{"id":"ebab810a-f2c7-41fc-a3d3-41c968d4672f","arxiv_id":"2506.20568","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The walls in the semistable cone of a quiver moduli problem equal the union, over all sub-dimension vectors e, of the intersections sst(e) ∩ sst(d-e).","lead":"The authors give an explicit formula for the stability parameters where quiver moduli spaces develop strictly semistable points: the walls are exactly the intersections of semistable cones for complementary sub-dimension vectors. This yields algorithms for computing GIT equivalence and the full GIT fan of a quiver, with software and worked examples.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1 is proven directly and appears sound, but the GIT fan corollaries rely on an unjustified transfer of [1]'s quasitorus framework to the non-abelian quiver action.","rationale":"The reader's weakest_assumption identifies the same underlying gap: the GIT fan structure imported from [1] is not justified for the non-abelian quiver action. My reading of the proof confirms that Theorem 1.1 itself is independent of this import and is correct, so the main wall-finding result is not threatened. However, the paper's advertised applications—GIT equivalence criterion, GIT fan algorithm, and the claim that the GIT fan is determined by its walls—rely essentially on [1, Theorem 3.2 and Proposition 5.2]. Since [1] is explicitly formulated for quasitori on varieties with a Cox ring, and since the paper gives no reduction argument for the action of the product of general linear groups on a representation space, this is a genuine correctness risk for those corollaries. The concern is not that the main theorem is false, but that the proofs of the secondary theorems are incomplete as written. A direct verification of [1]'s hypotheses, or a self-contained proof of wall-determinacy for quiver GIT, would settle the issue. Until that is supplied, the conditional verdict is appropriate.","tokens_in":12875,"tokens_out":19810,"duration_ms":219089,"concrete_test":"Check the hypotheses of [1, Theorem 3.2 and Proposition 5.2] against the action of G_d on R_d: either exhibit a Cox-ring/quasitorus model for quiver GIT under which the GIT fan of (Q,d) is authorized by [1], or add a self-contained proof that the GIT fan of (Q,d) is determined by its walls. As an independent computational check, discretize a small grid in ⊥d for Example 7.2, compute the semistable locus for each grid point using a representation-theoretic library, and verify that the resulting GIT equivalence classes coincide with the fan produced by Corollary 4.6; any mismatch would refute the transferred wall-determinacy claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central wall characterization, Theorem 1.1, is established in Section 3 without reference to [1] and its proof is correct: Lemma 3.2 reduces strictly semistable points to subrepresentations with θ(e)=0, and Proposition 3.3 identifies W_e with sst(e) ∩ sst(d−e) via Lemma 3.4. The load-bearing gap is in the derived claims: Corollaries 4.4–4.6 and Theorem 1.2 depend on [1, Theorem 3.2] and [1, Proposition 5.2] for the existence of the GIT fan and, crucially, for the property that it is determined by its walls. [1] develops GIT fans via Cox rings in the setting of actions of quasitori (diagonalizable groups) on normal varieties. The action of G_d = ∏_i GL(d_i) on R_d is not quasitoral when some d_i > 1, and the paper does not explain why the GIT fan of this reductive-group action falls under the hypotheses of [1]. This is not a merely pedantic point: Example 7.6 demonstrates that an arbitrary reductive-group action need not have a GIT fan determined by its walls, so Proposition 5.2 carries real content. If the transfer is invalid, the algorithmic core of Section 4 loses its foundation, although the wall characterization itself remains intact.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper gives an effective characterization of the walls in the variation of GIT problem for quiver moduli. The main result (Theorem 1.1) states that the locus of stability parameters for which R_d contains a strictly semistable representation is the union, over nonzero proper subdimension vectors e ≤ d, of W_e = sst(e) ∩ sst(d−e). This is proved directly in Section 3 using King's semistability and a short-exact-sequence lemma. The paper then derives three consequences: a GIT-equivalence criterion for stability parameters (Theorem 1.2 and Corollary 4.4), an algorithm for computing the GIT fan (Corollary 4.6), a geometric-phase criterion (Theorem 5.1), and a recursive description of special subdimension vectors (Proposition 6.3). The results are illustrated by several examples, including quiver mutation and a realization of the Segre cubic.","tokens_in":13117,"tokens_out":13941,"duration_ms":135802,"significance":"Assuming the GIT-fan framework transfers correctly, the paper is a useful contribution: it reduces wall computation to the semistable cones sst(e) and sst(d−e), which are algorithmically accessible via Schofield's recursion, and it provides explicit implementations. The central identity W_e = sst(e) ∩ sst(d−e) is proved from first principles and is robust. The geometric-phase characterization (Theorem 5.1) is clean, and the examples, especially the mutation sequence and the Segre cubic fan with its full f-vector, are valuable. The paper is also honest about its reliance on [1] and about parts of Theorem 1.3 being known in essence. The main gap is the unstated transfer of the GIT-fan formalism from [1]'s quasitorus setting to the non-abelian reductive group action; this affects the GIT-equivalence and fan-algorithm theorems, while leaving Theorem 1.1 intact.","major_comments":[{"comment":"The GIT-fan results depend on [1, Theorem 3.2] and [1, Proposition 5.2], but [1] is developed for actions of quasitori (diagonalizable groups) on normal varieties. The action of G_d = ∏_i GL(d_i) on R_d is not quasitoral when some d_i > 1, and the paper does not explain why the GIT fan of this reductive-group action falls under the hypotheses of [1]. This is load-bearing: if the transfer is invalid, Theorem 1.2 and the algorithm in Corollary 4.6 lose their foundation, even though Theorem 1.1 is proved directly. The authors should either cite a general reductive-group GIT-fan result that covers this setting, or prove the needed facts (existence of the fan and determination by its walls) for quiver moduli directly, for instance by a reduction to the abelianized quiver.","section":"§4, Corollaries 4.4–4.6 and Theorem 1.2"},{"comment":"In the proof that a non-generic e is special, the sentence 'Since e is not a generic subdimension vector of d, we can pick a nontrivial extension M of M'' by M''' is not justified. The existence of a nontrivial extension between general representations of dimensions e and d−e is equivalent to ext(e,d−e)>0, which follows from e not being generic via Theorem 2.6(iii) and the definition of ext in (14), but this argument is omitted. Please add a sentence or a reference so that the construction of M is rigorous.","section":"§6, Proposition 6.3"}],"minor_comments":[{"comment":"The equivalence is stated for 'every e ≤ d', but W_e is defined only for nonzero proper subdimension vectors; the quantifier should be restricted to nonzero proper e.","section":"§1, Theorem 1.2 and §4, Corollary 4.4"},{"comment":"The notation 'f' ̸=,→d−e' is confusing; since the text explains that it denotes a proper generic subdimension vector, a clearer phrasing such as '0 ⊊ f' ⊊ d−e with f' ,→ d−e' would be preferable.","section":"§6, Proposition 6.3"},{"comment":"The symbol W_e is overloaded: it first denotes a union of codimension-1 walls W_{ke}, and then is used in the algorithm as that union. Renaming this union, for instance ~{W}_e, would avoid conflict with Definition 3.1.","section":"§4, before Corollary 4.6"},{"comment":"The phrase 'θ can lays on 10 of them' should be 'θ can lie on 10 of them'.","section":"§7, Example 7.7"},{"comment":"The variable M is used both for the ambient representation in the pair (N, M) and for the semistable locus R_θ-sst_d; this is mildly confusing and could be clarified by denoting the semistable representation by X or N.","section":"§3, Definition 3.1"}],"recommendation":"major_revision","confidential_remarks":"The central wall characterization, Theorem 1.1, is sound and the paper is likely acceptable after the GIT-fan transfer issue is resolved. The authors should be encouraged to check whether [1, Proposition 5.2] can be applied directly to the reductive group action or whether a reduction to a torus action on an abelianized quiver is needed; if the latter, the proof should be written out. This is the main substantive concern and the reason for major revision rather than minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is the short version. The paper gives an effective description of the walls in quiver GIT: Theorem 1.1 says the wall locus is the union of W_e = sst(e) ∩ sst(d-e), and the proof in Section 3 is direct and correct. That is the main result and it is genuinely useful. The related criteria for GIT equivalence and for existence of a geometric phase are also solid, and the authors provide working code and a gallery of examples that illustrate the phenomena.\n\nThe real soft spot is the section on the GIT fan. Corollaries 4.4-4.6 and Theorem 1.2 depend on [1, Theorem 3.2 and Proposition 5.2] for the existence of the GIT fan and for the fact that it is determined by its walls. But [1] proves those statements for actions of quasitori on normal varieties. The action of ∏_i GL(d_i) on a quiver representation space is not quasitoral when any d_i > 1. The paper never addresses this mismatch. It is not a pedantic point: Example 7.6 is an example of a reductive-group action where the GIT fan is not determined by its walls. So either the transfer from [1] needs an explicit justification, or those corollaries need proofs that do not rely on [1]. The main wall theorem does not depend on this, so the core of the paper stands.\n\nThe rest of the paper looks fine. Lemma 5.3 and the proof of Theorem 5.1 are correct. The recursive characterization of special subdimension vectors in Proposition 6.3 is a real addition, and the counterexamples in Section 7 are meaningful. The authors are also honest about which parts of Theorem 1.3 are already in King's work, which is a good sign.\n\nWho should read it: people working on quiver moduli, variation of GIT, or computational algebra. It deserves a serious referee. The main theorem is worth publishing; the GIT fan section should be tightened before it is.","headline":"Clean main theorem, real computational payoff, and one citation gap in the GIT fan part worth fixing.","tokens_in":13693,"tokens_out":3547,"would_cite":true,"duration_ms":35265,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G20","14L24","14D20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The walls in quiver moduli are exactly intersections of semistable cones.","keywords":["quiver representations","stability parameters","semistable cone","walls","GIT fan","geometric invariant theory","Schur roots","geometric phase"],"falsifier":"For any concrete quiver and dimension vector, compute all W_e = sst(e) ∩ sst(d−e) using the paper's recursive algorithm and then test one stability parameter in each resulting chamber: the central claim would be falsified by a single θ outside all W_e that still admits a strictly semistable representation, or a θ inside some W_e for which every θ-semistable representation is stable.","tokens_in":12636,"feed_emoji":"🧱","tokens_out":5862,"duration_ms":62780,"temperature":0.7,"pith_summary":"Every stability parameter for a quiver representation of dimension vector d has a semistable cone sst(d) of parameters admitting semistable representations. This paper identifies the walls inside that cone, the parameters where strictly semistable representations appear, as a union of sets W_e = sst(e) ∩ sst(d−e), one for each nonzero proper subdimension vector e. Because the cones sst(e) can be computed recursively using Schofield's criterion for generic subdimension vectors, the walls become effectively computable. From this description the authors derive criteria for GIT equivalence of stability parameters, an algorithm for computing the GIT fan, and a geometric-phase criterion tied to indivisible Schur roots.","feed_headline":"Walls in quiver moduli are semistable-cone intersections","feed_subtitle":"A new theorem makes the strictly semistable locus computable and yields algorithms for GIT fans.","key_machinery":"The central object is the wall set W_e = sst(e) ∩ sst(d−e), defined for each nonzero proper subdimension vector e of d: it is the locus of stability parameters θ for which some θ-semistable representation of dimension d has a subrepresentation of dimension e with θ(e) = 0. Lemma 3.4, the extension lemma, allows the paper to pass between semistability of the middle term of a short exact sequence and semistability of the subobject and quotient, which is what makes the identification of the strictly semistable locus with the union of the W_e work. When e is a generic subdimension vector, W_e simplifies to the hyperplane H_e = {θ : θ(e) = 0} intersected with sst(d). Since sst(d) itself is computable from Schofield's recursive characterization of generic subdimension vectors, the entire wall system is algorithmically accessible.","core_discovery":"The central discovery is Theorem 1.1: for a quiver Q and dimension vector d, a stability parameter θ in the orthogonal complement of d admits a strictly θ-semistable representation of dimension d exactly when θ lies in W_e = sst(e) ∩ sst(d−e) for some proper nonzero subdimension vector e ≤ d. The proof uses a short-exact-sequence lemma: a representation is θ-semistable precisely when it is an extension of two θ-semistable representations of complementary dimension vectors with θ vanishing on the subobject. Combined with the recursive description of sst(d) via generic subdimension vectors, this gives an effective method for computing all walls. The paper goes on to show that two stability parameters are GIT equivalent exactly when their connecting segment does not cross any W_e transversely, and that a geometric phase exists exactly when d is an indivisible Schur root.","pith_inferences":["One could extend the same wall description to stability parameters with θ(d) ≠ 0 by rescaling, potentially giving a wall criterion on the full parameter space rather than only its orthogonal complement.","The recursive characterization of special subdimension vectors suggests a shortcut for detecting stable representations in small examples: one only needs to know generic subdimension vectors of proper subdimension vectors, not the full representation theory.","Because the walls are intersections of semistable cones, their codimensions can often be read off from dimensions of those cones; this could provide a quick sufficient check for the geometric-phase criterion without computing the whole fan.","The fact that the strictly semistable locus is a union of cone intersections may transfer to other linear actions of reductive groups that admit an analogous extension lemma, although the paper shows the analogous statement fails for arbitrary representations."],"forward_implications":["Two stability parameters θ and η are GIT equivalent if and only if for every e ≤ d the segment [θ, η] is either entirely contained in W_e or does not meet it, yielding a direct test for equivalence.","The GIT fan of (Q, d) can be computed by starting with the fan determined by the hyperplanes H_e and merging chambers across facets that are not contained in any W_e.","A geometric phase exists if and only if d is indivisible and a Schur root, which is equivalent to sst(d) spanning its ambient space and every wall having codimension one.","Even for acyclic quivers, not every special subdimension vector is orthogonal to a hyperplane in the wall system, and not every hyperplane in the wall system is a facet of the GIT fan."],"supporting_citations":[{"why":"Supplies the recursive characterization of generic subdimension vectors and the ext formula used to compute sst(d).","marker":"[16]"},{"why":"Provides the semistable cone formula sst(d) = {θ : θ(e) ≤ 0 for all e → d}, extended here to all quivers and real stability parameters.","marker":"[8]"},{"why":"Gives the GIT fan structure and the fact that the GIT fan is determined by its walls, which underpins the GIT-equivalence and fan algorithms.","marker":"[1]"},{"why":"Relates stability parameters to characters and supplies the King-semistability framework used in the geometric phase criterion.","marker":"[15]"},{"why":"Introduces the wall system and special subdimension vectors that Section 6 characterizes and tests.","marker":"[13]"}],"fun_headline_variants":["Quiver walls: semistable cones meet","Walls in quiver moduli are cone intersections","Effective walls: sst(d) via subdimensions","Semistable cone intersections define quiver walls","GIT walls for quivers: a cone-intersection rule"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The GIT-equivalence criterion and the GIT fan algorithm assume, without a fully spelled-out transfer, that the general GIT fan theory for diagonalizable group actions on normal varieties applies to the action of the product of general linear groups on quiver representation spaces; if that transfer fails, those corollaries lose their foundation, although the main wall description is proven directly.","fun_headline_variants_meta":{"raw":{"variants":["Quiver walls: semistable cones meet","Walls in quiver moduli are cone intersections","Effective walls: sst(d) via subdimensions","Semistable cone intersections define quiver walls","GIT walls for quivers: a cone-intersection rule"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000127,"raw_usage":{"total_tokens":1001,"prompt_tokens":719,"completion_tokens":282,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":335,"completion_tokens_details":{"reasoning_tokens":204}},"tokens_in":335,"tokens_out":282,"duration_ms":3377,"temperature":1.0,"reasoning_tokens":204,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:45:50.437827+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For any concrete quiver and dimension vector, compute all W_e = sst(e) ∩ sst(d−e) using the paper's recursive algorithm and then test one stability parameter in each resulting chamber: the central claim would be falsified by a single θ outside all W_e that still admits a strictly semistable representation, or a θ inside some W_e for which every θ-semistable representation is stable.","supporting_citations":[{"cited_title":"Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients","cited_arxiv_id":null,"evidence_quote":"Provides the semistable cone formula sst(d) = {θ : θ(e) ≤ 0 for all e → d}, extended here to all quivers and real stability parameters."},{"cited_title":"Stable representations of quivers","cited_arxiv_id":null,"evidence_quote":"Introduces the wall system and special subdimension vectors that Section 6 characterizes and tests."}],"review_version":1}