi-Yau categories","cited_arxiv_id":"2106.07692","evidence_quote":"proves vanishing of higher $A_\\infty$ products for $\\Sigma$-sequences in Calabi-Yau-2 categories, letting the paper compute the $A_\\infty$-structure used to define $F$."},{"cited_title":"On deformed preprojective algebras","cited_arxiv_id":null,"evidence_quote":"supplies the Calabi-Yau-2 property of preprojective algebras, the categorical framework in which the local-to-global functor is constructed."},{"cited_title":"Namikawa-Weyl groups of affinizations of smooth Nakajima quiver vari- eties","cited_arxiv_id":null,"evidence_quote":"is the earlier deformation-theoretic computation of the Namikawa-Weyl groups that the paper's monodromy calculation recovers."},{"cited_title":"Geometry of the moment map for representations of quivers","cited_arxiv_id":null,"evidence_quote":"provides the root-and-canonical-decomposition input used in the product theorem and in the enumeration of codimension-2 leaves."}],"review_version":1}