{"id":"15f75aff-8117-4cb2-8657-95dd0802a059","arxiv_id":"2607.07028","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Explicit formulas for Goodearl-Yakimov mutation matrices on symmetric Poisson CGL extensions are given as matrix products and shown to coincide, up to reordering, with BFZ mutation matrices associated to signed words.","lead":"The paper gives explicit matrix-product and entry-wise formulas for mutation matrices arising in the Goodearl-Yakimov cluster structure on symmetric Poisson CGL extensions. It connects these to Berenstein-Fomin-Zelevinsky mutation matrices from Lie theory, unifying two previously separate constructions.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"No significant objection identified. Algebraic formulas are independently derived; cluster-theoretic interpretation depends on [GY23] as the reader already notes.","rationale":"The paper's core algebraic contributions — the matrix product formula M = E^t ν^{-1} EΛ, the entry-wise descriptions via y-degrees and x-degrees, and the BFZ product formula — are derived through elementary linear algebra and explicit Poisson bracket computations that I was able to verify. The uniqueness argument (Corollary 3.3.2) is a clean linear algebra lemma. The existence argument (Proposition 3.4.3) correctly uses the UFD structure of k[x_1,...,x_n] and the primality of the y_j's. The extension to symmetric extensions (Theorem B) properly applies Theorem A to the re-ordered CGL extensions R_τ. The one external dependency for cluster-theoretic claims (mutation equivalence, upper cluster algebra equality) is [GY23], which the reader already flagged. The paper is transparent about this dependency and does not overclaim. The normality condition is shown to be achievable (Corollary 3.5.4) and does not affect the mutation matrix entries themselves. I find no internally inconsistent or incorrectly derived step in the paper's own arguments.","tokens_in":52047,"tokens_out":3628,"duration_ms":118455,"concrete_test":"Independently verify the entry-by-entry match between equation (5.16) in Theorem 5.4.5 and the BFZ matrix entries in [BZ05, (8.7)] for a concrete non-trivial signed word, e.g. i† = (1,-2,-2,2,3,3,-3,1,2,-1,3,-2,1,1) from Example 5.3.5. Compute fM(i†) = E(i†)^t Q(i†) E(i†)_{n×ex(i†)} directly from the matrix product formula and compare with eB(i†) as defined in [BZ05]. If any entry differs, the identification in Theorem 5.4.8 would need revisiting (though the product formula itself would remain valid).","verdict_should_be":"UNCHANGED","load_bearing_attack":"I traced the three main argument chains: (1) Uniqueness of M via Corollary 3.3.2 — elementary linear algebra, correct. (2) Existence via Proposition 3.4.3 and Lemma 3.4.2 — the key step is showing b_{s(j)} ≠ 0 (contradiction using primality of y_{s(j)} in the UFD k[x_1,...,x_n]) and that b_{s(j)} is a monomial (via Lemma 3.3.3 applied to the bracket relations from Lemma 3.4.2). The constant-term/linear-term comparison in y_j is valid because T'_{s(j)-1}[y_j] is a polynomial ring in y_j over T'_{s(j)-1}. (3) The BFZ identification (Theorem 5.4.8) rests on entry-by-entry comparison between Theorem 5.4.5(5.16) and [BZ05, (8.7)]; the matrix product formula (5.15) is independently proved from the Poisson CGL theory, so even if the BZ05 comparison had a minor sign convention issue, the product formula would stand. The normality condition (ι_{s(j)}=1) is needed only for Theorem 3.6.1 (R = U(y,M^ε)), not for the formulas for M_τ themselves. Corollary 3.5.4 shows normality is achievable by rescaling, and M_j (the exponent vector) is invariant under rescaling. The reader correctly identifies that the cluster-theoretic interpretation (mutation equivalence, R = A = U) depends on [GY23], but the explicit formulas and their algebraic properties are independently established.","agreement_with_reader":"agree"},"referee_report":{"model":"glm-5.2","summary":"This paper derives explicit formulas for the mutation matrices M_tau arising in the Goodearl-Yakimov cluster structure theory for symmetric Poisson CGL extensions. For an arbitrary (not necessarily symmetric) T-Poisson CGL extension R of length n, the authors show that the unique solution to the GSV Equations (1.9) is given by the matrix product M = E^t nu^{-1} E Lambda (Theorem 3.4.5, Corollary 3.3.2), and they verify existence by showing that the element b_{s(j)}/y_{s(j)} is a Laurent monomial whose exponent vector gives the j-th column of M (Proposition 3.4.3). For symmetric extensions, they obtain the more symmetric formula M = E^t Q E_{n x ex} (Theorem 4.3.3) and, for each tau in Xi_n, the formula M_tau = E^t_{tau bullet} Q tau^t_bullet E_{n x ex} (Theorem 4.7.5), together with an entry-wise description in terms of Cartan integers (Theorem 4.7.10). As an application, for symmetric CGL extensions R_{(A,i)} constructed from a symmetrizable generalized Cartan matrix A and a word i, they establish a bijection between signed words and admissible triples (Section 5.3) and prove that the BFZ mutation matrix eB(i dagger) equals a permutation of M_tau (Theorem 5.4.8), also yielding a matrix product formula for the nondegenerate cluster ensemble matrix bB(i dagger) (Theorem 5.5.1).","tokens_in":52553,"tokens_out":804,"duration_ms":193460,"significance":"The paper solves a concrete and well-motivated problem: the mutation matrices M_tau in the Goodearl-Yakimov theory were previously characterized only as unique solutions to a linear system (via an involved induction in [GY23, Theorem 11.1]), and this paper replaces that characterization with explicit closed-form expressions. The matrix product formula M = E^t nu^{-1} E Lambda is derived by elementary linear algebra (Lemma 3.3.1, Corollary 3.3.2), and the existence proof via the monomial property of b_{s(j)} (Proposition 3.4.3, Lemma 3.4.2) is self-contained. The connection to BFZ mutation matrices (Theorem 5.4.8) and cluster ensemble matrices (Theorem 5.5.1) provides a bridge between the Poisson CGL framework and the combinatorial theory of signed words, and the matrix product formula (5.15) for eB(i dagger) appears to be new. The entry-wise description of M_tau in terms of Cartan integers (Theorem 4.7.10) is a useful concrete output. The results are applicable to Poisson structures on Bott-Samelson cells.","major_comments":[],"minor_comments":[],"recommendation":"accept","confidential_remarks":"The paper is largely self-contained for its algebraic results. The cluster-theoretic interpretation (mutation equivalence of seeds, R = A = U) does depend on [GY23, Theorem 11.1], but the explicit formulas for M_tau and their algebraic properties are independently established here. The normality condition (iota_{s(j)} = 1) is needed only for Theorem 3.6.1 (the upper cluster algebra equality), not for the formulas for M_tau themselves; Corollary 3.5.4 shows normality can always be achieved by rescaling CGL generators. The paper fits well within the scope of a serious algebra journal."},"author_rebuttal":null,"desk_editor":{"model":"glm-5.2","letter":"The main thing to know: this paper takes the mutation matrices M_τ in the Goodearl-Yakimov theory for symmetric Poisson CGL extensions, which were previously characterized only as the unique solution to a system of linear equations (the GSV equations) via an involved induction, and gives them explicit closed-form expressions. The key formula is M = E^t ν^{-1} E Λ, derived by elementary linear algebra, plus an entry-wise description in terms of Cartan integers. They also identify these matrices, up to reordering, with the Berenstein-Fomin-Zelevinsky mutation matrices attached to signed words. That identification is a clean bridge between two bodies of work that people suspected were related but hadn't pinned down this precisely.","headline":"Solid explicit formulas for GY mutation matrices; clean identification with BFZ matrices","tokens_in":52926,"tokens_out":208,"would_cite":true,"duration_ms":51724,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["17B63","13F60","53D17"],"pacs":[],"model":"glm-5.2","headline":"Explicit matrix product formulas for Poisson CGL mutation matrices","keywords":["Poisson CGL extensions","cluster algebras","mutation matrices","GSV equations","symmetric Poisson structures","generalized Cartan matrices","BFZ mutation matrices","signed words"],"falsifier":"If the GSV Equations qM = -Λ, χ_y M = 0 admitted multiple solutions in Mat_{n×ex}(k) for some Poisson CGL extension, or if the matrix product E^t ν^{-1} E Λ failed to have integer entries for some valid CGL extension, the main formulas would break down. Concretely, one could test this on a specific symmetric Poisson CGL extension of small length and verify whether the Laurent monomial b_{s(j)} / y_{s(j)} predicted by the formula actually has the exponent vector given by the j-th column of M.","tokens_in":52323,"feed_emoji":"🔗","tokens_out":1579,"duration_ms":137776,"temperature":0.7,"pith_summary":"The paper solves a concrete computational problem inside the Goodearl-Yakimov theory of cluster structures on polynomial Poisson algebras. In that theory, a symmetric T-Poisson CGL extension — a polynomial ring with a Poisson bracket and torus action satisfying certain compatibility axioms — is shown to carry a family of cluster algebra seeds indexed by permutations τ in a subset Ξ_n of the symmetric group. Each seed comes with a mutation matrix M_τ that was previously known to exist and be unique only through an involved induction argument; its entries were not given by any closed formula. This paper provides those formulas. The key idea is to observe that the equations characterizing M_τ (the GSV Equations, relating the Poisson coefficient matrix of the cluster variables to their torus weights) can be solved directly by elementary linear algebra, yielding an explicit matrix product M = E^t ν^{-1} E Λ involving the predecessor/successor structure of the CGL extension, the Poisson bracket data, and a diagonal matrix of scalars. For the symmetric case, this simplifies to M = E^t Q E, where Q = ν^{-1} Λ encodes the Cartan integers of the log-canonical part of the Poisson bracket. For each re-ordering τ, the formula becomes M_τ = E^t_{τ•} Q τ^t_• E, a conjugation of the base matrix M by the change-of-basis matrix E^{-1}_{τ^{-1}_•} E. The paper then shows that when the symmetric Poisson CGL extension arises from a symmetrizable generalized Cartan matrix A and a word i in the index set, these mutation matrices coincide (up to re-indexing) with the Berenstein-Fomin-Zelevinsky mutation matrices associated to signed words, unifying two previously separate families of matrices in cluster algebra and Lie theory.","feed_headline":"Explicit matrix product formulas for Poisson CGL mutation matrices","feed_subtitle":"Closed-form expressions solve the Goodearl-Yakimov GSV equations, linking Poisson CGL extensions to BFZ mutation matrices from Lie theory.","key_machinery":"The GSV Equations — a system of linear equations qM = -Λ and χ_y M = 0 relating the Poisson coefficient matrix q of the log-canonical cluster variables y, the torus character matrix χ_y, and a diagonal scalar matrix Λ — are the central object. The paper shows these equations have a unique solution given by the matrix product M = E^t ν^{-1} E Λ, where E is built from the successor map s and ν is the upper-triangular matrix of Poisson bracket constants. The existence of an integer solution is proved by expanding certain elements of the Poisson algebra as Laurent polynomials in the homogeneous Poisson prime elements y and reading off the exponents, which directly give the columns of M.","core_discovery":"The mutation matrices M_τ in the Goodearl-Yakimov cluster structure on a symmetric Poisson CGL extension are given explicitly by the matrix product M_τ = (E^{-1}_{τ^{-1}_•} E)^t M (E^{-1}_{τ^{-1}_•} E)_{ex×ex}, where M = E^t Q E is the base mutation matrix, E encodes the predecessor/successor maps, and Q = ν^{-1} Λ is determined by the Poisson bracket and torus weights. For CGL extensions from generalized Cartan matrices, these matrices are exactly the BFZ mutation matrices associated to signed words, via an explicit bijection between signed words and admissible triples (i, τ, ε_1).","pith_inferences":["The matrix product formula M = E^t ν^{-1} E Λ suggests that mutation matrices in broader classes of cluster algebras with Poisson structures might admit similar factorizations through the combinatorics of predecessor/successor maps.","The connection between Poisson cohomology classes (via the vectors θ_{(j,s(j))} in the second T-invariant Poisson cohomology) and mutation matrix entries hints at a deeper cohomological interpretation of cluster exchange relations that could extend beyond the CGL setting.","The bijection between signed words and admissible triples (i, τ, ε_1) may serve as a dictionary for translating problems between the representation-theoretic side (quantum affine algebras, canonical bases) and the Poisson-geometric side (CGL extensions, Poisson brackets)."],"forward_implications":["The matrix product formula eB(i†) = E(i†)^t Q(i†) E(i†) for BFZ mutation matrices can be proved directly without Poisson CGL theory, providing a new algebraic tool for studying cluster structures on double Bruhat cells and Bott-Samelson cells.","The non-zero entries of every M_τ are constrained to be ±1 or ±a where a is a negative Cartan integer, giving strong structural constraints on cluster algebras arising from Poisson CGL extensions.","The identification of M_τ with BFZ matrices bridges the Goodearl-Yakimov Poisson-theoretic approach to cluster algebras with the Kashiwara-Kim exchange matrices from monoidal categorification, potentially allowing techniques to transfer between the two settings.","The nondegenerate cluster ensemble matrix bB(i†) is shown to equal the full square matrix cM(i†) = E(i†)^t Q(i†) E(i†), giving a matrix product decomposition for both its skew-symmetrizable and symmetrizable parts."],"fun_headline_variants":["Closed-form matrix products give Poisson CGL mutation matrices","Entry-wise formulas for mutation matrices in Goodearl-Yakimov cluster structures","Matrix product factorization of Poisson CGL mutation matrices","Explicit mutation matrices link Poisson CGL extensions to BFZ Lie theory"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The interpretation of the explicit formulas as genuine cluster mutation matrices relies on the Goodearl-Yakimov theory, which requires a normality condition (certain scalars ι_{s(j)} equal 1) and a rationality condition on the ratio of Poisson scalars. The algebraic identities and matrix product formulas themselves hold without these assumptions, but their role as mutation matrices in a mutation-equivalent family of seeds depends on them.","fun_headline_variants_meta":{"raw":{"variants":["Closed-form matrix products give Poisson CGL mutation matrices","Entry-wise formulas for mutation matrices in Goodearl-Yakimov cluster structures","Matrix product factorization of Poisson CGL mutation matrices","Explicit mutation matrices link Poisson CGL extensions to BFZ Lie theory"]},"model":"glm-5.2","effort":"low","cost_usd":0.0,"raw_usage":{"total_tokens":487,"prompt_tokens":427,"completion_tokens":60,"prompt_tokens_details":null},"tokens_in":427,"tokens_out":60,"duration_ms":43989,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-09T21:21:01.548900+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"If the GSV Equations qM = -Λ, χ_y M = 0 admitted multiple solutions in Mat_{n×ex}(k) for some Poisson CGL extension, or if the matrix product E^t ν^{-1} E Λ failed to have integer entries for some valid CGL extension, the main formulas would break down. Concretely, one could test this on a specific symmetric Poisson CGL extension of small length and verify whether the Laurent monomial b_{s(j)} / y_{s(j)} predicted by the formula actually has the exponent vector given by the j-th column of M.","supporting_citations":[],"review_version":1}