{"id":"41f9baf4-a725-429b-9b0f-367033f847d4","arxiv_id":"2506.13684","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For types A, B, C, and D, the sum of Schubert structure constants with Coxeter length k is eventually polynomial in n with leading term (2n)^k divided by k!.","lead":"This paper proves that, for each of the four classical families of Lie groups, the sum of Schubert structure constants with bounded Coxeter length is eventually a polynomial in the rank, with explicit leading term (2n)^k divided by k!. It also gives a conceptual proof of a recent result by Pak and Robichaux and extends it to all classical types.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.5 miscounts equivalence-class orbit sizes: the factor should be |Aut(∆(w))|/|Stab(w)|, not ∏ a_w(j)!, so the proof of Theorem 3.6 and the lead-term extraction rest on an invalid counting identity.","rationale":"The paper's central claim is likely correct: direct checks for k=1 and k=2 match the stated lead term, and the dominance argument (degree of N_w(n) equals p, maximized by the A1^k class) is sound once the orbit-size count is corrected. The flaw in Proposition 3.5 is real and load-bearing, because the proof of Theorem 3.6's exact formula and the extraction of the lead term invoke this proposition. However, the corrected count still gives the same lead term n^k/k! for the maximal class, so Theorem 1.2 survives a revision. The reader's conditional verdict is appropriate: the manuscript should be accepted after Proposition 3.5 is rewritten with the correct orbit-size multiplicity and a consistent relation between N_w and N'_w. I find no other objection that would change this assessment.","tokens_in":6369,"tokens_out":19743,"duration_ms":194734,"concrete_test":"In type A_5 (Weyl group S_6), enumerate all elements of length 2 with support A1^2 (e.g., w=s1s3) and support A2 (w=s1s2). Direct counts are 6 and 8 respectively. Proposition 3.5's formula N_w = N'_w·∏ a_w! gives 12 and 4 respectively, while the stated lead terms would predict (for n=5) values whose exact forms differ from these counts. This singles out the orbit-size multiplicity error and decisively checks whether the counting identity in Proposition 3.5 holds.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 3.5 is the load-bearing counting input for Theorem 3.6 and for the lead-term computation. Its displayed equality N_w(n) = N'_w(n)·∏ a_w(j)! is false in general. For w with support A1^k (a product of k commuting simple reflections), each induced subgraph determines exactly one element v, since all permutations of the commuting factors give the same element; the multiplicity is 1, not k!. For w=s1s2 in an A2 component, the Dynkin diagram automorphism swaps the two vertices and sends w to s2s1, so each edge support gives two distinct elements; the multiplicity is 2, while ∏ a_w! = 1. The correct multiplicity is the orbit size |Aut(∆(w))|/|Stab(w)|. Moreover, the proof's equation is internally inconsistent with the proposition's stated lead term: if N'_w has lead n^p, then N_w = N'_w·∏ a_w! would have lead n^p·∏ a_w!, not n^p/∏ a_w!. For the maximal class A1^k, the stated lead term n^k/k! coincides with the correct count C(n-k+1,k) ~ n^k/k!, so Theorem 1.2's lead term is plausible; nevertheless, the proof as written does not validly establish it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"Fix one of the four classical families of complex Lie groups, with Weyl groups of types A, B, C, and D. The paper studies γ_k(n), the sum of all Schubert structure constants c^w_{u,v} with ℓ(w)=k, and proves that for fixed k and all sufficiently large n, γ_k(n) is a polynomial in n with leading term (2n)^k/k!. The proof introduces an equivalence relation on Weyl group elements by isomorphism of Dynkin supports, then uses the equivariant restriction formula of Andersen–Jantzen–Soergel and a stability theorem of Robichaux–Yadav–Yong to show that all elements in one equivalence class contribute the same sum of structure constants. This reduces γ_k(n) to a finite sum over classes represented in W_{2k}, weighted by the class sizes N_w(n). Proposition 3.5 estimates these class sizes, Lemma 3.7 evaluates the sum of structure constants for the class A1^k, and the lead term is extracted from that class.","tokens_in":6648,"tokens_out":15323,"duration_ms":159688,"significance":"The proposed method is attractive and, if correct, would be a substantial improvement over the signed-puzzle proof of Pak–Robichaux: it gives the leading term for all classical types and cleanly separates the n-dependent counting from finitely many Schubert computations. The external tools used, including GKM localization and the AJS equivariant restriction formula, are standard and are applied in a plausible way; I see no circularity and no fitted parameters. The main obstacle is the counting in Proposition 3.5, which is currently incorrect; however, the small cases k=1 and k=2 indicate that the stated leading term is plausible, so the theorem may well survive a correct count. For these reasons the paper merits a major revision rather than rejection.","major_comments":[{"comment":"The displayed identity N_w(n) = N'_w(n)·∏ a_w(j)! is incorrect, and it is also inconsistent with the lead term stated in the same proposition. If N'_w(n) has leading term n^p, then the displayed product would have leading term n^p·∏ a_w(j)!, not n^p/∏ a_w(j)!. Concretely, for w = s_1...s_k with Δ(w) = A1^k, the k simple reflections commute, so each induced subgraph gives exactly one element equivalent to w; the true count is approximately n^k/k!, whereas the displayed product gives approximately n^k. For w = s_1s_2 with Δ(w) = A_2, the Dynkin automorphism sends w to s_2s_1, so each edge support contributes two equivalent elements, while ∏ a_w(j)! = 1. The correct multiplicity is the orbit size |Aut(Δ(w))|/|Stab(w)|, after first passing from labelled embeddings to induced subgraphs. Since Proposition 3.5 is the counting input used in Theorem 3.6, this error is load-bearing.","section":"Proposition 3.5"},{"comment":"Theorem 3.6 uses N_w(n) as the size of the equivalence class [w]∩W_n, and the final lead-term extraction uses the leading coefficient of N_w(n) for the class A1^k. Because Proposition 3.5's N_w(n) is not the true class size, neither statement is established as written. For example, for k=2 in type A, the true class A1^2 has approximately n^2/2 elements, not n^2 as the product formula gives; combined with Lemma 3.7's factor 4, this would change the coefficient of n^2 unless the counting is repaired. The intended value (2n)^2/2 = 2n^2 is consistent with the true count, so I expect the theorem remains true, but the proof must be reworked after fixing Proposition 3.5.","section":"Theorem 3.6 and proof of Theorem 1.2"},{"comment":"The counting convention in Lemma 3.4 should be stated explicitly. For Δ = A1^c, the lemma's formula is (k+1-c)!/(k+1-2c)! ≈ k^c, which counts labelled embeddings, one for each labelling of the c copies of A1. However, N'_w(n) is defined as the number of induced subgraphs isomorphic to Δ(w), which is an unlabeled count. The factor ∏ a_w(j)! in Proposition 3.5 is meant to mediate between these conventions, but it is applied in the wrong direction and it does not account for automorphisms of Δ(w) that do not come from permuting equal components, such as the automorphism of A_2. Please clarify the labelling convention and replace the identity with the correct orbit-size formula.","section":"Lemma 3.4 and Proposition 3.5"}],"minor_comments":[{"comment":"The word 'polynomality' should be 'polynomiality' throughout the manuscript.","section":"Throughout"},{"comment":"The statement that Δ(w) does not depend on the choice of reduced word I should be made part of the definition, not asserted afterward.","section":"Definition 3.1"},{"comment":"In the display following Definition 3.2, the index j is used both for the ambient dimension and for the component type, which makes the definition of a_w(j) unnecessarily confusing.","section":"Definition 3.2"},{"comment":"The computation in Example 2.3 would be easier to check if the two subwords J of I were written out explicitly before evaluating the restriction formula.","section":"Example 2.3"},{"comment":"For the exceptional D_n embeddings, the text lists the relevant shapes but does not explain why these are the only exceptional shapes; a sentence of justification would improve readability.","section":"Proposition 3.5, Figure 2"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the manuscript uses standard tools and I found no circularity. The flaw in Proposition 3.5 is local but load-bearing; because the intended leading term is plausible and the correction is identifiable, this feels like a major revision rather than a rejection. I would ask the author to replace the multiplicity identity with the correct orbit formula, re-run the B/C/D case analysis under that formula, and re-verify the coefficient in the lead term. I do not have concerns about scope or attribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe one thing you should know: this paper has a genuinely new proof idea and a likely-true lead term, but the counting at the center of the proof is wrong as written. The author groups Schubert elements by Dynkin support and uses stability to reduce γ_k(n) to a sum over W_{2k}. That is a real conceptual advance over Pak–Robichaux's signed-puzzle argument, and the extension to types B, C, D is new. The small cases k=1,2 work, and the asserted asymptotic (2n)^k/k! is almost certainly correct.\n\nNow the soft spot, and it is load-bearing. Proposition 3.5 claims N_w(n) = N'_w(n)·∏ a_w(j)!, then states the lead term as n^p / ∏ a_w(j)!. The two displays contradict each other, and the claimed identity is false. For w = s_1 s_2 in an A_2 component, each edge support gives two equivalent elements (s_1s_2 and s_2s_1), while ∏a_w! = 1. For w a product of k commuting simple reflections, each support gives exactly one element, not k!. The correct multiplicity is the orbit size |Aut(Δ(w))|/|Stab(w)|, not the factorial product. Lemma 3.4 also miscounts embeddings: for k isolated vertices in A_n it gives roughly n^k, while the true count is C(n−k+1,k) ≈ n^k/k!. These errors are not cosmetic; Theorem 3.6 and the lead-term extraction use the degree and coefficient of N_w(n).\n\nThat said, the errors appear to cancel in a way that the final answer survives. For the dominant class A_1^k, the true embedding count has leading n^k/k!, and the inner Schubert sum is 2^k, so (2n)^k/k! comes out right. A careful revision replacing the factorial product with the orbit-size formula and correcting Lemma 3.4 should make the proof rigorous. The author should also check the other classical families, where the same mistake will affect the coefficient, not just the degree.\n\nWorth a serious referee? Yes. The idea is good, the intended theorem is interesting, and the existing literature result is genuinely extended. The referee should be asked to verify the enumeration carefully; this is a revision, not a rejection.","headline":"New proof strategy and likely-true lead term, but the load-bearing count in Proposition 3.5 (and Lemma 3.4) is wrong as written and needs a serious revision.","tokens_in":7179,"tokens_out":6456,"would_cite":false,"duration_ms":61294,"reading_group":"yes","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E14","14N15","20F55"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that for each classical Lie family, the sum of Schubert structure constants over triples with Coxeter length $k$ is eventually a polynomial in $n$ with leading term $(2n)^k/k!$.","keywords":["Schubert structure constants","equivariant cohomology","Dynkin support equivalence","Weyl groups","Coxeter length","classical Lie types","polynomiality","GKM theory"],"falsifier":"In type A, fix $k$ and take $w$ to be a product of $k$ pairwise non-adjacent simple reflections. The elements equivalent to $w$ in $W_n$ are exactly the ways to choose $k$ non-adjacent nodes from the $n$-node Dynkin diagram, so the exact class count is $\\binom{n-k+1}{k}$, with leading term $n^k/k!$. Checking whether Proposition 3.5 reproduces this count, rather than a $k!$-multiple of it, for any fixed $k$ and large $n$ settles the proof's central counting step.","tokens_in":6135,"feed_emoji":"🧮","tokens_out":14777,"duration_ms":135481,"temperature":0.7,"pith_summary":"The paper's goal is to show that the sums $\\gamma_k(n)$ of Schubert structure constants over triples $u,v,w$ with $\\ell(w)=k$ are eventually polynomial in $n$, with the single leading term $(2n)^k/k!$, for each of the four classical Lie families (types A, B, C, and D). The method is to group Weyl-group elements $w$ by an equivalence relation: two elements are equivalent when their reduced words draw on isomorphic Dynkin subdiagrams and the isomorphism carries one element to the other. Stability of Schubert structure constants under Dynkin-diagram embeddings makes the inner sum over $u,v$ depend only on the class, so $\\gamma_k(n)$ becomes a finite sum of a class constant times a class count $N_w(n)$. The leading class is the one supported on $k$ isolated $A_1$ nodes, which corresponds to products of $k$ commuting simple reflections; its structure-constant sum is $2^k$ and its count contributes $n^k/k!$. If correct, this answers the request for a conceptual proof and fixes the exact asymptotic size of bounded-length Schubert sums.","feed_headline":"Schubert sums have leading term (2n)^k/k! in every classical type","feed_subtitle":"A symmetry among Weyl-group elements turns bounded-length Schubert sums into a polynomial count.","key_machinery":"The central object is the Dynkin-support equivalence $w \\sim w'$: two Weyl-group elements are equivalent when their reduced words involve induced subdiagrams of the relevant classical Dynkin diagram that are isomorphic, with the isomorphism sending $w$ to $w'$. The load-bearing mechanism is the equivariant restriction formula, which expresses the restriction of an equivariant Schubert class at a fixed point as a sum over reduced subwords, together with the stability property that follows from it: an embedding of Dynkin diagrams $\\iota: \\Delta \\to \\Delta'$ forces the equivariant structure constants $C^w_{u,v}$ and $C^{\\iota(w)}_{\\iota(u),\\iota(v)}$ to agree up to relabelling. These tools let the paper replace a sum over all of $W_n$, which grows with $n$, by a finite sum over equivalence classes represented inside $W_{2k}$, reducing the $n$-dependence to a polynomial class count $N_w(n)$.","core_discovery":"The paper's central claim, stated as Theorem 1.2, is that for fixed $k$ and any one of the families $\\{SL_{n+1}\\}$, $\\{SO_{2n+1}\\}$, $\\{SP_{2n}\\}$, $\\{SO_{2n}\\}$, the number $\\gamma_k(n) = \\sum_{u,v,w \\in W_n, \\ell(w)=k} c^w_{u,v}$ is, for all sufficiently large $n$, a polynomial in $n$ whose leading term is $(2n)^k/k!$. The proof splits $\\gamma_k(n)$ by the Dynkin-support equivalence relation, obtaining $\\gamma_k(n) = \\sum_{[w], w \\in W_{2k}, \\ell(w)=k} (\\sum_{u,v \\le w} c^w_{u,v}) N_w(n)$, where $N_w(n)$ counts elements equivalent to $w$ in $W_n$. Because the inner structure-constant sum is stable under Dynkin embeddings, the only $n$-dependence is in $N_w(n)$; because $N_w(n)$ is eventually polynomial of degree equal to the number of type-A components of the support, the largest degree occurs when the support is $A_1^k$, where the inner sum equals $2^k$ and the count has leading term $n^k/k!$. The paper concludes that the leading term is therefore $(2n)^k/k!$.","pith_inferences":["A natural extension not pursued in the paper is to track the first subleading term of $\\gamma_k(n)$; the degree-$k-1$ coefficient should be computable from classes whose Dynkin support has $k-1$ components or one non-$A$ component, and it may be the first place where Lie type shows up.","The same Dynkin-support grouping should apply to sums weighted by $\\ell(u)$ or $\\ell(v)$, or to equivariant structure constants at specializations, since the stability property used here is equivariant; one can test whether the leading $n$-dependence remains $(2n)^k/k!$.","The explicit thresholds in Theorem 3.6 (roughly $n \\ge k$ in the classical types) make the formula computationally checkable: for each small $k$, direct computation of $\\gamma_k(n)$ in types A, B, C, D for several $n$ above the threshold would either confirm the polynomial or expose the discrepancy."],"forward_implications":["For each fixed $k$ and each classical type, $\\gamma_k(n)$ is eventually a degree-$k$ polynomial in $n$, so bounded-length structure-constant sums have a stable, type-independent asymptotic shape.","The leading term $(2n)^k/k!$ is the same for the $SL$, $SO$, and $Sp$ families, so the growth rate of these sums does not distinguish Lie type at top order.","The leading contribution comes entirely from elements $w$ whose Dynkin support is $A_1^k$, i.e. products of $k$ commuting simple reflections; every other support contributes at most degree $k-1$.","Because $\\gamma_k(n)$ is a finite linear combination of fixed structure-constant sums from $W_{2k}$ with polynomial coefficients $N_w(n)$, the full polynomial can in principle be computed once the finite $W_{2k}$ data and the counts are known."],"supporting_citations":[{"why":"Supplies the equivariant restriction formula used throughout to compare structure constants across Weyl groups.","marker":"[1]"},{"why":"Provides the GKM localization theorem identifying equivariant cohomology with fixed-point data, the basis for the restriction setup.","marker":"[2]"},{"why":"The previous result whose proof is replaced and strengthened; its polynomiality theorem is the baseline for the paper's extension.","marker":"[3]"},{"why":"Records the stability properties and Bruhat-order support facts for equivariant Schubert classes used in the grouping argument.","marker":"[4]"}],"fun_headline_variants":["Schubert sums polynomial: lead term (2n)^k/k! for all classical types","Bounded-length Schubert sums: (2n)^k/k! leading term in every classical type","Classical Schubert sums: lead term (2n)^k/k! via stable counts","Schubert structure constants: bounded-length sums have lead (2n)^k/k!","Every classical type: Schubert sums lead term (2n)^k/k!"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the count of Weyl-group elements equivalent to a given element $w$ under the Dynkin-support relation: if that count is wrong by a multiplicative factor, the claimed leading term is wrong by the same factor.","fun_headline_variants_meta":{"raw":{"variants":["Schubert sums polynomial: lead term (2n)^k/k! for all classical types","Bounded-length Schubert sums: (2n)^k/k! leading term in every classical type","Classical Schubert sums: lead term (2n)^k/k! via stable counts","Schubert structure constants: bounded-length sums have lead (2n)^k/k!","Every classical type: Schubert sums lead term (2n)^k/k!"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000694,"raw_usage":{"total_tokens":3113,"prompt_tokens":891,"completion_tokens":2222,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":507,"completion_tokens_details":{"reasoning_tokens":2106}},"tokens_in":507,"tokens_out":2222,"duration_ms":14919,"temperature":1.0,"reasoning_tokens":2106,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:00:25.448888+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In type A, fix $k$ and take $w$ to be a product of $k$ pairwise non-adjacent simple reflections. The elements equivalent to $w$ in $W_n$ are exactly the ways to choose $k$ non-adjacent nodes from the $n$-node Dynkin diagram, so the exact class count is $\\binom{n-k+1}{k}$, with leading term $n^k/k!$. Checking whether Proposition 3.5 reproduces this count, rather than a $k!$-multiple of it, for any fixed $k$ and large $n$ settles the proof's central counting step.","supporting_citations":[{"cited_title":"Representations of quantum groups at a pth root of unity and of semisimple groups in characteristic p: independence of p , Astrisque No","cited_arxiv_id":null,"evidence_quote":"Supplies the equivariant restriction formula used throughout to compare structure constants across Weyl groups."},{"cited_title":"Equivariant cohomology, Koszul duality, and the localization theorem, Invent","cited_arxiv_id":null,"evidence_quote":"Provides the GKM localization theorem identifying equivariant cohomology with fixed-point data, the basis for the restriction setup."},{"cited_title":"The A·B·C·Ds of Schubert calculus, S´ em","cited_arxiv_id":null,"evidence_quote":"Records the stability properties and Bruhat-order support facts for equivariant Schubert classes used in the grouping argument."}],"review_version":2}