{"id":"72bb747e-ca61-4dc5-aa98-359fcbb23756","arxiv_id":"2506.20038","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For odd d and any d-admissible signature, the mixed Plücker ring R_sigma(V) is shown to equal a cluster algebra A_sigma built from Demazure weaves.","lead":"This paper constructs natural cluster algebra structures on mixed Grassmannians, which are configuration spaces of vectors and covectors, for odd dimensions and a broad class of orderings. The result unifies and extends earlier constructions for Grassmannians, tensor diagrams, and mixed plabic graphs.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The cited classification theorem for Demazure weaves may not cover the marked-boundary variant used to define A_sigma, so the well-definedness of the cluster algebra could rest on an unproved hypothesis.","rationale":"The reader identified the classification theorem and the amalgamation/algebraic-independence lemmas as the weakest assumptions. I agree that the construction is externally load-bearing, but I sharpen the concern: the paper's own Remark 2.38 says its Demazure weaves differ from the literature by having marked boundary vertices on the top boundary. The classification theorem cited as Theorem 2.62 is asserted for weaves 'with the same marked boundary vertices', but it is not demonstrated that the cited theorem in [Cas+24] covers exactly this marking convention. If it does not, then Theorem 3.54 has no proof for the weaves used in Definition 3.58, and the independence of the cut (Proposition 3.65) collapses, making A_sigma undefined. This is not a claim that the paper is wrong; it is a precise hypothesis-checking concern that can be settled by reading the cited theorem and, if needed, by the explicit reduction test I propose. The deferred lemmas 3.63 and 4.52 also deserve scrutiny, but I do not see an evident circularity from the available text: the algebraic independence of the extended cluster appears to follow from the dimension count in Corollary 4.35 together with the later proof that the extended cluster contains all Weyl generators. Thus the most actionable and load-bearing issue is the applicability of the classification theorem to marked weaves. Because the concern is concrete and checkable, and because it concerns the definition of the central object rather than a minor technical step, I move the verdict from unconditional ACCEPT to CONDITIONAL: the acceptance should be contingent on verifying the marked-boundary version of Theorem 2.62 or supplying a proof of it.","tokens_in":74544,"tokens_out":4795,"duration_ms":59578,"concrete_test":"Check the exact statement of [Cas+24, Theorem 4.12] (or the version in [Cas+25]) for whether it permits arbitrary marked boundary vertices on the top boundary, where marked vertices are allowed to be origins of Lusztig cycles. If it does not, carry out the following reduction: add a small trivalent vertex immediately below each marked top boundary vertex, converting each marked-origin frozen cycle into an ordinary frozen cycle ending at the bottom boundary; then verify that this conversion commutes with the equivalence moves and mutation moves used in Theorem 2.62, and that the resulting unmarked weaves have the same top and bottom words.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The construction of A_sigma depends on Theorem 3.54, which says that any two reduced Demazure weaves with the same marked top word and same (reverse-paired) bottom word give mutation-equivalent seeds. This in turn relies entirely on the external classification theorem Theorem 2.62, cited from [Cas+24, Theorem 4.12]. However, the paper explicitly states in Remark 2.38 that its notion of Demazure weave differs from the literature by allowing a designated set of marked boundary vertices on the top boundary, which generate extra frozen cycles in Definition 2.44. If Theorem 2.62 was proved only for the unmarked or differently marked weaves of [Cas+24], then the mutation equivalence of the weaves w1 and w2 used in Definition 3.58 has not been established for the objects actually used here. Since Proposition 3.65 (independence of the valid cut) and Definition 3.66 both rely on this mutation equivalence, the central object A_sigma may not be well-defined unless the marked-boundary version of the classification theorem is supplied. The deferred technical lemmas (Lemma 3.63 and Lemma 4.52) are also load-bearing, but the marked-boundary hypothesis mismatch is the most concrete, externally checkable risk: it concerns whether a cited theorem applies to the paper's generalized weaves.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs, for any odd d, any d-admissible signature σ of type (a,b), and n = a+b > d^2, an explicit cluster algebra A_σ inside the fraction field of the mixed Plücker ring R_σ(V), and proves that R_σ(V) = A_σ. The construction cuts the cyclic word β_σ at a valid cut (p,q), builds two reduced Demazure weaves with reversed bottom words, amalgamates their seeds along common frozen variables, and shows the resulting cluster algebra is independent of the cut. The proof combines the classification of Demazure weaves (Theorem 2.62), a detailed analysis of an initial weave and its quiver, and two applications of results from [GLS13], including the Starfish lemma. The paper also recovers Scott's Grassmannian cluster structure, Fomin–Pylyavskyy's tensor-diagram structures for d = 3, and Carde's mixed plabic graph structures for separated signatures.","tokens_in":74792,"tokens_out":5230,"duration_ms":60828,"significance":"If the central result is correct, it settles the Fomin–Pylyavskyy conjecture for all odd dimensions and all d-admissible signatures under the mild size condition n > d^2, and it gives a unified weave-theoretic framework for previously disparate cluster structures. The paper is remarkable for its explicitness: the initial seed, the cluster and frozen variables, and the exchange relations are written down in closed form in Propositions 4.33 and 4.40, and the quiver mutations are described locally. The two proofs of the main theorem, one via [GLS13, Theorem 1.4] and one via the Starfish lemma, are conceptually clean. The manuscript does not ship machine-checked proofs, but the combinatorial arguments are organized so that the remaining verification is largely local and checkable.","major_comments":[{"comment":"The well-definedness of A_σ depends on the claim that any two reduced Demazure weaves with the same marked top word and reversed bottom words give mutation-equivalent seeds. This is Theorem 3.54, whose proof is entirely delegated to the external classification theorem Theorem 2.62, cited from [Cas+24, Theorem 4.12]. However, Remark 2.38 explicitly states that the paper's notion of Demazure weave differs from the literature by allowing a designated set of marked boundary vertices on the top boundary, which generate extra frozen cycles. If [Cas+24, Theorem 4.12] is proved only for the unmarked or differently marked weaves of that paper, then the mutation equivalence of the two weaves w1 and w2 in Definition 3.58 has not been established for the objects actually used here. Since Proposition 3.65 and Definition 3.66 both rest on this equivalence, the central object A_σ may not be well-defined unless a marked-boundary version of the classification theorem is supplied. The manuscript should either prove that version or give a precise statement of where [Cas+24] treats marked boundary vertices and how the marking is transported.","section":"§2.4, Remark 2.38; §3.3–3.4, Theorems 2.62 and 3.54, Definitions 3.58 and 3.66"},{"comment":"Definition 3.59 defines the amalgamated seed Σ_σ(p,q) only after Lemma 3.63, which asserts that the two seeds Σ_1 and Σ_2 can be amalgamated along z0, and Lemma 4.52, which asserts that the resulting extended cluster is algebraically independent. Both lemmas are deferred to later sections, and Remark 3.48 explicitly concedes that Σ(w) is not yet known to be a seed at that point. This is acceptable only if the later proofs do not themselves rely on Theorem 3.67 or on the well-definedness of A_σ. The paper should state the dependency graph explicitly and indicate the exact place where algebraic independence of the amalgamated cluster is proved without circularity.","section":"§3.4, Definition 3.59, Lemma 3.63; §4.4, Lemma 4.52"},{"comment":"The text states that for r = d−1 and i ∈ [2, m1−r−1] the local pictures are analogous to Propositions 4.39 and 4.40 and that their explicit examination is omitted. This family is not a negligible special case: the second proof of Theorem 3.67 invokes the once-mutated cluster variables calculated in Section 4.3 to apply the Starfish lemma, and the initial seed for the amalgamated cluster algebra includes the bottom strip. If the exchange relations for this family are not written down, the claim that all once-mutated cluster variables lie in R_σ is not fully verified. Please include the missing local pictures and exchange relations, or give a precise reduction showing that the omitted cases are covered verbatim by the formulas already proved.","section":"§4.3, after Proposition 4.40"}],"minor_comments":[{"comment":"The title contains a typo: 'Grassmanianns' should presumably be 'Grassmannians' or 'Grassmannian'.","section":"Title and Abstract"},{"comment":"In the 0-Hecke monoid relation, 'τ^2_i = τ' is missing a subscript; it should be 'τ_i^2 = τ_i'.","section":"§2.4, Definition 2.57"},{"comment":"The name 'Casals Roger' should be 'Roger Casals'.","section":"Acknowledgment"},{"comment":"The word 'troplicalizing' in the computation is a typo for 'tropicalizing'.","section":"§2.5, proof of Lemma 2.55"},{"comment":"The isomorphism R_σ(V) ≅ R_{a,b}(V) is stated but not proved; a one-line argument identifying the permutation of factors would help the reader.","section":"§2.2, Definition 2.21"}],"recommendation":"major_revision","confidential_remarks":"The paper is a serious and detailed contribution, and the reader's positive assessment is understandable. My recommendation of major_revision is driven by one externally checkable risk: Theorem 2.62 is cited for Demazure weaves with marked boundary vertices, but Remark 2.38 states that this marked-boundary variant is new. If the classification theorem in [Cas+24] does not cover the marked case, the object A_σ is not known to be well-defined. This is fixable within the manuscript's scope by adding a proof or a precise reference, but it is load-bearing. The omitted r = d−1 exchange relations in Section 4.3 should also be supplied before the Starfish lemma proof can be considered complete."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a substantial paper. It proves a new theorem: for odd d and d-admissible signatures, the mixed Plücker ring carries an explicit, natural cluster structure whose cluster and frozen variables include all Weyl generators. That is a genuine step beyond Fomin–Pylyavskyy (d=3) and Carde (separated signatures), and the worked d=3, n=8 example reproduces the expected T433 mutation type. The manuscript is serious and well organized: definitions are explicit, proofs are structured, and the author states the limitations (even d and the n>d^2 condition) honestly.\n\nThe construction is clever: cut the cyclic word into two halves, build reduced Demazure weaves on each half, amalgamate them along the reverse-paired bottom word, and use the classification theorem for Demazure weaves plus GLS13 to show the resulting cluster algebra equals the invariant ring. The decorated flag machinery and the mixed wedge notation are well chosen; the exchange relations reduce to familiar Plücker-type identities.\n\nThe main risk is the marked-boundary variant. Remark 2.38 explicitly says the paper's Demazure weaves differ from the literature by allowing marked boundary vertices on the top boundary, generating extra frozen cycles. Theorem 2.62, the classification of reduced Demazure weaves, is then cited as covering \"the same marked boundary vertices.\" If [Cas+24, Theorem 4.12] was proved only for unmarked or differently marked weaves, then the independence of the cut (Proposition 3.65) and the well-definedness of A_sigma do not follow as written. This is checkable and fixable, but it is the first thing a referee should verify. The deferred amalgamation and independence lemmas (3.63 and 4.52) are also load-bearing; they look plausible given the cycle descriptions in Section 4.2, but the full details matter.\n\nI could not verify the entire proof. The paper is long and case-heavy. Still, the internal consistency is high, and the explicit computations give real evidence for the construction. This deserves a serious referee. If the marked-boundary classification claim checks out, it is a strong contribution.","headline":"A serious, detailed construction of cluster structures on mixed Grassmannians for odd d that deserves a careful referee, with the main external-input risk concentrated in the marked-boundary variant of Demazure weaves.","tokens_in":75323,"tokens_out":2392,"would_cite":true,"duration_ms":31108,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13F60","05E99","13A50","14M15","15A72","15A75"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every d-admissible mixed Plücker ring, odd d and n>d^2, is a cluster algebra.","keywords":["cluster algebra","mixed Grassmannian","mixed Plücker ring","Demazure weave","signature","Lusztig cycle","amalgamation","invariant theory"],"falsifier":"Compute all algebraic relations among the claimed cluster and frozen variables for a small d-admissible signature with odd d and n > $d^{2}$, and check whether the extended cluster has full transcendence degree d(n−d)+1 and whether its generated ring equals R_σ; a single nontrivial relation among the initial cluster variables, or a Weyl generator that never appears in any mutated seed, would refute the theorem.","tokens_in":74339,"feed_emoji":"🕸️","tokens_out":6168,"duration_ms":65759,"temperature":0.7,"pith_summary":"The paper aims to prove that every mixed Grassmannian — the configuration space of a vectors and b covectors in a d-dimensional complex vector space, modulo the special linear group — has a natural cluster algebra structure whenever the cyclic ordering of the entries (the signature) is d-admissible. The main theorem states that for odd d and n = a + b > $d^{2}$, the mixed Plücker ring R_σ is exactly the explicit cluster algebra A_σ built from an amalgamated pair of Demazure weaves. This matters because cluster structures organize the invariant ring's generators and relations into mutation dynamics, and earlier constructions covered only dimension 3 or separated signatures; the result extends them to all odd dimensions and all admissible orderings. The construction is natural in a strong sense: every Weyl generator (the Plücker coordinates, dual Plücker coordinates, and pairings that generate R_σ) appears as a cluster or frozen variable, and the structure specializes to the classical Grassmannian, tensor-diagram, and mixed plabic graph cases.","feed_headline":"Mixed Grassmannians now have cluster structures in all odd dimensions","feed_subtitle":"The invariant ring of vectors and covectors equals an explicit cluster algebra built from two stitched weaves.","key_machinery":"The central machinery is the Demazure weave: a planar graph in a rectangle, with edges colored by 1,...,d−1, oriented from top to bottom, whose only allowed internal vertices are trivalent, 4-valent, and 6-valent configurations. A reduced Demazure weave yields a seed whose vertices are Lusztig cycles (certain oriented weighted subgraphs), whose arrows are intersection pairings between cycles, and whose cluster variables are cycle weights; a classification theorem ensures that any two reduced weaves with the same top and bottom words are mutation equivalent. To reach mixed Grassmannians, the paper cuts the signature word into two halves, forms two such weaves, and stitches them along reversed bottom words, amalgamating the seeds; odd d is exactly what makes the frozen variables match without a sign error. A supporting algebraic device is the mixed wedge operator, which is a wedge or an intersection depending on total degree and simplifies the cluster-variable formulas and exchange relations.","core_discovery":"At the center of the paper is the equality R_σ = A_σ. For a d-admissible signature σ with d odd and n > $d^{2}$, the paper defines A_σ by cutting the cyclic word β_σ at any valid cut (p,q), choosing two reduced Demazure weaves whose bottom words are reverses of each other, amalgamating their seeds along common frozen variables, and declaring the resulting cluster algebra independent of the choices. The paper proves that the amalgamated seed is well defined, counts its cluster variables, computes all exchange relations, shows that every Weyl generator occurs among cluster and frozen variables, and then argues by two routes, one via a factorial-subalgebra criterion and one via the Starfish lemma, that the cluster variables generate all of R_σ. Thus the mixed Plücker ring carries an explicit, signature-dependent cluster structure, not merely a birational one.","pith_inferences":["The n > d^2 hypothesis looks like a proof-technical bound rather than a structural boundary; the paper's own d = 3 example with n = 8 works below d^2, so a plausible next step is lowering the bound for all d by finding the Weyl generators more efficiently.","The reliance on odd d is tied to a sign in cyclic symmetry; a sign-curve adaptation, mentioned in the paper as a possibility, would be the natural route to extend the theorem to even d.","If seed independence holds generally, the amalgamated-weave method could be pushed to the conjectural setting of invariant rings of extensors of arbitrary levels, where the paper proposes a parallel cluster structure.","A concrete robustness test is whether the exchange relations computed in the paper, together with the once-mutated variables, generate R_σ without invoking the full Demazure classification theorem."],"forward_implications":["For every odd d and n > d^2 with d-admissible σ, the invariant ring R_σ is generated by an explicit cluster seed, so all its generators and relations are controlled by quiver mutation.","The same construction recovers the standard Grassmannian cluster structure when b = 0, the d = 3 tensor-diagram cluster structures, and the mixed plabic graph cluster structures for separated signatures.","The cluster structure is independent of arbitrary choices: different valid cuts and different reduced weaves give mutation-equivalent seeds, and the structure is invariant under cyclic shifts of the signature.","For separated signatures with a, b ≥ d−1 and n ≥ 2d, an analogous cluster structure exists, relaxing the n > d^2 bound.","The extended cluster of the initial seed has d(n−d)+1 elements, matching the dimension of the mixed Grassmannian."],"supporting_citations":[{"why":"Introduces cluster algebras and the Laurent phenomenon, providing the ambient theory the paper works within.","marker":"[FZ02]"},{"why":"Supplies the earlier conjecture and the d = 3 tensor-diagram cluster structures that this paper generalizes.","marker":"[FP16]"},{"why":"Gives the separated-signature mixed plabic graph construction, the structural template for amalgamating vector and covector sides.","marker":"[Car14]"},{"why":"Provides the classification theorem for reduced Demazure weaves that makes the cluster algebra independent of weave presentation.","marker":"[Cas+24]"},{"why":"Develops the construction of cluster structures from Demazure weaves on braid varieties, which the paper adapts and amalgamates.","marker":"[Cas+25]"},{"why":"Supplies the factorial-subalgebra criterion and Starfish-type tools used in both proofs that R_σ = A_σ.","marker":"[GLS13]"},{"why":"Gives finite generation and factoriality of invariant rings, used to identify R_σ as a UFD.","marker":"[VP89]"},{"why":"Establishes the standard cluster structure on Grassmannians that is recovered when b = 0.","marker":"[Sco06]"}],"fun_headline_variants":["Odd-dimensional mixed Grassmannians get explicit cluster algebras","Cluster structure proven for all odd d-admissible signatures","Mixed Plücker ring equals cluster algebra from two weaves","New proof: R_σ = A_σ for odd d via Demazure weaves","All odd-dimensional mixed Grassmannians carry cluster structures"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on the cited classification of Demazure weaves, that any two reduced weaves with the same top and bottom words are mutation equivalent, and on the lemmas that the two halves can be amalgamated with algebraically independent cluster variables; if that classification or those lemmas fail for the weaves used here, the cluster algebra A_σ would not be well defined.","fun_headline_variants_meta":{"raw":{"variants":["Odd-dimensional mixed Grassmannians get explicit cluster algebras","Cluster structure proven for all odd d-admissible signatures","Mixed Plücker ring equals cluster algebra from two weaves","New proof: R_σ = A_σ for odd d via Demazure weaves","All odd-dimensional mixed Grassmannians carry cluster structures"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000206,"raw_usage":{"total_tokens":1307,"prompt_tokens":767,"completion_tokens":540,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":383,"completion_tokens_details":{"reasoning_tokens":454}},"tokens_in":383,"tokens_out":540,"duration_ms":5291,"temperature":1.0,"reasoning_tokens":454,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:57:06.757679+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute all algebraic relations among the claimed cluster and frozen variables for a small d-admissible signature with odd d and n > $d^{2}$, and check whether the extended cluster has full transcendence degree d(n−d)+1 and whether its generated ring equals R_σ; a single nontrivial relation among the initial cluster variables, or a Weyl generator that never appears in any mutated seed, would refute the theorem.","supporting_citations":[],"review_version":1}