{"id":"4fd4f3ae-7cc2-4a9a-8c5a-9fe993505a32","arxiv_id":"2505.07806","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new left-moving family of colored lattice models is proven solvable and dual to the right-moving family, with the crystal limit yielding a Schützenberger involution bijection on Gelfand-Tsetlin patterns.","lead":"The authors construct a new family of left-moving colored lattice models and prove they are dual to existing right-moving models, including models for Iwahori and metaplectic Whittaker functions. The duality refines in the crystal limit to a bijection of states governed by the Schützenberger involution, and it supplies the missing fourth R-matrix left open in earlier work.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unreleased SageMath check in Lemma 3.8 is load-bearing for the left-right duality; until it is supplied the proof is incomplete.","rationale":"The reader's weakest_assumption identifies the same load-bearing concern: Lemma 3.8's undocumented SageMath verification. I examined possible alternatives and found no stronger objection. Lemma 3.6 is terse but its structural argument is plausible; the decisive gap is that the finite check it enables is not reproducible from the manuscript. I checked the train argument in Theorem 3.10 and the powers of z in Lemmas 3.11 and 3.12; they are consistent. The crystal section is supported by explicit bijections and does not inherit the SageMath gap. Thus the appropriate status remains conditional: the central partition-function duality should be accepted only after the computational check is released and independently run. If the script reveals a failure or an incomplete case enumeration, the verdict would need to move to REJECT or UNVERDICTED; if the check passes, ACCEPT becomes appropriate.","tokens_in":40008,"tokens_out":6712,"duration_ms":65799,"concrete_test":"Release the SageMath verification used in Lemma 3.8 as an ancillary file and run it from a clean session: for each combination of row types in Theorem 3.1 not involving RR_L, with palette size m = 3 and the Iwahori specialization, enumerate all admissible configurations with fixed boundary spins and symbolically verify the Yang-Baxter equalities (3.2) and (3.3) in C(z1,z2,z3,v). Additionally, run an exhaustive search with m = 4 on at least one RTT and one RRR equation to test Lemma 3.6: if any admissible state has an internal color absent from the boundary, the m = 3 reduction is incomplete and the check must be rerun on the full state space.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.10 is derived from solvability via the train argument in Lemma 3.11, and solvability is Theorem 3.1. The proof of Theorem 3.1 is reduced, by Corollary 3.5 and Lemma 3.6, to a finite verification, and Lemma 3.8 then states: 'We have now reduced the problem to checking a fixed, finite number of equalities ... which we have verified with a symbolic computer algebra system (SageMath).' No script, output, or independent derivation is included. This computational assertion is exactly the load-bearing step: if any of the finitely many Yang-Baxter equalities is false, or if Lemma 3.6's boundary-color reduction missed a configuration, then Theorem 3.1 and hence Theorem 3.10 lack proof. The rest of the paper is coherent: Proposition 3.7 is a parameter-transfer argument, the train argument has no visible normalization error, and the crystal-limit results in Section 4 are proved combinatorially and do not depend on Lemma 3.8. The concern is reproducibility, not plausibility.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs a new family of solvable, colored lattice models, the left-moving counterparts to the right-moving family studied in previous work, and proves that the two families have equal partition functions (the left-right duality, Theorem 3.10). The equality is established by Yang–Baxter equations that mix left- and right-moving rows, proved in Theorem 3.1. The paper then specializes to the Iwahori models and takes a crystal limit, where the partition-function duality is refined to a state-by-state weight-respecting bijection. In this limit the row swaps are identified with Berenstein–Kirillov involutions on Gelfand–Tsetlin patterns, and the composition of these swaps is shown to be the Schützenberger involution (Theorem 4.10). The paper also addresses problems raised in prior work by constructing the missing mixed R-matrix in the non-crystal setting and explaining the failure of the crystal limit to retain it.","tokens_in":40259,"tokens_out":2263,"duration_ms":24036,"significance":"If the results are correct, this is a substantial contribution to the theory of colored lattice models and to combinatorial representation theory. The left-right duality generalizes the known Gamma-Delta duality for metaplectic ice, unifies the treatment of Iwahori and metaplectic Whittaker models, and provides a new left-moving model for Demazure characters. The refinement to a state-by-state bijection in the crystal limit, with the individual steps identified as Berenstein–Kirillov/Bender–Knuth involutions and the total map as the Schützenberger involution, is an elegant and nontrivial result. The proof of Theorem 4.10 is combinatorial and self-contained, and the train argument in Section 3.3 is standard and clearly presented. The paper is also explicit about its relationship to prior work, including the questions raised in [BBF11b] and [BS22]. The main weakness is that the proof of the central solvability theorem rests on a finite SageMath verification that is neither documented nor supplied.","major_comments":[{"comment":"The proof of Theorem 3.1, and hence of the central duality Theorem 3.10, depends on the assertion in Lemma 3.8 that a finite number of rational-function equalities in C(z1,z2,z3,v) have been verified with SageMath. No script, input file, output, or independent derivation is included. Because Lemma 3.6 reduces the Yang–Baxter equations to these finite checks, an unverified computational assertion here is load-bearing: if any of the finitely many equalities were false, or if the reduction in Lemma 3.6 missed a configuration, Theorems 3.1 and 3.10 would lack proof. I request that the authors supply the SageMath code and the output of the verification as supplementary material, or replace the finite check with a human-readable proof. The rest of the paper's arguments appear coherent, but this gap must be closed before the claim of full solvability can be accepted.","section":"Section 3, Lemma 3.8"},{"comment":"Theorem 3.9, which establishes solvability of the crystal models used in Section 4, is derived as a limit of Theorem 3.1. Consequently it inherits the dependence on the unverified SageMath check in Lemma 3.8. Even though the crystal-limit results in Section 4 are proved combinatorially and may be independently checkable, the statement that the crystal RTT- and RRR-equations hold for all row types relies on the same computational assertion. The authors should clarify this dependency and include the verification for the limiting case as part of the requested supplementary material.","section":"Section 3.2, Theorem 3.9"}],"minor_comments":[{"comment":"The statement of Lemma 3.8 contains a grammatical error: 'for a particular the Iwahori specialization' should read 'for a particular Iwahori specialization' or 'for the particular Iwahori specialization'.","section":"Section 3, Lemma 3.8"},{"comment":"In the proof of Proposition 3.7, the phrase 'It is easy to verify by hand as in [BBBG24b]' leaves some details of the ϕ-factor equality to the reader. Since the paper explicitly notes in footnote 1 that a related case was previously omitted in [BBBG24b], the authors should either display the verification or provide a short appendix with the conservation-equation computation for the RHS of (3.10).","section":"Section 3, Proposition 3.7"},{"comment":"The convention that column numbers are numbered from right to left starting at 0 in the unfused model (Section 2.2) is stated, but the figures in Section 2.4 and later sometimes display columns from left to right without an explicit arrow. Adding a coordinate axis or a clarifying remark to Figure 4 and Figure 6 would improve readability.","section":"Section 2.4"},{"comment":"In the proof of Theorem 4.10, the phrase 'the resulting patterns in (4.22) and (4.23) would only consist of a single column with only the top generator' is slightly ambiguous when i = r-1; it may help to spell out that the bottom row of the short pattern is absent and that the monoid word then reduces to the indicated generator.","section":"Section 4.3"}],"recommendation":"major_revision","confidential_remarks":"The missing SageMath certification is the single blocking issue; it is a reproducibility problem that seems fixable by supplying the code and output. I would encourage the editor to ask for this as a condition of revision. The mathematical architecture of the paper is otherwise convincing, and the combinatorial results in Section 4 appear solid. There is some overlap with the authors' prior work, but the left-right duality for the full family and the Schützenberger refinement are clearly new and within the scope of the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Gustafsson–Westerlund is a serious contribution. The new left-moving family, the mixed R-matrices, and the crystal-limit refinement of the duality to the Schützenberger involution are genuinely new and mostly well argued. The main thing to ask for before accepting is the SageMath verification in Lemma 3.8: it is not included, and the proof of Theorem 3.1 reduces to it.\n\nWhat works. The paper does a lot of things right. The left-moving model is not a mirror of the right-moving one; the partition function equality is non-trivial and the train argument in Lemma 3.11 seems sound. Section 4 is particularly nice: the bijection between mixed states and Gelfand–Tsetlin patterns is proved directly, and the Berenstein–Kirillov/Bender–Knuth refinement is a clean combinatorial result. The appendix proving Proposition 3.4 is detailed and, as far as I can tell, correct. The paper also addresses the previously missing fourth R-matrix in the crystal limit, giving a useful response to a question from Buciumas–Scrimshaw. The citation pattern is honest, with clear separation of what is new from what comes from BBBG24b and earlier work.\n\nSoft spots. The load-bearing step is Lemma 3.8. The reduction to a finite check via Lemma 3.6 is clever, but the actual check is only described as “verified with SageMath,” with no script, output, or independent derivation. That is a real reproducibility gap. If any of the finitely many rational identities is wrong, or if Lemma 3.6 missed a configuration, Theorems 3.1 and 3.10 lack proof. This is not a plausibility problem – I have no reason to think the check fails – but it is a completeness problem. It should be fixed before publication, either by releasing the script or by writing out the finite verification. It is finite, after all.\n\nBottom line: this paper deserves a serious referee, and with the verification supplied I would accept it. As is, I would not rely on Theorem 3.10 without independently checking Lemma 3.8.","headline":"New left-moving lattice models and a nice crystal-limit refinement, but the main duality theorem currently rests on an unreleased SageMath check in Lemma 3.8.","tokens_in":40778,"tokens_out":2533,"would_cite":true,"duration_ms":24723,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82B23","16T25","05E10","05A19","05E05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs a new family of solvable left-moving colored lattice models, proves their partition functions equal the right-moving family, and shows the crystal-limit refinement is the Schützenberger involution.","keywords":["solvable lattice models","colored six-vertex models","Gamma-Delta duality","Iwahori Whittaker functions","metaplectic ice","Gelfand-Tsetlin patterns","Berenstein-Kirillov involutions","Schützenberger involution"],"falsifier":"For $m=4$, evaluate the left and right sides of the RTT equation (3.2) with row types $X=L$, $Y=R$, boundary colors $\\{c_1,c_2,c_3\\}$ on the six boundary edges, and search for an admissible state whose internal vertical edge carries $c_4$; exhibiting such a state, or finding a nonzero difference between the two sides, would disprove Lemma 3.6 and therefore the solvability proof, since the paper's reported finite symbolic check is not included.","tokens_in":39804,"feed_emoji":"🧊","tokens_out":7995,"duration_ms":69404,"temperature":0.7,"pith_summary":"This paper constructs a new family of solvable colored lattice models whose paths move down and left, and proves that its partition functions equal those of the existing right-moving family for all boundary data. The equality, stated as Theorem 3.10, covers the metaplectic ice models and the Iwahori Whittaker model, and supplies the previously missing left-moving Iwahori model. The proof goes through Yang–Baxter equations that mix left- and right-moving rows, including a mixed R-matrix whose dependence on the number of colors cancels by telescoping sums. In the crystal limit $v\\to 0$, the duality refines to a weight-respecting bijection of states: row swaps act as Berenstein–Kirillov involutions on Gelfand–Tsetlin patterns, and the composition of all swaps is the Schützenberger involution on semistandard Young tableaux.","feed_headline":"New left-moving ice models match the right-moving ones","feed_subtitle":"The duality becomes the Schützenberger involution on tableaux in the crystal limit.","key_machinery":"The central object is a family of six-vertex colored lattice models: a right-moving family (paths move down and right) and a new left-moving family (paths move down and left), defined first in an expanded $m$-column form and then fused into blocks. Solvability is carried by Yang–Baxter equations of two kinds, RTT and RRR, with R-matrices $R^L_L$, $R^R_R$, $R^L_R$, $R^R_L$ mixing row types; the train argument repeatedly applies these equations to swap adjacent rows. The crystal-limit refinement uses Gelfand–Tsetlin patterns whose row-pair inequalities are left-strict or right-strict according to the row type, and the Berenstein–Kirillov involutions $t_i$, defined by reflecting an entry in its admissible interval, which are transferred to lattice-model states and shown to swap row types and boundary colors. The composition of these involutions in the order of the longest element $w_0$ is the Schützenberger involution on semistandard Young tableaux.","core_discovery":"On its own terms, the paper's discovery is a duality of partition functions: for any top-boundary data $\\mu$, any horizontal boundary colors $\\sigma$, and row parameters $z$, the right-moving partition function equals $z^N$ times the left-moving partition function with reversed boundary colors and reversed row parameters, $Z^R_{\\mu,\\sigma}(z)=z^N Z^{L,N}_{\\mu,w_0\\sigma}(w_0 z)$, and this holds for the metaplectic and Iwahori specializations. In the crystal limit of the Iwahori specialization, the duality becomes a genuine bijection of states $S^\\Theta_{\\lambda+\\rho,\\sigma}\\to S^{s_i\\Theta}_{\\lambda+\\rho,s_i\\sigma}$ given by the Berenstein–Kirillov involutions on Gelfand–Tsetlin patterns; applying the involutions in the order dictated by the proof of Theorem 3.10 yields the Schützenberger involution. The paper also proves the whole family is solvable: all four types of R-matrices satisfy the RTT and RRR Yang–Baxter equations, resolving a question left open for the alternating Gamma/$\\Delta$ crystal models used for type B and C Demazure characters.","pith_inferences":["The non-crystal duality does not refine to a state bijection; following the packet idea noted in the introduction, one could try to define duality packets as invariants under the Drinfeld twists that interpolate between specializations, and the crystal-limit bijection would be the singleton case.","Because the crystal-limit R-matrix $R^\\Gamma_\\Delta$ degenerates to a single nonzero weight, the alternating type-B/C models may need a different mechanism at nonzero $v$; the telescoping color-loop sums in Proposition 3.4 suggest where to look.","The Coxeter-monoid flag action used to track boundary colors could generalize to other Cartan types or to higher-rank crystals, where the Schützenberger involution is replaced by the corresponding canonical involution.","The omission of the symbolic verification script means an immediate testable extension is to expose the finite three-color checks in a computer algebra file; until then the finite check is an asserted computational fact."],"forward_implications":["Theorem 3.10 yields a left-moving Iwahori ice model with the same partition function as the known right-moving model, which the paper identifies as the missing dual needed for a future metaplectic-Iwahori Whittaker duality.","The same theorem reproduces the Γ-∆ duality for metaplectic ice as a special case of one uniform left-right equality.","In the crystal limit, the partition-function equality upgrades to a state-by-state weight-respecting bijection, so the Demazure character and atom model now has a left-moving counterpart.","The Yang–Baxter solvability result supplies all four R-matrices for the alternating Γ/∆ models, providing the fourth R-matrix that was missing in the quasi-solvable models for type B and C Demazure characters.","The boundary colors transform by simple transpositions under each Berenstein–Kirillov step, so the full duality is compatible with horizontal boundary conditions at the level of states."],"supporting_citations":[{"why":"Defines the right-moving family and its metaplectic ∆′ model; the paper's left-right duality is proved against this family.","marker":"[BBBG24b]"},{"why":"Introduces the Iwahori ice model and the fusion lemma (Lemma 5.4) that transfers unfused Yang–Baxter equations to fused models.","marker":"[BBBG24a]"},{"why":"Establishes the right-moving crystal model for Demazure characters and atoms of type A, which the paper extends with a left-moving dual.","marker":"[BBBG21]"},{"why":"Proves the metaplectic Γ-∆ Yang–Baxter duality and supplies the train-argument template referenced in Theorem 3.10.","marker":"[BBB19]"},{"why":"Contains the original Γ-∆ duality and the packet construction for Gelfand–Tsetlin patterns that the crystal-limit bijection refines to singletons on special subsets.","marker":"[BBF11b]"},{"why":"Defines Berenstein–Kirillov involutions and the factorization of the Schützenberger involution used in Theorem 4.10.","marker":"[KB95]"},{"why":"Identifies the Berenstein–Kirillov involutions with Bender–Knuth involutions on semistandard Young tableaux, connecting the lattice states to tableaux.","marker":"[BK72]"},{"why":"Constructs quasi-solvable alternating Γ/∆ models for Sp_{2n} and SO_{2n+1}; the missing fourth R-matrix is the problem the paper's solvability theorem resolves in the non-crystal setting.","marker":"[BS22]"}],"fun_headline_variants":["New ice models match old ones, duality becomes Schutzenberger involution","New solvable lattice models dual to known ones, crystal limit: Schutzenberger","Duality of ice models becomes Schutzenberger involution on tableaux","Solving lattice models: new duality and Schutzenberger connection","Metaplectic and Iwahori lattice models unified by new duality"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument leans on the claim that in every Yang–Baxter equation without $R^R_L$ vertices, any color on an internal edge already appears on the boundary, reducing the proof to a finite three-color check; that finite check is reported but its computer verification is not included.","fun_headline_variants_meta":{"raw":{"variants":["New ice models match old ones, duality becomes Schutzenberger involution","New solvable lattice models dual to known ones, crystal limit: Schutzenberger","Duality of ice models becomes Schutzenberger involution on tableaux","Solving lattice models: new duality and Schutzenberger connection","Metaplectic and Iwahori lattice models unified by new duality"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00081,"raw_usage":{"total_tokens":3629,"prompt_tokens":1093,"completion_tokens":2536,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":709,"completion_tokens_details":{"reasoning_tokens":2438}},"tokens_in":709,"tokens_out":2536,"duration_ms":15582,"temperature":1.0,"reasoning_tokens":2438,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:07:46.585777+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $m=4$, evaluate the left and right sides of the RTT equation (3.2) with row types $X=L$, $Y=R$, boundary colors $\\{c_1,c_2,c_3\\}$ on the six boundary edges, and search for an admissible state whose internal vertical edge carries $c_4$; exhibiting such a state, or finding a nonzero difference between the two sides, would disprove Lemma 3.6 and therefore the solvability proof, since the paper's reported finite symbolic check is not included.","supporting_citations":[],"review_version":1}