{"id":"d65c3f15-b885-4e64-96a7-7dc4b9c8a19e","arxiv_id":"2506.10382","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Open Richardson varieties in symmetrizable Kac-Moody flag varieties, including twisted-product cases, are shown to carry upper cluster algebra coordinate rings.","lead":"The authors prove that coordinate rings of open Richardson varieties, and their twisted-product generalizations, are upper cluster algebras in any symmetrizable Kac-Moody type. The result extends finite-type theorems of Casals-Gorsky-Gorsky-Le-Shen-Simental and Galashin-Lam-Sherman-Bennett-Speyer to the full Kac-Moody setting.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.13's quotient diagram may be ill-defined: deleting a vertex imposes A_i=1, but Proposition 5.9 identifies the defining relation T=1 with an exchange-ratio monomial, so U(s_m) may be an over-ring unless that monomial is 1.","rationale":"Reader's weakest assumption is plausible but not the most load-bearing: the unpublished fibration [3, Prop. 3.2] affects only Theorem 6.5 and Theorem 3 for twisted products; Theorem 1 for ˚B_{v,w} does not depend on it. The soft spot in Theorem 5.13 is internal and earlier: the only bridge from the combinatorial seed s_m to the geometric equations defining ˚B_{v,w} is Proposition 5.9. That proposition states the defining function T is an exchange ratio, not a cluster variable. In the cluster-theoretic operations used (freezing and deletion, [32, Prop. 3.7]), deleting a frozen vertex imposes A_i=1, not X_i=1. So for the quotient diagram to be valid, the frozen monomial multiplying X must be a unit in U(s_m). The proof neither computes it nor shows it is 1; Remark 5.10 declines to compute it. This is exactly the kind of omitted verification that can hide a false isomorphism. If the monomial is nontrivial, the stated upper cluster structure would be on a localization or further quotient rather than the coordinate ring, so the central claim would not be established. A concrete A_2 computation can settle the issue. Hence the verdict should stay CONDITIONAL, gated on checking this step and, separately, the unpublished [3, Prop. 3.2] used for the twisted-product extension.","tokens_in":34191,"tokens_out":27924,"duration_ms":342940,"concrete_test":"Compute the A_2 case w=s_1s_2s_1, v=s_1 (and, as a second datum, v=s_1s_2) with the seed from Definition 5.11. Use Proposition 5.9 to obtain the explicit image of T_{r(l),v(l)} in U(s_m) after setting A_{(i,0)}=1 and deleting the vertices (i,n_i-a_l+1); check whether this image is 1. Independently compute C[˚B_{v,w}] from Proposition 3.6 and U(s_{v,w}) from the seed (e.g., with a small Gröbner basis or the coordinate formulas in [31]); compare. If the image of T is a nonunit frozen monomial, the displayed square in Theorem 5.13 does not commute and the proof's final identification C[˚B_{v,w}]≅U(s_m) is unjustified.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 5.3, proof of Theorem 5.13. Proposition 3.6 presents C[˚B_{v,w}] as the quotient of O(Conf_w(A)) by A_{(i,0)}=1 and T_{r(l),v(l)}=1 for l=1,...,m. Proposition 5.9 identifies each T_{r(l),v(l)} in U(˜s_m) as X_{(i,n_i-a_l+1)} times a monomial of frozen variables. For the vertex (i,n_i-a_l+1), the exchange ratio X is a monomial in adjacent vertices, not in the deleted variable itself (ε_{ii}=0 and p_{ii}=0 in (4.4)). Deleting that vertex means imposing A_{(i,n_i-a_l+1)}=1, not X=1. The 'quotient map U(˜s_m)→U(s_m)' in the displayed diagram is defined by sending A^{-1}_{(i_{k_l},n_{i_{k_l}}-a_l+1)} to a frozen monomial and by sending T_{r(l),v(l)} to 1; but a ring homomorphism can only send T to 1 if T already has image 1 in U(s_m). The paper never proves that the frozen monomial in Proposition 5.7/5.9 becomes 1 after setting A_{(i,0)}=1 and deleting the earlier vertices; Remark 5.10 even says the monomial is not needed. If it is not 1, the diagram produces C[˚B_{v,w}] as a further quotient of U(s_m), not as U(s_m), so Theorem 1 does not follow.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims that for any symmetrizable Kac-Moody group G and any v ≤ w in its Weyl group, the coordinate ring of the open Richardson variety ˚B_{v,w} in the full flag variety is an upper cluster algebra (Theorem 5.13), and that the same holds for open Richardson varieties in twisted products of flag varieties (Theorem 6.5), covering reduced double Bruhat cells, Bott-Samelson varieties, and braid varieties. It also identifies the cluster positive locus with the Lusztig totally nonnegative part (Propositions 7.4-7.5) and proves full-rank and reddening-sequence properties. The strategy is to start from Shen-Weng's cluster structure on double Bott-Samelson configuration spaces, use rational quasi-cluster maps to realize the defining functions T_{r(l),v(l)} as exchange-ratio monomials, and then pass to ˚B_{v,w} by freezing and deletion.","tokens_in":34667,"tokens_out":16984,"duration_ms":204409,"significance":"If the proof can be repaired, these results are significant: they would establish the Kac-Moody analogue of Leclerc's conjecture, provide a uniform seed construction generalizing recent finite-type work, extend the theory to twisted products, and give a new identification of cluster positive loci with Lusztig totally nonnegative parts. The paper also provides an explicit seed algorithm, full-rank and reddening-sequence statements, and a concrete comparison with prior seeds. The main reservations are technical: the quotient step in the proof of Theorem 5.13 has a load-bearing gap, and the twisted-product results depend on an unpublished fibration theorem. These issues prevent me from recommending acceptance before revision.","major_comments":[{"comment":"The bullet defining the quotient map U(˜s_m) → U(s_m) is not justified. Proposition 5.9 asserts that T_{r(l),v(l)} equals X_{(i,n_i-a_l+1)} times a monomial of frozen variables. Since ε_{d,d}=0 and p_{d,d}=0 in (4.4), the exchange ratio X_d is a monomial in variables adjacent to d and does not contain A_d. Therefore, once the image of A^{-1}_d is chosen, the image of T_{r(l),v(l)} is forced; it cannot also be declared to be 1. The displayed diagram thus does not show that C[˚B_{v,w}] is U(s_m); it only exhibits C[˚B_{v,w}] as a quotient of U(s_m) unless the frozen monomial in Proposition 5.9 is identically 1. Remark 5.10 says that the precise monomial is not needed, but this is exactly the point where it is needed.","section":"Theorem 5.13, proof (Section 5.3)"},{"comment":"The deletion step in Definition 5.11 is applied to the vertex (i,n_i-a_l+1). For a_l>1 this vertex is mutable in s(w), since it satisfies 0 < n_i-a_l+1 < n_i, and the definition only freezes its neighbours, not the vertex itself. Proposition 2.4 provides a deletion isomorphism only for frozen vertices. The paper does not explain why the vertex is frozen before deletion, so the passage from U(˜s_m) to U(s_m) in Theorem 5.13 is not a direct consequence of the cited seed operations.","section":"Definition 5.11 / Proposition 2.4"},{"comment":"In the induction step for i=j, the equality Φ*_l(A'_{(j,m'_j)}) = Φ*_{l-1}(X''_{(j,m''_j)}) is used to conclude that this element is a monomial of frozen variables. The induction hypothesis applied to Φ_{l-1} only says that Φ*_{l-1}(X''_{(j,m''_j)}) is X_{(j,m_j)} times a frozen monomial, not that it is itself a frozen monomial. No argument is given that X_{(j,m_j)} is a frozen monomial in this situation. This gap is load-bearing because Proposition 5.7 feeds directly into Proposition 5.9.","section":"Proposition 5.7, proof"},{"comment":"Theorem 6.5 and Proposition 7.5 rest on Proposition 6.3, quoted from the unpublished preprint [3], which asserts a locally trivial (C×)^{n-1} fibration Z' → Z with section th. This is load-bearing for all twisted-product results. The manuscript should either include a proof of Proposition 6.3 or explicitly state that Theorem 2 and Proposition 7.5 are conditional on [3]. The current citation is not sufficient for a referee to verify the main theorem.","section":"Proposition 6.3 / Theorem 6.5"}],"minor_comments":[{"comment":"The notation for the seed s(i) with reversed exchange matrix is not typeset distinctly; the sentence 'The seed s(i) for i ∈ I is the same as s(i) but with exchange matrix (−εij)' is confusing and should use an overline or another label.","section":"§2.7"},{"comment":"Lemma 3.5 is used later in Proposition 3.6, but its proof is a single sentence referring to Lemma 3.3 after displaying diagrams; this should be expanded for the reader.","section":"Lemma 3.5"},{"comment":"The phrase 'there is a secret shift on the index sets' is informal; since the shift is essential to the correspondence of vertices, it should be replaced by a precise statement.","section":"Remark 4.17"},{"comment":"The symmetrizable case is dismissed with 'standard folding arguments; see [11]'; either give the precise folding argument or state the proposition only under the symmetric assumption.","section":"Proposition 7.2"},{"comment":"Reference [23] is listed as 'Unpublished'; if it is used, provide a preprint reference, and if not, remove the citation.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The referee believes the main theorem may be true, but the manuscript needs substantial revision: the quotient map in Theorem 5.13 must be repaired, the deletion of potentially mutable vertices must be justified, and the dependence on the unpublished thickening fibration [3] must be clarified. If the authors can supply the missing arguments, the paper would be a strong candidate for acceptance. The journal may also wish to consider whether the current level of reliance on unpublished work is acceptable for the claims made."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this paper proves a major result—coordinate rings of open Richardson varieties are upper cluster algebras in full symmetrizable Kac–Moody type—and offers a genuinely new seed construction generalizing Menard's finite-type algorithm. The total positivity identification is new beyond type A, and the theta-basis corollary is a nice payoff. The framework of rational quasi-cluster maps is a real tool, not just a repackaging.\n\nThe soft spots are real but not fatal. The proof of Theorem 5.13 hinges on a quotient diagram where the map U(˜s_m) → U(s_m) is said to send A^{-1} to a frozen monomial and T to 1. As written, this is under-specified: T's image is determined by the images of the cluster variables, and it is not shown that a choice of frozen monomial makes T map to 1. A stress-test note raises exactly this concern, and I think the concern is legitimate. The paper's Remark 5.10 explicitly declines to give the monomial, but the main theorem's proof needs it, or at least needs an argument that such a homomorphism exists. This is fixable, but it is load-bearing.\n\nAlso, Theorem 6.5 depends on a fibration theorem from the authors' own unpublished [3]. That is a heavy dependency for a headline result. Proposition 7.2 handles only the symmetric case and dismisses the symmetrizable case as \"standard folding\"; a referee cannot quickly verify that step from this text. Lemma 3.5's one-line proof is another spot where the paper is too terse.\n\nWhat the paper does well: the strategy is coherent and explanatory—config spaces, rational quasi-cluster maps, explicit seed algorithm with worked example, then freezing and deletion. The total positivity section is clean once the cluster structure is in hand.\n\nBottom line: the core claim is plausible and important, but the written proof has at least one genuine gap and several compressed dependencies. This deserves a serious referee, but the referee should require a rewritten proof of Theorem 5.13 and careful statements of what is imported from [3]. I would not cite the main theorem as proven until the gap is closed.","headline":"A serious, important generalization of cluster structures on Richardson varieties, but the proof of the main theorem has a gap in its quotient diagram that needs explicit fixing.","tokens_in":35072,"tokens_out":12752,"would_cite":false,"duration_ms":140007,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13F60","17B67","14M15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the coordinate ring of every open Richardson variety in a symmetrizable Kac-Moody flag variety is an upper cluster algebra, and extends the result to twisted products.","keywords":["Kac-Moody groups","Richardson varieties","upper cluster algebras","total positivity","twisted products of flag varieties","braid varieties","Bruhat atlas","rational quasi-cluster maps"],"falsifier":"For a symmetrizable indefinite Kac-Moody group, such as the rank-2 example with Cartan matrix $\\begin{pmatrix}2&-2\\\\-2&2\\end{pmatrix}$, compute the rational function $T_{r,v}$ after the mutation sequence in Definition 5.2 and check whether its exchange ratio acquires a non-frozen monomial factor; if it does, the induction in Proposition 5.7 fails and the upper cluster structure on $\\mathring{B}_{v,w}$ is not established. Alternatively, test the fibration in Proposition 6.3 directly for this example by examining whether the locally trivial $(\\mathbb{C}^\\times)^{n-1}$ bundle and its section exist.","tokens_in":34016,"feed_emoji":"💠","tokens_out":6107,"duration_ms":64576,"temperature":0.7,"pith_summary":"The paper establishes that, for any symmetrizable Kac-Moody group, the coordinate ring of every open Richardson variety in the full flag variety is an upper cluster algebra, and that the same holds for open Richardson varieties on twisted products of flag varieties. This class includes reduced double Bruhat cells, Bott-Samelson varieties, and braid varieties. The seed is built explicitly from a reduced expression of $w$ and the left-most subexpression for $v$, by mutating, freezing, and deleting vertices in the known cluster structure on $B_{e,w}$. The paper also proves that the cluster positive locus of each variety equals its Lusztig totally nonnegative part, extending results previously known only in finite type.","feed_headline":"Open Richardson varieties are upper cluster algebras in Kac-Moody type","feed_subtitle":"Every open Richardson variety gets an explicit upper cluster structure, including twisted products and braid varieties.","key_machinery":"The load-bearing tool is the rational quasi-cluster map: a rational map between cluster algebras that sends each cluster variable to a cluster variable up to a Laurent monomial in frozen variables and preserves exchange ratios. Through the Bruhat atlas isomorphisms, which locally factor a Richardson variety into a product of smaller Richardson varieties, these maps let the authors transfer the known upper cluster structure on $\\mathring{B}_{e,w}$ to $\\mathring{B}_{v,w}$. For twisted products, the thickening map embeds $\\mathring{Z}_{v,w}$ into a Richardson variety inside a larger Kac-Moody flag variety, where the extra torus directions become frozen cluster variables that can be deleted.","core_discovery":"The central claim is Theorem 5.13: for any $v \\le w$ in the Weyl group of a symmetrizable Kac-Moody group, the coordinate ring $\\mathbb{C}[\\mathring{B}_{v,w}]$ is isomorphic to the upper cluster algebra $\\mathcal{U}(s(v,w))$ for an explicitly constructed seed. Theorem 6.5 extends this to open Richardson varieties $\\mathring{Z}_{v,w}$ on twisted products of flag varieties, and Propositions 7.4 and 7.5 identify the cluster positive locus with the totally nonnegative part $B_{v,w,>0} = B^{\\mathrm{cl}}_{v,w,>0}$ and $Z_{v,w,>0} = Z^{\\mathrm{cl}}_{v,w,>0}$. The proof realizes $\\mathring{B}_{v,w}$ as a quotient of $\\mathring{B}_{e,w}$ by setting certain rational functions $T_{r(l),v(l)}$ to $1$, then transfers the known upper cluster structure on $\\mathring{B}_{e,w}$ through the Bruhat atlas using rational quasi-cluster maps, so that freezing and deleting the right vertices yields a seed for $\\mathring{B}_{v,w}$.","pith_inferences":["If the paper's Conjecture 5.15 on local acyclicity holds, the upper cluster algebra would coincide with the ordinary cluster algebra, strengthening the isomorphism from an upper cluster structure to a full cluster structure.","The thickening method suggests that any variety fibered over a Richardson variety with a torus factor and a section should inherit an upper cluster structure whenever the thickened variety admits one, potentially yielding cluster structures on new families of moduli spaces.","The equality of cluster positivity and total nonnegativity may give a practical way to compute totally nonnegative parts in indefinite Kac-Moody types, where canonical bases are far less explicit than in finite type.","The full-rank and reddening-sequence results invite applying the theta-basis and quantization machinery to these Kac-Moody Richardson varieties, going beyond the canonical-basis consequence the paper itself states."],"forward_implications":["Every open Richardson variety in a symmetrizable Kac-Moody flag variety carries an explicit upper cluster algebra whose seed is read off from reduced words.","The same holds for open Richardson varieties on twisted products, so reduced double Bruhat cells, Bott-Samelson varieties, and braid varieties all have upper cluster structures in Kac-Moody type.","The cluster positive locus equals the Lusztig totally nonnegative part, so the cluster-theoretic notion of positivity reproduces the canonical-basis notion of positivity.","The constructed seeds have full rank and admit reddening sequences, which yields canonical theta bases parametrized by integral tropical points.","The result contains the finite-type theorems of the paper's references [5] and [18] as special cases, with matching seeds in finite type."],"supporting_citations":[{"why":"Supplies the upper cluster structure on double Bott-Samelson cells and the explicit cluster variables used on $\\mathring{B}_{e,w}$.","marker":"[36]"},{"why":"Provides the parametrizations and explicit formulas for the functions $T_{r,rs_i}$ and for total positivity of $\\mathring{B}_{e,w}$.","marker":"[31]"},{"why":"Gives the freezing and deletion operations on seeds and the comparison of cluster and upper cluster algebras under them.","marker":"[32]"},{"why":"Establishes that cluster variables are irreducible in the upper cluster algebra, a key step in proving the containment $\\mathcal{U}(s_m) \\subset \\mathbb{C}[\\mathring{B}_{v,w}]$.","marker":"[19]"},{"why":"Supplies the factoriality criteria used to conclude that the upper cluster algebra embeds into the coordinate ring of the Richardson variety.","marker":"[12]"},{"why":"Establishes the Laurent phenomenon, which underlies the definition and comparison of cluster and upper cluster algebras.","marker":"[13]"},{"why":"Supplies the thickening fibration and the total positivity results for twisted products, the key geometric input for Theorem 6.5 and Proposition 7.5.","marker":"[3]"},{"why":"Establishes the finite-type braid variety cluster structures that the present seeds extend and agree with in finite type.","marker":"[5]"},{"why":"Establishes the finite-type braid variety cluster structures that the present seeds extend and agree with in finite type.","marker":"[18]"}],"fun_headline_variants":["Open Richardson varieties are upper cluster algebras in all Kac-Moody types","Kac-Moody Richardson varieties: explicit upper cluster seeds","Upper cluster algebras unify Bruhat, Bott-Samelson, and braid varieties","Every Kac-Moody open Richardson variety gets a canonical upper cluster structure","Generalization of Richardson cluster algebras to symmetrizable Kac-Moody"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The twisted-product theorem rests on an unpublished fibration statement (Proposition 6.3) asserting that the thickened Richardson variety fibers over the twisted-product Richardson variety with a torus factor and a section; if that fibration does not hold in the Kac-Moody setting, the twisted-product upper cluster structure does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Open Richardson varieties are upper cluster algebras in all Kac-Moody types","Kac-Moody Richardson varieties: explicit upper cluster seeds","Upper cluster algebras unify Bruhat, Bott-Samelson, and braid varieties","Every Kac-Moody open Richardson variety gets a canonical upper cluster structure","Generalization of Richardson cluster algebras to symmetrizable Kac-Moody"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000294,"raw_usage":{"total_tokens":1685,"prompt_tokens":895,"completion_tokens":790,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":511,"completion_tokens_details":{"reasoning_tokens":690}},"tokens_in":511,"tokens_out":790,"duration_ms":8485,"temperature":1.0,"reasoning_tokens":690,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:28:09.219921+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a symmetrizable indefinite Kac-Moody group, such as the rank-2 example with Cartan matrix $\\begin{pmatrix}2&-2\\\\-2&2\\end{pmatrix}$, compute the rational function $T_{r,v}$ after the mutation sequence in Definition 5.2 and check whether its exchange ratio acquires a non-frozen monomial factor; if it does, the induction in Proposition 5.7 fails and the upper cluster structure on $\\mathring{B}_{v,w}$ is not established. Alternatively, test the fibration in Proposition 6.3 directly for this example by examining whether the locally trivial $(\\mathbb{C}^\\times)^{n-1}$ bundle and its section exist.","supporting_citations":[{"cited_title":"Shen and D","cited_arxiv_id":null,"evidence_quote":"Supplies the upper cluster structure on double Bott-Samelson cells and the explicit cluster variables used on $\\mathring{B}_{e,w}$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the parametrizations and explicit formulas for the functions $T_{r,rs_i}$ and for total positivity of $\\mathring{B}_{e,w}$."},{"cited_title":"Muller, Locally acyclic cluster algebras , Adv","cited_arxiv_id":null,"evidence_quote":"Gives the freezing and deletion operations on seeds and the comparison of cluster and upper cluster algebras under them."},{"cited_title":"Geiss, B","cited_arxiv_id":null,"evidence_quote":"Establishes that cluster variables are irreducible in the upper cluster algebra, a key step in proving the containment $\\mathcal{U}(s_m) \\subset \\mathbb{C}[\\mathring{B}_{v,w}]$."},{"cited_title":"Fomin and A","cited_arxiv_id":null,"evidence_quote":"Establishes the Laurent phenomenon, which underlies the definition and comparison of cluster and upper cluster algebras."},{"cited_title":"Casals, E","cited_arxiv_id":null,"evidence_quote":"Establishes the finite-type braid variety cluster structures that the present seeds extend and agree with in finite type."}],"review_version":1}