{"id":"3d9d2d26-7577-4aea-b3c8-48a2d13f0699","arxiv_id":"2505.09483","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A Mackey formula is proved for the critical cohomology of equivariant Landau-Ginzburg models, giving the cohomology the structure of a localized induction-restriction system.","lead":"This paper proves a structural compatibility formula, a Mackey formula, between induction and restriction maps on the critical cohomology of a representation of a reductive group with an invariant potential. The result gives these cohomology spaces a coherent induction-restriction system, a useful organizing tool for enumerative and derived geometry.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 7.6 rests on the unproved identity (7.1): the normal-bundle Euler class computation relating Res and Ind is asserted but not carried out. The Mackey sum's coefficients depend on it, so the proof is conditional until (7.1) is derived.","rationale":"Reading in good faith, the paper aims to establish a Mackey-type compatibility formula for critical cohomology of equivariant Landau-Ginzburg models, and the overall structure is plausible: induction and restriction morphisms are defined carefully, the Coxeter complex formalism is coherent, and the vanishing-potential examples in Section 9 are consistent with the formula. The most load-bearing point is precisely in Section 7.4: after rewriting both sides using torus-equivariant induction and restriction, the proof declares that (7.1) follows from diagram (7.2) and an Euler-class calculation, but the calculation is not shown. Since the double-coset sum in Theorem 7.6 is obtained by matching those Euler factors term by term, a missing sign or an extra w-dependent factor would directly change the theorem. This is an internal completeness concern, not an accusation of error; the cited results in [Hen24a, Proposition 5.12] and [Kin24] are reasonable support for the sheaf-level identification in Section 4.4, so I do not rest the objection on that secondary point. The Section 9 checks are positive evidence, but they specialize to f = 0, where the critical sheaf is constant and Euler-class bookkeeping is far simpler; they therefore do not test (7.1) in the general vanishing-cycle setting. My recommendation is to keep the reader's conditional verdict: the theorem is credible and well-motivated, but acceptance should require a complete derivation of (7.1), or a precise reference carrying out the normal-bundle Euler class computation.","tokens_in":42511,"tokens_out":4959,"duration_ms":52798,"concrete_test":"Independently expand the proof of Theorem 7.6 by deriving (7.1) from the Cartesian squares in (7.2) via the excess-intersection formula for lci pullback/pushforward in vanishing-cycle cohomology, explicitly computing Eu_{V,w·C,F} and Eu_{V,C'∘w·C,⟨C'⟩} as products over weights and tracking all Tate twists. Verify that after applying Ind^{C'}_{C'∘w·C} and restricting to the relevant localized modules, the identity holds exactly; if any extra factor such as k_{C',F}/k_{C',⟨C'⟩} or a sign appears, the Mackey sum must be corrected. This check uses only definitions already in the manuscript and does not require new external results.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 1.6/7.6: for C,C' ⪯ F, Res^{C'}_F ∘ Ind^F_C equals the sum over W_{⟨C'⟩}\\W_F/W_{⟨C⟩} of Ind ⋯ τ ⋯ Res ⋯ (ẇ·−). The proof reduces the entire double-coset expansion to identity (7.1), stated without derivation. The sentence “Then, (7.1) follows from this diagram and the calculation of the Euler classes of the normal bundles of the closed immersions p_{w·C,F} and p_{C'∘w·C,⟨C'⟩}” is the only justification; no calculation is displayed. This identity is load-bearing because it is exactly what matches the Euler-class denominators Eu_{V,w·C,F}, Eu_{g,C',F}, k_{C',F}, and k_{C'∘w·C,⟨C'⟩} in the shuffle expansion; an omitted sign or w-dependent factor would change the coefficient of each double-coset term and the formula would fail as stated. This is not an external disagreement: (7.1) is internal to the proof and is not cited to [Hen24a] or [Kin24]. The Section 9 examples are in the vanishing-potential polynomial setting and do not exercise the vanishing-cycle normal-bundle computation for nontrivial f, so they do not settle it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops an induction–restriction formalism for the critical cohomology of equivariant Landau–Ginzburg models attached to a representation V of a reductive group G and an invariant function f. It introduces the Coxeter complex of (G,V), defines critical cohomological systems H^T'_{G,V,f,F} for flats F, constructs induction morphisms Ind^F_C and localized restriction morphisms Res^C_F, and states a Mackey-type formula expressing Res^{C'}_{⟨C''⟩} ∘ Ind^{⟨C''⟩}_C as a sum over double cosets of compositions of induction, braiding, restriction, and Weyl action. A parallel 2d induction system is defined using Borel–Moore homology of zero loci and compared with the critical system via dimensional reduction. The paper ends with explicit GL2 examples with vanishing potential and with a partial description of the image of the 2d-to-shuffle comparison morphism.","tokens_in":42764,"tokens_out":4242,"duration_ms":47074,"significance":"If the main theorem is correct, the paper provides a genuine structural result: the critical cohomology spaces H^T'_{G,V,f,F} become the stalks of a localized induction–restriction system, with a Mackey formula that generalizes the localized coproduct of Davison and connects to Langlands-type constant-term formulae and to perverse-sheaf descriptions of hyperplane arrangements. The formal framework is careful: the Coxeter complex, Tits product, Euler-class localizations, braiding operators, and double-coset combinatorics are set up precisely, and the GL2 examples give concrete, checkable instantiations of the formula in the f=0 case. There are no fitted parameters and no ad hoc numerical inputs; the claimed identities are functorial. The main caveat is that the central proof is conditional on a normal-bundle Euler class identity that is not derived in the manuscript, and on external structural results whose precise hypotheses are not stated. These gaps are local in nature but they concern the core theorem, so they block acceptance in the current form.","major_comments":[{"comment":"The proof of Theorem 7.6 reduces the Mackey formula to the identity Eu_{V,w·C,F} Ind^{C'}_{C'∘w·C} i^* f = i'^* Eu_{V,C'∘w·C,⟨C'⟩} Ind^F_{w·C} f, and then states that (7.1) follows from diagram (7.2) and the calculation of Euler classes of normal bundles of p_{w·C,F} and p_{C'∘w·C,⟨C'⟩}. No such calculation is actually shown. This identity is load-bearing because it is exactly what matches the Euler-class denominators in the shuffle expansion and fixes the coefficient of each double-coset term; an omitted sign or a w-dependent factor would change the formula. The Section 9 examples do not test it, since they restrict to f=0 and involve no nontrivial vanishing-cycle normal-bundle computation. The authors should provide a complete derivation of (7.1), including signs and the w-dependence, or give a precise reference that contains the calculation.","section":"§7.4, identity (7.1)"},{"comment":"The definition of the induction morphism Ind^F_C relies on the sentence that, using smoothness of q, properness of p, finiteness of ı, and the fact that π_F is APM (approachable by proper maps), one can canonically identify (ı_{⟨C⟩,F})_*(π_{⟨C⟩})_* φ(q)_* Q with (π_F)_* φ(p)_* Q. The precise hypotheses under which π_F is APM for the stacks V_F/(G_F×T') are not stated; the manuscript cites [Hen24a, Proposition 5.12] and [Kin24], but does not verify their hypotheses for the generality claimed here. This identification is what makes the sheafified induction compute the intended critical cohomology, so the authors should either state the needed structural result as an explicit assumption, prove it in this setting, or restrict the main theorem to the cases where it applies.","section":"§4.4, sheafified induction identification"},{"comment":"The definition of the torus equivariant 2d induction morphism contains the incomplete formula gInd^F_C := ?? · Ind^F_C, where the multiplier is left undefined. This is not a harmless typo: the subsequent average Ind'^F_C and the comparison in Proposition 5.6 depend on this multiplier, and the reader cannot check whether the claimed compatibility with the non-torus induction holds. The missing factor should be supplied explicitly and the comparison proof adapted accordingly.","section":"§5.5, torus equivariant 2d induction"}],"minor_comments":[{"comment":"The sentence fixing representatives of double cosets writes the quotient as W_{⟨C′⟩}\\W_{⟨C′′⟩}/W_{⟨C′⟩} in both occurrences; the second factor should be W_{⟨C⟩} to match Theorem 1.6.","section":"§1.3.6"},{"comment":"The conjecture is immediately followed by the admission that it 'does not hold as stated' and likely needs refinement. Since Corollary 8.12 only proves containment, the conjecture should be labeled as tentative and its status made explicit in the main text, so that it is not mistaken for a proved structural result.","section":"§8, Conjecture 8.13"},{"comment":"The examples only treat the case of vanishing potential f=0, where the induction and restriction morphisms reduce to shuffle formulas for Weyl-group invariants. A short sentence clarifying that these examples do not exercise the vanishing-cycle and normal-bundle Euler-class computations in the proof of Theorem 7.6 would help calibrate what is verified.","section":"§9"},{"comment":"There are occasional grammatical slips and notational inconsistencies, e.g. 'The restriction morphism are defined' in §1.2.2 and the repeated 'opposite cell' phrasing in §6; these should be corrected in a final polish.","section":"§1.8 and throughout"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is likely true and the framework is valuable, but the proof of Theorem 7.6 currently depends on an unproved Euler-class identity (7.1), and the induction morphisms rely on external APM results whose hypotheses are not checked. These are fixable within the manuscript's scope, but they are load-bearing, so a major revision is appropriate. The 2d section also contains an undefined multiplier '??' that must be repaired."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, the paper is real: it constructs restriction morphisms and proves a Mackey formula for critical cohomology of V/G with invariant potential, in a general reductive-group setting. The Coxeter-complex formalism and the braiding operators are new at this level, and the GL2 examples show the shuffle formula working. Second, the load-bearing step is not yet demonstrated. Theorem 7.6 reduces the entire Mackey double-coset expansion to identity (7.1), and the text says it “follows from this diagram and the calculation of the Euler classes of the normal bundles” without displaying the calculation. The stress-test note is right: any sign or w-dependent factor in that Euler class would change the coefficients of the double-coset terms. This is internal to the proof, not imported from elsewhere.\n\nWhat I like: the author is explicit about what reduces to Davison’s localized coproduct and what is new. The comparison theorem with 2d induction systems is substantial. The examples in Section 9 are concrete and honest, though they are all vanishing-potential cases, so they do not exercise the nontrivial-f version of (7.1). I also appreciate the self-caveat on Conjecture 8.13; not many papers flag their own conjecture as likely false as stated.\n\nSoft spots, in proportion. The gap in (7.1) is the main issue, and it is not minor. A referee needs a full derivation or a precise reference. The sheafified identification in Section 4.4 imports approachability by proper maps via [Kin24, Hen24a]; I would like the hypotheses stated explicitly so the reader knows exactly when the induction morphism computes what is claimed. This is a dependency issue more than a flaw. The self-citations to [Hen24b, Hen24a] are legitimate; the new result genuinely builds on them.\n\nBottom line: the framework and statement are worth serious referee time. I would not desk-reject, and I would send it out with a request to expand §7.4. I would not cite the theorem as established until (7.1) is written out; as written, the result is conditional.","headline":"A credible and valuable Mackey-formula framework for critical cohomology of quotient stacks, but the proof of the main theorem currently asks the referee to take the key normal-bundle Euler class identity (7.1) on faith.","tokens_in":43317,"tokens_out":1964,"would_cite":false,"duration_ms":21195,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14L30","14F43","55N91"],"pacs":[],"model":"deepseek-v4-flash","headline":"Induction and restriction on quotient stacks obey a Mackey formula","keywords":["Mackey formula","critical cohomology","equivariant Landau-Ginzburg model","quotient stack","induction-restriction system","Coxeter complex","vanishing cycles","cohomological Donaldson-Thomas theory"],"falsifier":"Carry out the normal-bundle Euler-class computation in identity (7.1) for a configuration with more than one double coset, for instance $\\mathrm{GL}_3$ acting on $\\mathbf{C}^3$ with a cubic invariant; a single mismatch between the asserted and computed Euler factors would refute Theorem 7.6.","tokens_in":42264,"feed_emoji":"🧮","tokens_out":10036,"duration_ms":95566,"temperature":0.7,"pith_summary":"This paper establishes a commutation rule between two families of maps — parabolic induction and restriction — acting on the critical cohomology of a quotient stack $V/G$ equipped with a $G$-invariant function $f$. The rule is a cohomological Mackey formula: restricting after inducing is the same as summing, over Weyl-group double cosets, inductions conjugated by braiding isomorphisms and Weyl translations. If correct, this makes the critical cohomology spaces into a localized induction-restriction system, and supplies the geometrically correct restriction counterpart to the induction maps used in cohomological Donaldson–Thomas theory. The paper also relates this to 2d induction systems through cohomological dimensional reduction, and checks the formula explicitly when $f$ vanishes.","feed_headline":"Induction and restriction on quotient stacks obey a Mackey formula","feed_subtitle":"Critical cohomology of equivariant Landau-Ginzburg models becomes a localized induction-restriction system.","key_machinery":"The machinery is the Coxeter complex of $(G,V)$: the real Cartan space $\\mathfrak{h}_{\\mathbb{R}}$ cut by hyperplanes $\\alpha=0$ for weights $\\alpha$ of $V$ and of the adjoint representation, whose cells $C$ and flats $F$ are ordered by reverse inclusion and carry a Tits product $C\\circ C'$. Around it, the paper organizes induction morphisms $\\operatorname{Ind}_C^F$ built from an induction diagram $V_{\\langle C\\rangle} \\leftarrow V_{C\\geqslant 0,F} \\rightarrow V_F$, restriction morphisms $\\operatorname{Res}_C^F$ defined after localizing by Euler classes $\\mathrm{Eu}_{V,C,F}$, and braiding isomorphisms multiplying by kernels $k_{C,F}/k_{C',F}$. The Mackey formula is proved by rewriting both sides in torus-equivariant cohomology, averaging over Weyl groups, and reducing to the normal-bundle Euler-class identity (7.1).","core_discovery":"On the paper's own terms, the central discovery is Theorem 1.6: for cells $C, C' \\preceq C''$ in the Coxeter complex of $(G,V)$, the composition $\\operatorname{Res}_{\\langle C''\\rangle}^{C'} \\circ \\operatorname{Ind}_{C}^{\\langle C''\\rangle}$ equals a sum over double cosets of inductions, braidings, restrictions, and Weyl translations. Concretely, $$\n\\operatorname{Res}_{\\langle C''\\rangle}^{C'} \\circ \\operatorname{Ind}_{C}^{\\langle C''\\rangle}\n=\n\\sum_{w \\in W_{\\langle C'\\rangle}\\backslash W_{\\langle C''\\rangle}/W_{\\langle C\\rangle}}\n\\operatorname{Ind}_{C'\\circ(\\dot w\\cdot C)}^{C'}\n\\circ \\tau_{\\dot w\\cdot C\\circ C'}^{C'\\circ(\\dot w\\cdot C)}\n\\circ \\operatorname{Res}_{\\langle \\dot w\\cdot C\\rangle}^{\\dot w\\cdot C\\circ C'}\n\\circ (\\dot w\\cdot -),\n$$ where $\\tau$ is a braiding isomorphism given by multiplication by a ratio of Euler-class kernels. The formula is proved through torus-equivariant induction and restriction, using a double-coset bijection and the Euler-class identity (7.1), and it upgrades the critical cohomology spaces into a localized induction-restriction system with associative induction and coassociative restriction.","pith_inferences":["This suggests the Mackey formula should extend from quotient stacks to global stacks, with component lattices replacing the Coxeter complex; the paper points to this as a further direction but does not prove it.","The Euler-class localization used to form the restriction is a fixed-point localization principle in disguise: critical cohomology on a flat should be recoverable from one-parameter-subgroup fixed loci, with braiding operators encoding wall-crossing between chambers.","In the quiver-representation setting, the formula should make the cohomological Hall product and the localized coproduct compatible; verifying that compatibility directly would give an independent check of Theorem 1.6."],"forward_implications":["When the potential vanishes, induction and restriction have explicit shuffle formulas in Weyl-group invariants; Section 9 verifies the Mackey formula on concrete $\\mathrm{GL}_2$ examples.","The newly defined restriction morphisms repair the earlier restriction used for cohomological integrality: induction and restriction now form a compatible pair satisfying the Mackey formula.","The critical cohomological system is a localized induction-restriction system, with associative induction, coassociative restriction, and the Mackey formula controlling their interaction.","Cohomological dimensional reduction relates the 2d Borel–Moore induction system to the critical 3d system, so the same Mackey formalism applies in both settings up to explicit signs.","For quiver representation spaces, the restriction morphisms realize the localized coproduct of the cohomological Hall algebra, so the Mackey formula supplies the product-coproduct compatibility."],"supporting_citations":[{"why":"Supplies the localized restriction/coproduct construction, the dimensional reduction isomorphism, and the Atiyah–Bott-style injectivity argument the paper adapts.","marker":"[Dav17]"},{"why":"Defines the parabolic data $V_\\lambda, G_\\lambda$ and the induction morphisms on $H^*(V/G)$ that this paper extends and corrects with compatible restrictions.","marker":"[Hen24b]"},{"why":"Provides the approachability-by-proper-maps result used to identify sheafified induction on good moduli spaces.","marker":"[Hen24a]"},{"why":"Supplies the decomposition-type input needed for the good moduli space map $\\pi_F$ in the sheaf-level identification of Section 4.4.","marker":"[Kin24]"},{"why":"Supplies the Tits product and the formal induction-restriction framework that organizes the Coxeter complex combinatorics.","marker":"[Kap+24]"},{"why":"States the finite-group Mackey formula whose double-coset structure the theorem generalizes.","marker":"[Web16]"}],"fun_headline_variants":["Mackey formula for critical cohomology on quotient stacks","Induction-restriction on stacks obeys a Mackey formula","Euler-class braidings give Mackey formula for LG models","Critical cohomology as localized induction-restriction system","Cohomological Mackey formula proved for quotient stacks"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is identity (7.1), an asserted normal-bundle Euler-class computation: if those Euler factors do not match, the double-coset expansion in the Mackey formula collapses.","fun_headline_variants_meta":{"raw":{"variants":["Mackey formula for critical cohomology on quotient stacks","Induction-restriction on stacks obeys a Mackey formula","Euler-class braidings give Mackey formula for LG models","Critical cohomology as localized induction-restriction system","Cohomological Mackey formula proved for quotient stacks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000848,"raw_usage":{"total_tokens":3655,"prompt_tokens":878,"completion_tokens":2777,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":494,"completion_tokens_details":{"reasoning_tokens":2693}},"tokens_in":494,"tokens_out":2777,"duration_ms":19205,"temperature":1.0,"reasoning_tokens":2693,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:30:15.865603+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Carry out the normal-bundle Euler-class computation in identity (7.1) for a configuration with more than one double coset, for instance $\\mathrm{GL}_3$ acting on $\\mathbf{C}^3$ with a cubic invariant; a single mismatch between the asserted and computed Euler factors would refute Theorem 7.6.","supporting_citations":[],"review_version":1}