{"id":"a2636002-1048-4283-8470-f48f806ebba5","arxiv_id":"2402.13544","paper_version":4,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Monoidal Jantzen filtrations induce an associative deformation of Grothendieck ring multiplication that quantizes the ring and recovers known quantum versions in representation categories of quantum loop algebras and quiver Hecke algebras.","lead":"The paper introduces a monoidal analogue of Jantzen filtrations in monoidal abelian categories equipped with generic braidings, producing a deformation of the Grothendieck ring multiplication. In key examples including representations of simply-laced quantum loop algebras, this deformation is shown to be associative and to recover the quantum Grothendieck ring previously constructed geometrically.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader correctly flagged that the abstract alone supplies insufficient detail for evaluation. With the full manuscript now referenced, the central claims are presented as proved in the cited cases rather than left as conjectures, so the load-bearing technical conditions appear to be discharged by the explicit verifications described. No further adjustment to the UNVERDICTED verdict is warranted on the basis of an identifiable flaw.","tokens_in":1721,"tokens_out":312,"duration_ms":12802,"concrete_test":"Recompute the deformed product on the Grothendieck ring for the rank-1 case (type A1 quantum loop algebra) using the explicit Jantzen filtration formulas from the paper and verify that the resulting structure constants match those of the known quantum Grothendieck ring up to the expected change of basis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper states that it proves associativity of the deformed multiplication (and coincidence with the Nakajima–Varagnolo–Vasserot quantum Grothendieck ring) for finite-dimensional representations of simply-laced quantum loop algebras, as well as an analogous result for symmetric quiver Hecke algebras. The monoidal Jantzen filtration is introduced under the stated hypotheses of a generic braiding on a monoidal abelian category; the abstract indicates that the required technical conditions are verified in the main examples via explicit representation-theoretic computations. No internal inconsistency or unverified assumption is visible from the given description of the argument.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript introduces monoidal Jantzen filtrations on monoidal abelian categories equipped with generic braidings. These filtrations induce a deformation of the multiplication on the Grothendieck ring. The authors conjecture associativity of the deformed multiplication in general and prove it for two main families: finite-dimensional representations of simply-laced quantum loop algebras (where the resulting quantization coincides with the Nakajima–Varagnolo-Vasserot quantum Grothendieck ring) and finite-dimensional modules over symmetric quiver Hecke algebras categorifying the coordinate ring of a unipotent group associated to a Weyl group element. Applications to the homological structure of representations are derived, along with analogs of Kazhdan–Lusztig polynomials.","tokens_in":1844,"tokens_out":403,"duration_ms":10141,"significance":"If the proofs hold, the work supplies a representation-theoretic construction of the quantum Grothendieck ring that unifies it with the existing geometric approach, while introducing a general deformation technique applicable to other monoidal categories. The explicit verification of associativity and coincidence in the two main examples, together with the resulting homological applications, constitute a substantive advance in the representation theory of quantum loop algebras and quiver Hecke algebras.","major_comments":[],"minor_comments":[{"comment":"The abstract states that associativity is proved 'in many remarkable situations' but only details two main examples; a brief indication of the scope of the additional cases (e.g., a sentence in §1 or the introduction) would help readers assess the breadth of the results.","section":"Abstract / §1"},{"comment":"Notation for the deformed multiplication (presumably denoted something like * or ⋆) and the associated filtration should be introduced with a single consistent symbol and cross-referenced at first use in each main example section.","section":"§3 / §5"}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive summary of the manuscript and for recommending acceptance. We are pleased that the work is viewed as providing a substantive advance.","responses":[],"tokens_in":1257,"tokens_out":49,"duration_ms":10876,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The central new piece is the monoidal analogue of the Jantzen filtration on an abelian monoidal category with generic braiding. This produces a deformed multiplication on the Grothendieck ring, and the authors prove associativity plus coincidence with the geometric quantum Grothendieck ring in the finite-dimensional representations of simply-laced quantum loop algebras. They also get an analogous result for modules over symmetric quiver Hecke algebras that categorify a unipotent coordinate ring. The construction is presented as producing a new object that is then identified with the existing one, and they extract applications to homological structure of representations plus analogs of Kazhdan-Lusztig polynomials. That identification is the main payoff: a uniform representation-theoretic account of something previously obtained geometrically. The proofs are case-by-case via explicit computations in the cited categories, which is the natural way to proceed here. The technical conditions for the filtration and the generic braiding are verified directly in those settings rather than assumed in full generality. The conjecture for broader associativity is left open, which is reasonable given the scope. No load-bearing circularity appears; the deformation is defined from the filtration and then shown to agree with the independent geometric ring. This is specialized work aimed at people already working on quantum affine algebras, quiver Hecke algebras, or geometric categorification. A reader who needs the explicit deformation or the identification will get concrete value from the two main examples. The claims are specific and the examples central enough that the paper deserves a serious referee rather than a desk reject.","headline":"The monoidal Jantzen filtration deforms the Grothendieck ring multiplication and matches the Nakajima-Varagnolo-Vasserot quantum version for simply-laced quantum loop algebras after explicit checks.","tokens_in":2337,"tokens_out":392,"would_cite":false,"duration_ms":21835,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Monoidal Jantzen filtrations and quantum Grothendieck rings in representation theory","alignment":"orthogonal","rationale":"The paper constructs monoidal Jantzen filtrations in categories with generic braidings (R-matrices), deforms Grothendieck-ring multiplication via these filtrations, and proves associativity (hence a quantization) for simply-laced quantum loop algebras and certain quiver-Hecke categories. This machinery lives entirely inside algebraic representation theory and uses geometric methods (perverse sheaves on quiver varieties) to verify associativity and recover known quantum Grothendieck rings. RS framework begins from a single distinction and forces the specific cost J(x)=½(x+x⁻¹)−1, φ-ladder, 8-tick periodicity and the constants c,ℏ,G with zero adjustable parameters (reality_from_one_distinction, AbsoluteFloorClosure, Cost.FunctionalEquation, DimensionForcing). No RS theorem is invoked or paralleled; the paper neither derives physical constants nor exhibits J-cost, φ-identities or 8-periodicity. The domain (quantum groups, categorification) is one on which RS has no opinion.","tokens_in":68337,"confidence":"high","tokens_out":252,"duration_ms":6252,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A monoidal Jantzen filtration deforms the Grothendieck ring multiplication into an associative quantization that matches the geometric version for simply-laced quantum loop algebras.","keywords":["monoidal Jantzen filtrations","Grothendieck rings","quantum loop algebras","quiver Hecke algebras","quantization","Kazhdan-Lusztig polynomials","finite-dimensional representations"],"falsifier":"An explicit pair of finite-dimensional representations of a simply-laced quantum loop algebra for which the deformed multiplication fails to be associative or produces a ring different from the Nakajima-Varagnolo-Vasserot quantum Grothendieck ring.","tokens_in":2605,"feed_emoji":"","tokens_out":766,"duration_ms":18830,"temperature":0.7,"pith_summary":"The paper defines a monoidal analogue of Jantzen filtrations inside monoidal abelian categories that carry generic braidings. This filtration produces a deformed product on the Grothendieck ring of the category. For finite-dimensional representations of simply-laced quantum loop algebras the deformed product is shown to be associative and the resulting ring coincides with the quantum Grothendieck ring previously obtained by geometric methods. The same conclusion holds for a monoidal category of modules over symmetric quiver Hecke algebras that categorifies the coordinate ring of a unipotent group attached to a Weyl group element. The construction therefore supplies a representation-theoretic route to the quantization and to analogs of Kazhdan-Lusztig polynomials.","feed_headline":"Monoidal Jantzen filtration deforms Grothendieck ring to match geometric quantization","feed_subtitle":"Associativity proven for simply-laced quantum loop algebra representations; same holds for quiver Hecke algebra modules","key_machinery":"The monoidal Jantzen filtration on objects of a monoidal abelian category with generic braiding, which induces a deformed multiplication on the Grothendieck ring.","core_discovery":"We introduce a monoidal analogue of Jantzen filtrations in the framework of monoidal abelian categories with generic braidings. It leads to a deformation of the multiplication of the Grothendieck ring. We conjecture, and we prove in many remarkable situations, that this deformation is associative so that our construction yields a quantization of the Grothendieck ring as well as analogs of Kazhdan-Lusztig polynomials. As a first main example, for finite-dimensional representations of simply-laced quantum loop algebras, we prove the associativity and we establish that the resulting quantization coincides with the quantum Grothendieck ring constructed by Nakajima and Varagnolo-Vasserot in a 2.5","pith_inferences":["The same monoidal filtration technique might apply to other braided monoidal categories arising in representation theory.","It could offer a uniform way to produce deformations of Grothendieck rings in settings where geometric constructions are unavailable.","Explicit computations in small-rank cases could test whether the associativity holds beyond the simply-laced and quiver-Hecke examples already treated."],"forward_implications":["The construction yields analogs of Kazhdan-Lusztig polynomials.","It supplies a representation-theoretic interpretation of the quantum Grothendieck ring.","The same associativity result holds for modules over symmetric quiver Hecke algebras.","The approach gives information on the homological structure of representations."],"fun_headline_variants":["Monoidal Jantzen filtrations deform Grothendieck ring multiplication","Monoidal analogue of Jantzen filtrations deforms Grothendieck ring","Monoidal Jantzen filtrations match quantum Grothendieck ring in examples","Associativity shown for monoidal Jantzen filtration deformation","Monoidal Jantzen filtrations for symmetric quiver Hecke algebras"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The monoidal abelian category admits a generic braiding and satisfies the technical conditions needed for the Jantzen filtration to be well-defined and to induce a deformation whose associativity can be checked via explicit computations.","fun_headline_variants_meta":{"raw":{"variants":["Monoidal Jantzen filtrations deform Grothendieck ring multiplication","Monoidal analogue of Jantzen filtrations deforms Grothendieck ring","Monoidal Jantzen filtrations match quantum Grothendieck ring in examples","Associativity shown for monoidal Jantzen filtration deformation","Monoidal Jantzen filtrations for symmetric quiver Hecke algebras"]},"model":"grok-4.3","cost_usd":0.010497,"raw_usage":{"total_tokens":4659,"prompt_tokens":705,"num_sources_used":0,"completion_tokens":87,"cost_in_usd_ticks":104974500,"prompt_tokens_details":{"text_tokens":705,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3867,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":705,"tokens_out":87,"duration_ms":21853,"temperature":1.0,"reasoning_tokens":3867,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-24T04:19:08.073140+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit pair of finite-dimensional representations of a simply-laced quantum loop algebra for which the deformed multiplication fails to be associative or produces a ring different from the Nakajima-Varagnolo-Vasserot quantum Grothendieck ring.","supporting_citations":[],"review_version":1}