{"id":"bbb01453-9b7c-407a-b0f6-997708d2a345","arxiv_id":"2502.06410","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A multiplication formula for cluster characters of gentle algebras is proved and used to interpret surface cluster exchange relations and type B/type A variable relations categorically.","lead":"This paper proves a formula that multiplies two cluster characters of a gentle algebra and rewrites the product as a sum of two simpler characters, extending a known result from acyclic quivers to a wider class of algebras. The formula gives a representation-theoretic reading of exchange relations in cluster algebras coming from triangulated surfaces, and it interprets known relations between type B and type A cluster variables categorically.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.0.4's proof counts χ(Gr^B(M)) via successor-closed subquivers (Remark 2.0.4), but the constructed B can fail to be a string algebra—Example 4.0.10(iii) violates G2 at arrow e—so the Euler-characteristic equality is unproved in the Ext^1(S,X/X)=0 cases.","rationale":"The reader's CONDITIONAL verdict is appropriate, and this analysis does not move it. The reader's weakest assumption identified the auxiliary algebra B as under-specified and potentially non-gentle/non-string; I agree, and I sharpen that to a precise failure of the proof mechanism. The specific issue is that the proof of Theorem 4.0.4 counts quiver Grassmannians by successor-closed subquivers (Remark 2.0.4), which requires the algebra to be string and the module to be string or band. Example 4.0.10(iii) provides an explicit case where the constructed B is not a string algebra: adding a_R:3→5 gives two arrows into vertex 5, both composing non-trivially with e:5→1, violating condition G2. Thus the proof's bijective count of subquivers is not justified for exactly the cases where Ext^1(S,X/X)=0 and an extra arrow must be added. The paper's examples and computations are valuable and suggest the formula may be true, but the written proof of the central theorem has a concrete gap in these cases. I do not see evidence that the theorem is false; it is a question of missing justification. Rigidity of M (Remark 4.0.5) is a separate unproved input affecting only Theorem 4.0.11, so it is secondary to the main multiplication formula. Hence CONDITIONAL remains the right verdict, and the reader's assessment is unchanged.","tokens_in":24976,"tokens_out":11847,"duration_ms":97350,"concrete_test":"Use the second construction in Example 4.0.10(iii): let Q' = Q ∪ {a_R:3→5}, I' = I = ⟨ca,bd,ec⟩, B = kQ'/I'. First verify G2 fails at arrow e (both de and a_R e are nonzero). Then compute χ(Gr^B_{e-dimS}(M)) for M = M(3/5) directly as Euler characteristics of quiver Grassmannians over this finite-dimensional non-string algebra, and compare with the number of successor-closed subquivers of the diagram of M for each dimension vector. If these disagree, the proof's use of Remark 2.0.4 is invalid in exactly the situation the example exhibits; if they agree, the theorem needs a different proof of the identification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main formula (4.1) is proved by reducing χ(Gr_e(X⊕S)) to a count of successor-closed subquivers, using Remark 2.0.4. That remark applies only to string modules over string algebras. In the proof of Theorem 4.0.4, when Ext^1(S,X/X)=0 and only condition (1) (or (2)) holds, the module M is constructed as M(w_L a_L v_L^{-1}) ⊕ M(α_R) ⊕ M(β_R) in an algebra B = kQ'/I obtained by adding an arrow a_L (or a_R) to Q. The paper explicitly concedes in Remark 4.0.6 and Example 4.0.10(iii) that B need not be gentle or even string. Concretely, in the second construction of Example 4.0.10(iii) the added arrow a_R:3→5 produces two incoming arrows d:4→5 and a_R:3→5 at vertex 5 that both compose non-trivially with e:5→1 (de and a_R e are not in I=<ca,bd,ec>), violating condition G2. Therefore B is not a string algebra, and Remark 2.0.4 cannot be invoked to identify χ(Gr^B_e(M)) with the number of successor-closed subquivers of the diagram of M. The proof only asserts that L/S gives such a subquiver; it does not establish that all B-submodules of M arise this way. The statement also never specifies the ideal I' or proves B finite-dimensional, though the example is finite-dimensional. Thus the central equality is not proved in the cases where B is non-string; this is a proof gap, not a demonstrated falsehood.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a multiplication formula for cluster characters (Caldero–Chapoton maps) over gentle algebras. For A-modules X, S with dim Ext^1(S,X)=1 and a generating extension 0→X→Y→S→0, Theorem 4.0.4 asserts χ(Gr_e(X⊕S)) = χ(Gr_e(Y)) + χ(Gr^B_{e-dimS}(M)), where M is an Ext-minimum extension in a finite-dimensional algebra B ⊇ A. The paper then derives a cluster-character multiplication formula (Corollary 4.0.8), shows that B=A for gentle algebras coming from unpunctured surface triangulations, and upgrades the formula to an exchange relation for rigid indecomposable modules (Theorem 4.0.11). Section 5 applies these results to give a representation-theoretic proof of a known type-B/type-A cluster variable formula (Theorem 5.0.14).","tokens_in":25306,"tokens_out":8389,"duration_ms":66894,"significance":"If fully established, the main theorem would generalize the acyclic-quiver result of Cerulli Irelli, Esposito, Franzen, and Reineke to all gentle algebras and would provide a uniform representation-theoretic interpretation of exchange relations in cluster algebras from unpunctured surfaces. The paper is clearly organized, the combinatorial case-by-case analysis in the overlap-extension cases with Ext^1(S,X/X)≠0 is plausible and is supported by several worked examples (Examples 4.0.10, 4.0.12), and the application to type B is natural. However, several load-bearing points in the proof of Theorem 4.0.4 and in the derivation of Theorem 5.0.14 are asserted rather than proved. The significance of the paper depends on completing those arguments.","major_comments":[{"comment":"The proof invokes Remark 2.0.4 to compute χ(Gr^B_{e-dimS}(M)) as the number of successor-closed subquivers. Remark 2.0.4 is stated for string algebras and string/band modules. In the cases treated under conditions (1) or (2), the algebra B obtained by adjoining a_L or a_R is, as the paper itself concedes in Remark 4.0.6 and Example 4.0.10(iii), not gentle, and in the second construction of Example 4.0.10(iii) it is not even a string algebra (two incoming arrows at vertex 5 compose non-trivially with e, violating condition G2). Therefore Remark 2.0.4 cannot be applied to identify the Euler characteristic with the subquiver count. The proof shows that L/S gives a successor-closed subquiver of M but does not prove that every B-submodule of M arises in this way. Equality (4.1) is thus not established in these cases.","section":"§4, proof of Theorem 4.0.4, Ext^1(S,X/X)=0 cases"},{"comment":"The algebra B = kQ'/I ⊇ A is never completely defined. The proof only says that one adjoins an arrow a_L or a_R to Q and does not specify the ideal I of the quotient kQ'/I, nor does it prove that the resulting algebra is finite-dimensional. Since M is defined as the Ext-minimum extension between S/S and X in B and the term χ(Gr^B_{e-dimS}(M)) is taken in B, the theorem is incomplete without a construction of B and a proof of its finite dimensionality.","section":"Theorem 4.0.4 statement and proof"},{"comment":"The definition of X and S depends on the choice of the 'non-zero morphism f: X→τS that does not factor through an injective A-module' (and dually for g). The paper does not prove that such a morphism is unique; if several exist, ker f and im g, and hence the modules X and S, may depend on the choice. The argument in the proof of Theorem 4.0.4 selects the morphism associated with a maximal overlap but does not show that any other non-zero morphism not factoring through an injective would lead to the same submodules. Since X and S appear throughout (4.1), the well-definedness of these submodules is load-bearing.","section":"Definition 4.0.1"},{"comment":"The assertion 'The module M is rigid' is stated without proof. This rigidity is used in the proof of Theorem 4.0.11 to conclude that X ⊕ Y ⊕ M and S ⊕ Y ⊕ M are rigid and hence that (4.2) is an exchange relation via the cluster character bijection. A proof of Ext^1_B(M,M)=0, or a precise reference, is required; without it the exchange-relation theorem is not established.","section":"Remark 4.0.5"},{"comment":"The theorem is stated for arbitrary A-modules, but the proof treats only string modules in detail, saying that if one or both are bands 'the argument is the same with minor adaptations.' For band modules the extension theory differs (there are no arrow extensions, only overlap extensions, cf. Theorem 1.1.8 and the subsequent remarks), and band modules are not rigid, so the reduction to the string case is not automatic. A separate argument for bands is needed to support the claimed generality.","section":"§4, proof of Theorem 4.0.4, band module case"},{"comment":"After applying Remark 4.0.9 and Proposition 5.0.1, the proof concludes 'M = L(a,¯b) ⊕ L(ρ(a),ρ(¯b))' from an equality of F-polynomials. F-polynomials do not determine modules in general, and the proof does not invoke or prove a uniqueness property of the Ext-minimum extension in this setting. Since this identification is used to derive (5.9) and (5.10), a missing step needs to be supplied.","section":"§5, proof of Theorem 5.0.14(ii)"}],"minor_comments":[{"comment":"The notation B = kQ'/I ⊇ A reuses I for the ideal of B, while I is already the ideal of A; this is confusing since the ideal of B is generally different. Please use a different symbol, e.g., I'.","section":"Theorem 4.0.4 statement"},{"comment":"In the first construction, the paper states that the resulting B is still gentle, but it does not verify conditions (G1)–(G4) after the addition of the arrow a_L. A short check would make the example more convincing.","section":"Example 4.0.10(iii)"},{"comment":"The phrase 'Res(N) = (Vi, ϕa) is indecomposable as ordinary module' should read 'as an ¯A-module'; the term 'ordinary module' is vague.","section":"Theorem 5.0.14(i)"},{"comment":"The symbols a, b, c, d are sometimes arrows and sometimes the empty set (e.g., 'a = ∅'). This notation is nonstandard and should be clarified, for instance by introducing a convention for absent arrows.","section":"Definition 1.1.6(2)"},{"comment":"The sentence 'We show the proof for X, S both string modules' appears after a reduction to indecomposables; if the reduction is only valid for string modules, this should be stated explicitly before the reduction.","section":"Proof of Theorem 4.0.4"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a natural and timely question, and the main formula is supported by several explicit computations. The concerns listed in the major comments are all in the proof of the central theorem or in the derivation of the advertised application; they are substantive but appear to be within the scope of the paper's own methods. I see no indication of a contradiction with known results, and the missing arguments appear repairable with additional work. The manuscript is not ready for acceptance in its present form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Thanks for sharing this paper. My take: it's a genuine advance, but the proof has a hole that needs to be fixed before the main theorem can be accepted as stated.\n\nWhat's new: Ciliberti extends the Cerulli Irelli–Esposito–Franzen–Reineke multiplication formula from acyclic quivers to gentle algebras. The new definitions of X and S using the Auslander–Reiten translation are natural, and the surface case gives a representation-theoretic route to exchange relations for unpunctured surfaces. The type B application is honest—it interprets an earlier formula rather than pretending to reprove it independently. The paper is clearly written, and the examples are helpful; Remark 4.0.6 even admits that the auxiliary algebra can leave the gentle class.\n\nThe soft spot is real. The proof of Theorem 4.0.4 treats the cases where Ext^1(S,X/X)=0 by constructing an auxiliary algebra B = kQ'/I' obtained by adding an arrow, and then counts successor-closed subquivers using Remark 2.0.4. But that remark only applies to string algebras, and B need not be string—Example 4.0.10(iii) demonstrates this. The paper acknowledges it, but doesn't supply an alternative argument for why the Euler characteristic of Gr^B(M) should equal the subquiver count in those cases. So the equality χ(Gr^A_e(X⊕S)) = χ(Gr^A_e(Y)) + χ(Gr^B_{e-dimS}(M)) is unproved for exactly the cases where B is non-string. The statement also never fully specifies the ideal I' or proves B finite-dimensional. This is a proof gap, not a counterexample to the formula—the surface case (where B=A) is safe because A is gentle.\n\nTwo smaller issues: Remark 4.0.5 asserts M is rigid without proof, and Theorem 4.0.11 depends on that to invoke the cluster character bijection. And the final step of Theorem 4.0.11 is compressed into a reference to cluster category bijections; it needs a precise statement.\n\nDespite the gaps, this paper deserves a serious referee. The result is significant if correct, the gaps are addressable, and the author is clearly in control of the material. I'd recommend sending to peer review and asking for a full proof of the non-string cases, or a restriction of Theorem 4.0.4 to situations where B is known to be string (e.g., the surface setting).","headline":"Genuine generalization with a real proof gap: the main formula is plausible, but the counting argument breaks down in exactly the cases where the auxiliary algebra B is not string.","tokens_in":25908,"tokens_out":3822,"would_cite":true,"duration_ms":38985,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G20","13F60"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a generating extension in a gentle algebra splits the quiver Grassmannian of X⊕S into two pieces, yielding a multiplication formula for cluster characters and, for surface algebras, an exchange relation.","keywords":["cluster character","gentle algebra","quiver Grassmannian","exchange relation","surface triangulation","Ext-order","F-polynomial","string module"],"falsifier":"Find a gentle algebra A and a generating extension with one-dimensional Ext space for which the constructed B is infinite-dimensional or M has a nonzero self-extension; then the equality $\\chi(\\mathrm{Gr}^A_e(X\\oplus S))=\\chi(\\mathrm{Gr}^A_e(Y))+\\chi(\\mathrm{Gr}^B_{e-\\dim S}(M))$ would fail for some e, or the exchange-relation upgrade in Theorem 4.0.11 would not follow.","tokens_in":24648,"feed_emoji":"📐","tokens_out":14983,"duration_ms":110735,"temperature":0.7,"pith_summary":"Cluster characters turn modules into Laurent polynomials whose products are meant to mirror cluster variables. For acyclic quivers a generating short exact sequence gives a two-term multiplication formula, but the proof uses heredity. This paper extends the formula to gentle algebras, the class arising from unpunctured surface triangulations, by defining submodules of X and S through the Auslander–Reiten translation and inserting an Ext-minimal extension module M as the second term. The payoff is a two-term product identity for cluster characters that reproduces exchange relations in principal-coefficient cluster algebras and, for symmetric modules, a type-B/type-A comparison.","feed_headline":"Cluster character products split into two terms in gentle algebras","feed_subtitle":"One extra term, the minimal extension M, turns cluster character products into exchange relations.","key_machinery":"The load-bearing object is the cluster character $\\mathrm{CC}(L)=\\sum_e \\chi(\\mathrm{Gr}_e(L))x^{Be+g_L}y^e$, built from Euler characteristics of quiver Grassmannians. To obtain a two-term product, the proof cuts a generating extension $0\\to X\\to Y\\to S\\to 0$ using the submodules $\\underline{X}=\\ker(f)$ and $\\underline{S}=\\mathrm{im}(g)$ defined by the morphism $X\\to\\tau S$ that does not factor through an injective module and the morphism $\\tau^{-1}X\\to S$ that does not factor through a projective module. The correction term is the $\\leq_{\\mathrm{Ext}}$-minimum extension $M$ between $S/\\underline{S}$ and $\\underline{X}$, where the Ext-order is generated by nonsplit short exact sequences; for surface gentle algebras the extra arrows needed for $M$ already lie in $Q$, so $B=A$. The proof runs on string combinatorics: extensions of string modules are generated by arrow and overlap extensions, and $\\chi(\\mathrm{Gr}_e(L))$ counts successor-closed subquivers of the string diagram.","core_discovery":"Let $A=kQ/I$ be gentle, let $X,S$ be $A$-modules with $\\dim\\mathrm{Ext}^1_A(S,X)=1$, and let $\\xi:0\\to X\\to Y\\to S\\to 0$ be a generating extension. Theorem 4.0.4 asserts that for every dimension vector $e$, $\\chi(\\mathrm{Gr}^A_e(X\\oplus S))=\\chi(\\mathrm{Gr}^A_e(Y))+\\chi(\\mathrm{Gr}^B_{e-\\dim S}(M))$, where $M$ is the $\\leq_{\\mathrm{Ext}}$-minimum extension between $S/\\underline{S}$ and $\\underline{X}$ in a finite-dimensional algebra $B\\supseteq A$; when $A$ is the gentle algebra of a triangulation of an unpunctured surface, $B=A$. This recovers the earlier acyclic formula when $A$ is hereditary. The induced cluster-character identity (Corollary 4.0.8) is $\\mathrm{CC}(X)\\mathrm{CC}(S)=\\mathrm{CC}(Y)x^{g_X+g_S-g_Y}+y^{\\dim S}\\mathrm{CC}(M)x^{B\\dim S+g_X+g_S-g_M}$, and Theorem 4.0.11 upgrades it to an exchange relation when $X$ and $S$ are rigid indecomposables and $A$ is the gentle algebra of an unpunctured marked surface. Section 5 applies the formula to orthogonal modules over the symmetric algebra of a reflection-invariant triangulation of a regular polygon, proving the type-B/type-A F-polynomial and g-vector comparison.","pith_inferences":["The same two-term shape may hold for wider classes of string algebras or for Jacobian algebras of punctured surfaces, since the proof only needs a combinatorial basis of extensions and a finite-dimensional ambient algebra; the paper's Remark 4.0.13 already gestures at this.","If the auxiliary algebra $B$ is made explicit and proved finite-dimensional, the formula becomes an effective algorithm for computing cluster character products from string data alone.","The type-B/type-A application suggests a general categorical reading of restriction: symmetric-module categories over symmetric gentle algebras should categorify restriction maps between cluster algebras, with the Ext-minimal extension encoding the subtraction term in F-polynomial comparisons."],"forward_implications":["For every gentle algebra and every generating extension, the Euler characteristic of $\\mathrm{Gr}_e(X\\oplus S)$ is determined by $Y$ and one minimal extension module $M$, giving cluster character multiplication a two-term shape.","For unpunctured surface triangulations the correction term is defined inside the same gentle algebra, so the identity is intrinsic and not an artifact of an auxiliary algebra.","Specializing $x_i=1$ gives the F-polynomial identity $F_XF_S=F_Y+y^{\\dim S}F_M$, a purely combinatorial statement about successor-closed subquivers of string diagrams.","For rigid indecomposable modules over surface gentle algebras, the identity is an exchange relation in the cluster algebra with principal coefficients, giving a module-theoretic proof of those exchanges.","For reflection-invariant triangulations of polygons, the formula proves the type-B/type-A restriction identities for F-polynomials and g-vectors of all orthogonal indecomposable modules, without a heredity assumption."],"supporting_citations":[{"why":"Supplies the acyclic-quiver multiplication formula and the original definitions of the submodules that this paper generalizes.","marker":"[Cer+21]"},{"why":"Provides the arrow and overlap extension basis of Ext^1 for gentle algebras used throughout the proof of Theorem 4.0.4.","marker":"[ÇPS21]"},{"why":"Gives the string-module description and Auslander–Reiten sequences underlying the combinatorial proof.","marker":"[BR87]"},{"why":"Counts morphisms between string modules by admissible overlaps, used to identify the submodules and the minimal extension.","marker":"[Cra89]"},{"why":"Shows Jacobian algebras of unpunctured surface triangulations are gentle, giving the B=A case.","marker":"[Ass+10]"},{"why":"Supplies the cluster category and rigid-object bijection used to upgrade the identity to an exchange relation.","marker":"[BZ11]"},{"why":"Establishes the type-B/type-A cluster variable comparison and the symmetric-module correspondence that Section 5 reproves categorically.","marker":"[Cil25]"},{"why":"Provides the orthogonal and symplectic module framework and the twisted dual used in the type-B application.","marker":"[BC25]"}],"fun_headline_variants":["Cluster character product splits into sum of two terms","Minimal extension yields exchange relations in gentle algebras","Multiplying cluster characters: minimal extension term appears","Product formula for cluster characters in gentle algebras","Cluster characters multiply with an extra minimal extension term"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The formula's second term assumes that the algebra B built by adding one or two arrows is finite-dimensional and that the minimal extension module M is rigid; the paper only sketches B and asserts rigidity without a proof.","fun_headline_variants_meta":{"raw":{"variants":["Cluster character product splits into sum of two terms","Minimal extension yields exchange relations in gentle algebras","Multiplying cluster characters: minimal extension term appears","Product formula for cluster characters in gentle algebras","Cluster characters multiply with an extra minimal extension term"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000324,"raw_usage":{"total_tokens":1830,"prompt_tokens":968,"completion_tokens":862,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":584,"completion_tokens_details":{"reasoning_tokens":801}},"tokens_in":584,"tokens_out":862,"duration_ms":7517,"temperature":1.0,"reasoning_tokens":801,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T15:31:55.752966+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a gentle algebra A and a generating extension with one-dimensional Ext space for which the constructed B is infinite-dimensional or M has a nonzero self-extension; then the equality $\\chi(\\mathrm{Gr}^A_e(X\\oplus S))=\\chi(\\mathrm{Gr}^A_e(Y))+\\chi(\\mathrm{Gr}^B_{e-\\dim S}(M))$ would fail for some e, or the exchange-relation upgrade in Theorem 4.0.11 would not follow.","supporting_citations":[],"review_version":1}