{"id":"ae03ae19-d123-4d12-88d7-53cf9f75592f","arxiv_id":"1908.11224","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new shifted analogue of edge labeled tableaux is defined and conjecturally gives a Littlewood-Richardson rule for the D. Anderson-W. Fulton ring in equivariant Schubert calculus.","lead":"This mathematics paper introduces shifted edge labeled tableaux, a new combinatorial model conjectured to compute equivariant Schubert calculus of isotropic Grassmannians. It surveys the existing edge labeled tableau theory for Grassmannians and gives evidence for a conjectural Littlewood-Richardson rule.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Structure constants d^ν_{λ,μ} depend on rectification order (Examples 9.1, 9.9); no proof that the chosen row order matches the Anderson–Fulton ring, so Conjecture 10.1 may rest on an arbitrary convention.","rationale":"The paper's central claim is explicitly conjectural, and the reader's conditional verdict is appropriate. The most load-bearing weakness is the convention-dependence of shEqRect: the rule is not well-defined independently of rectification order, and the paper selects row order by fiat. Example 9.9 demonstrates that counts differ under column order. While the paper's commutativity proof (Theorem 9.5) uses row order and the n ≤ 4 checks support row order, no theoretical justification connects row order to the Anderson–Fulton ring. Additionally, equation (34) linking the AF ring to equivariant Schubert calculus is only a private communication, so the geometric interpretation is not independently established in this manuscript. A direct comparison for n = 5 would settle whether the row-rectification rule continues to match the AF constants. I agree with the reader's identification of the weakest assumption and see no reason to change the conditional verdict.","tokens_in":35142,"tokens_out":6418,"duration_ms":59640,"concrete_test":"Compute the Anderson–Fulton structure constants D^ν_{λ,μ} for all λ,μ,ν ⊆ ρ_5 directly from the Pfaffian presentation in Section 10.1 (or from L^ν_{λ,μ} via (34)), and compare with the row-rectification tableau counts. Focus on triples where row and column rectification differ; if column rectification matches AF instead, or if neither matches, Conjecture 10.1 is refuted; if row matches all cases, the row-rectification convention is empirically validated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The definition of d^ν_{λ,μ} in Section 9.2 uses shEqRect with row rectification order, but shEqRect is not independent of this order when edge labels are present (Example 9.1). Example 9.9 shows that column rectification gives 20 vs 16 for the same (λ,μ,ν), so the convention matters. Since the Anderson–Fulton structure constants are canonical, a valid Littlewood–Richardson rule should not depend on an arbitrary rectification order; the paper gives no argument that row order is the geometrically correct one. The only link to the AF ring is equation (34), credited to a private communication, so the numerical checks for n ≤ 4 cannot establish that row rectification is correct beyond small cases. If the true geometric rule corresponds to a different order, or if (34) is incorrect, Conjecture 10.1 fails.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper is a survey-announcement on equivariant Schubert calculus and edge labeled tableaux. It reviews the Thomas–Yong edge labeled jeu de taquin rule for equivariant structure constants of Grassmannians, the associated ballot rule, and applications to nonvanishing, saturation, Horn inequalities, and computational complexity. The new contribution is a shifted analogue of edge labeled tableaux: Section 9 defines shifted edge labeled tableaux with labels on diagonal edges, a jeu de taquin slide, and a rectification using a fixed “row rectification order.” It then defines d^ν_{λ,μ} as the number of such tableaux of shape ν/λ with |μ| labels that rectify to the superstandard tableau S_μ, and D^ν_{λ,μ} = 2^{ℓ(λ)+ℓ(μ)-ℓ(ν)-(|λ|+|μ|-|ν|)} z^{|λ|+|μ|-|ν|} d^ν_{λ,μ}. The authors prove commutativity of the resulting Z[z]-algebra R_n (Theorem 9.5), conjecture associativity (Conjecture 9.10), and conjecture that R_n is isomorphic to the Anderson–Fulton ring P with [λ]↦σ_λ (Conjecture 10.1). They verify special cases λ=(p) and λ=μ=ρ_n (Theorems 10.3 and 10.6) and report exhaustive checks for n≤4. The connection to equivariant Schubert calculus of the Lagrangian Grassmannian is made through identity (34) attributed to Anderson–Fulton private communication.","tokens_in":35358,"tokens_out":6591,"duration_ms":64357,"significance":"If true, Conjecture 10.1 would supply the first explicit Littlewood–Richardson-type rule for the equivariant Schubert structure constants of isotropic Grassmannians, a problem highlighted in Section 8. The paper’s proved results—commutativity of R_n, Theorem 8.2 relating the O and L structure constants, and the exact Pieri and top-class checks—provide nontrivial supporting evidence, and the survey portions are useful and clearly organized. The paper is also transparent about which statements are conjectural, and the numerical evidence is explicitly documented for small cases. However, the significance is conditional on an unresolved identification with the Anderson–Fulton ring and on the geometric correctness of the chosen rectification convention.","major_comments":[{"comment":"The structure constants d^ν_{λ,μ} and hence the ring R_n depend on the choice of row rectification order, as the paper itself shows in Example 9.1, and Example 9.9 records different counts (20 versus 16) under column rectification. Since the Anderson–Fulton structure constants are canonical, a proposed Littlewood–Richardson rule should be independent of such an arbitrary convention, or the paper should supply a geometric argument that row rectification is the correct convention. As written, the central conjecture 10.1 is tied to one fixed choice, and no evidence is given that this choice is forced by the geometry rather than by convenience.","section":"9.1–9.2, Examples 9.1 and 9.9"},{"comment":"The identity L^ν_{λ,μ}(α1↦z, α2↦0, ..., αn↦0) = D^ν_{λ,μ} is attributed to a private communication with Anderson and Fulton. This identity is load-bearing: it is used in Proposition 10.2 and in Lemma 10.5 to connect the tableaux counts to the equivariant geometry, and it converts Conjecture 10.1 into a statement about Schubert calculus. A private communication is not a verifiable reference. The paper should either provide a proof or a detailed derivation of (34), or clearly restate the affected results as conditional on an unpublished identity.","section":"10.1, Eq. (34)"},{"comment":"The proof of Theorem 10.6, which gives the check d^{ρ_n}_{ρ_n,ρ_n} = 2^{\\binom{n}{2}} = d^{ρ_n}_{ρ_n,ρ_n}, rests on Claim 10.9, whose proof is deferred with the statement “The proof of this claim is lengthy and will appear elsewhere.” As a result, the claimed verification of Conjecture 10.1 in a nontrivial maximal case is conditional. The theorem should be reworded as conditional, or the proof of Claim 10.9 should be included or made available in a cited preprint.","section":"10.2, Claim 10.9 and Theorem 10.6"}],"minor_comments":[{"comment":"References [65] and [67] are identical; one should be removed or the two entries should be distinguished.","section":"References"},{"comment":"In the expansion of P_{(2,1)}(x_1,x_2,x_3), the monomial x_1x_2^2 appears twice; the first occurrence should presumably be x_1^2x_2.","section":"7.3, displayed expansion of P_{(2,1)}"},{"comment":"The sentence “This has one diagonal box but three diagonal edges” is confusing; the definition of a diagonal edge should be stated precisely and illustrated, since a single diagonal box would normally have one southern edge.","section":"9.1, definition of diagonal edge"},{"comment":"The counts 20 and 16 are stated without explanation; a short description of the computation or a table of the relevant tableaux would make the order-dependence check reproducible.","section":"Example 9.9"},{"comment":"The row rectification order is described informally in Example 9.1 as “choosing the southmost inner corner”; Section 9.2 should give a precise algorithm for this order, especially because the definition of shEqRect depends on it.","section":"9.1, row rectification order"},{"comment":"The variables α_i in Eq. (34) are not reintroduced at the start of Section 10.1; the substitution should be stated explicitly, including the role of the indeterminate z.","section":"10.1, Eq. (34)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is an honest research announcement, and the survey portions are valuable. However, the main tableaux construction depends on a non-canonical rectification order, and the central numerical bridge to the Anderson–Fulton ring is an unpublished identity from a private communication. I would want these points addressed, or at least made fully explicit as caveats with a proof sketch of (34), before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, this is a survey-and-announcement chapter from Yong's school. The genuinely new thing is the shifted edge labeled tableau model and the ring R_n built from it. Conjecture 10.1 identifies R_n with the Anderson-Fulton ring, which would give the long-sought equivariant Littlewood-Richardson rule for isotropic Grassmannians. That is a substantial research program, and the paper gives concrete evidence: the ring is proved commutative via a Fomin growth-diagram argument, the Pieri-type and maximal-case coefficients are computed in closed form, and there is a new proof of the O/L structure constant relation. The small cases are checked up to n=4 and many n=5 cases. The writing is good, and the survey parts are a useful entry point.\n\nThe soft spot is the one the stress-test flags. The definition of d^ν_{λ,μ} in Section 9.2 uses row rectification order, and shEqRect is not independent of that order when edge labels are present. Example 9.1 shows the two orders give different rectification target shapes, and Example 9.9 gives different counts (20 vs 16) for the same triple. The paper simply declares row rectification to be the rule. There is no argument that this convention is the one that matches the Anderson-Fulton ring. The only bridge to the geometry is equation (34), attributed to a private communication, so the numerics cannot distinguish between conventions beyond small cases. If the correct geometric rule requires a different order, the conjecture fails. That is a genuine load-bearing concern, and the authors are upfront about it, but they don't resolve it.\n\nWhat is proved is real. The commutativity proof is nontrivial, the special cases are proven, and the O/L theorem (Theorem 8.2) is a solid contribution. The paper is honest about what is conjectural: Conjecture 9.2 (integrality), Conjecture 9.10 (associativity), and Conjecture 10.1 (isomorphism) are all clearly marked.\n\nProportionately: this is an announcement, not a complete theory. The soft spots are inherent to that genre. I would like to see a more serious attempt to justify the rectification order, or at least to check whether column order can ever match the AF ring. That would either strengthen or kill the conjecture, and it is the first thing I would ask a future version to address.\n\nWho is this for? Anyone working on Schubert calculus, especially equivariant and isotropic cases. It deserves a serious referee, not a desk reject. I would recommend peer review with the expectation of a revised version. I'd also cite it if I work in this area, primarily as the source of the shifted edge labeled model and the conjecture.","headline":"New shifted edge labeled tableaux give a plausible conjectural LR rule for isotropic Grassmannians, but the central ring isomorphism is open and depends on a rectification-order convention with no geometric justification.","tokens_in":35853,"tokens_out":2623,"would_cite":true,"duration_ms":24804,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E10","14M15","05E05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proposes a conjectural Littlewood-Richardson rule for the equivariant Schubert calculus of isotropic Grassmannians, built from shifted edge labeled tableaux and the Anderson-Fulton ring.","keywords":["shifted edge labeled tableaux","equivariant Schubert calculus","isotropic Grassmannians","jeu de taquin","Littlewood-Richardson rule","Anderson-Fulton ring","growth diagrams","commutative ring"],"falsifier":"For any strict partitions $\\lambda,\\mu,\\nu \\subseteq \\rho_5$ not among the cases already checked, compute $d^\\nu_{\\lambda,\\mu}$ by the row-rectification count and compare it with the Anderson-Fulton coefficient $D^\\nu_{\\lambda,\\mu}$ obtained from the Pfaffian formula in Section 10; a single disagreement would disprove Conjecture 10.1.","tokens_in":34949,"feed_emoji":"🧮","tokens_out":8193,"duration_ms":74402,"temperature":0.7,"pith_summary":"The paper is partly a survey of edge labeled tableaux as a model for equivariant Schubert calculus of Grassmannians, and partly an announcement of a shifted analogue aimed at maximal orthogonal and Lagrangian Grassmannians. It introduces shifted edge labeled tableaux, in which edge labels are allowed only on the southern edges of diagonal boxes, and defines a product on formal basis elements $[\\lambda]$ by counting tableaux of shape $\\nu/\\lambda$ that row-rectify to a fixed superstandard tableau of shape $\\mu$. The central conjecture is that this product makes the ring $R_n$ isomorphic to the Anderson-Fulton ring $P$, whose structure constants specialize to equivariant Schubert structure constants of the Lagrangian Grassmannian. If true, this supplies the missing combinatorial Littlewood-Richardson rule for the equivariant cohomology of the isotropic Grassmannians and opens the same circle of questions (nonvanishing, saturation, complexity) that has been answered for ordinary Grassmannians. The paper proves commutativity of $R_n$ by growth diagrams and verifies the isomorphism in all cases $n \\le 4$ and many $n = 5$ cases.","feed_headline":"Shifted tableaux could solve isotropic Schubert calculus","feed_subtitle":"A conjectural ring-counting rule would yield the missing Littlewood-Richardson formula for equivariant isotropic Schubert calculus.","key_machinery":"The key machinery is the shifted edge labeled tableau together with its shifted jeu de taquin slides: the usual two-box slides (J1) and (J2) are supplemented by diagonal-edge moves (J3') and (J4') that move a box label past an edge-label set or replace the smallest edge label. Rectification is fixed to row order, and the count of tableaux rectifying to the superstandard tableau $S_\\mu$ produces the structure constants. A growth-diagram formalism based on e-partitions (partitions with edge-label multiplicities recorded on diagonal edges) is then used to prove that the resulting product is commutative, by reflecting the diagram across its antidiagonal.","core_discovery":"The central discovery is a new combinatorial object, the shifted edge labeled tableau: a filling of a shifted skew shape in which every label appears once, rows and columns increase, and the southern edge of each diagonal box may carry a subset of labels. Counting such tableaux of shape $\\nu/\\lambda$ that rectify to a fixed superstandard tableau under a row-by-row jeu de taquin order defines coefficients $d^\\nu_{\\lambda,\\mu}$, packaged as $D^\\nu_{\\lambda,\\mu} = 2^{L(\\nu;\\lambda,\\mu)-\\Delta(\\nu;\\lambda,\\mu)} z^{\\Delta(\\nu;\\lambda,\\mu)} d^\\nu_{\\lambda,\\mu}$. The formal span of symbols $[\\lambda]$ with product $[\\lambda] \\star [\\mu] = \\sum_\\nu D^\\nu_{\\lambda,\\mu}[\\nu]$ forms a ring $R_n$ that the authors conjecture is isomorphic to the Anderson-Fulton ring $P$ with basis $\\sigma_\\lambda$. Under that isomorphism, the structure constants become the equivariant Schubert structure constants of the Lagrangian Grassmannian after specializing $\\alpha_1 = z$ and $\\alpha_i = 0$ for $i > 1$. Evidence includes a growth-diagram proof of commutativity, matching special cases $d^\\lambda_{\\lambda,(p)} = \\binom{\\ell(\\lambda)}{p}2^{p-1}$ and $d^{\\rho_n}_{\\rho_n,\\rho_n} = 2^{\\binom{n}{2}}$, and exhaustive verification for $n \\le 4$.","pith_inferences":["The dependence of rectification on row order, highlighted by Example 9.1, is the most delicate point in the construction: if the geometrically correct jeu de taquin uses a different order, the entire conjectural dictionary would shift, and Example 9.9 shows the order sensitivity is real.","The equality $d^{\\rho_n}_{\\rho_n,\\rho_n} = 2^{\\binom{n}{2}}$ invites a bijective explanation: staircase-shaped shifted edge labeled tableaux rectifying to the superstandard tableau may correspond to labeled graphs on $n$ vertices, Aztec diamond tilings, or Gelfand-Tsetlin patterns, exposing recursive structure that could help with the general conjecture.","Associativity of $R_n$ is a finite check for each fixed $n$ because $R_n$ is finitely generated as a $\\mathbb{Z}[z]$-module; verifying it for $n = 5$ computationally would be a concrete intermediate test that does not require the geometric comparison.","If the conjecture holds, the specialization $\\alpha_1 = z$ suggests that shifted edge labeled tableaux may also serve as a combinatorial model for a one-dimensional torus action on the Lagrangian Grassmannian, potentially giving a GKM-style proof independent of the Anderson-Fulton algebra."],"forward_implications":["If Conjecture 10.1 holds, the coefficients $D^\\nu_{\\lambda,\\mu}$ give an explicit Littlewood-Richardson rule for the equivariant Schubert structure constants of the Lagrangian Grassmannian after the specialization $\\alpha_1 = z$, $\\alpha_i = 0$ for $i > 1$.","The conjectured isomorphism implies Conjecture 9.2: every $D^\\nu_{\\lambda,\\mu}$ is an integer polynomial in $z$, with nonnegative coefficients inherited from equivariant positivity.","Commutativity of $R_n$ is already established, and the Pieri-type cases match the Anderson-Fulton ring, so the remaining gap to the full isomorphism is associativity together with a general comparison of structure constants.","A proof of associativity combined with the Pieri rule would complete the standard associativity-argument route to the isomorphism, mirroring the classical proof of the Littlewood-Richardson rule."],"supporting_citations":[{"why":"Introduces the ordinary edge labeled tableau model and its jeu de taquin rule for equivariant Schubert calculus of Grassmannians, which the shifted construction directly extends.","marker":"[66]"},{"why":"Supplies excited Young diagram formulas for restriction on the Lagrangian Grassmannian, used to identify the Anderson-Fulton coefficients and to compute the Pieri-type special cases.","marker":"[28]"},{"why":"Proves the equivalence between orthogonal and Lagrangian equivariant structure constants, the relation the shifted tableau ring is designed to capture.","marker":"[55]"},{"why":"Establishes that Schur P- and Q-functions represent the Schubert classes of the maximal orthogonal and Lagrangian Grassmannians, providing the algebraic setting for the conjecture.","marker":"[50]"},{"why":"Develops shifted tableaux and ballot rules for the classical shifted Littlewood-Richardson coefficients, the base theory on which the new edge-labeled generalization is built.","marker":"[63]"},{"why":"Provides Fomin's growth diagram formulation of jeu de taquin, adapted here to e-partitions to prove commutativity of the ring $R_n$.","marker":"[62]"},{"why":"Formulates the equivariant Horn-type conjecture for orthogonal Grassmannians and reports computational checks for small staircases, framing the open problem that the new rule targets.","marker":"[44]"}],"fun_headline_variants":["Shifted tableaux yield conjectural Schubert rule","New tableau rule for isotropic Schubert calculus","Isotropic Schubert numbers via shifted tableaux","Edge-labeled shift: key to equivariant Schubert constants"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The product coefficients are defined using a fixed row-by-row order of sliding moves, but the final tableau can depend on that order once edge labels are present; if the correct geometric rule needs a different order, the conjectured ring would not be the Anderson-Fulton ring.","fun_headline_variants_meta":{"raw":{"variants":["Shifted tableaux yield conjectural Schubert rule","New tableau rule for isotropic Schubert calculus","Isotropic Schubert numbers via shifted tableaux","Edge-labeled shift: key to equivariant Schubert constants"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000936,"raw_usage":{"total_tokens":4010,"prompt_tokens":960,"completion_tokens":3050,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":576,"completion_tokens_details":{"reasoning_tokens":2989}},"tokens_in":576,"tokens_out":3050,"duration_ms":22039,"temperature":1.0,"reasoning_tokens":2989,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:20:46.043096+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For any strict partitions $\\lambda,\\mu,\\nu \\subseteq \\rho_5$ not among the cases already checked, compute $d^\\nu_{\\lambda,\\mu}$ by the row-rectification count and compare it with the Anderson-Fulton coefficient $D^\\nu_{\\lambda,\\mu}$ obtained from the Pfaffian formula in Section 10; a single disagreement would disprove Conjecture 10.1.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the ordinary edge labeled tableau model and its jeu de taquin rule for equivariant Schubert calculus of Grassmannians, which the shifted construction directly extends."},{"cited_title":"Ikeda and H","cited_arxiv_id":null,"evidence_quote":"Supplies excited Young diagram formulas for restriction on the Lagrangian Grassmannian, used to identify the Anderson-Fulton coefficients and to compute the Pieri-type special cases."},{"cited_title":"The A.B.C.Ds of Schubert calculus","cited_arxiv_id":"1906.03646","evidence_quote":"Proves the equivalence between orthogonal and Lagrangian equivariant structure constants, the relation the shifted tableau ring is designed to capture."},{"cited_title":"Pragacz, Algebro-geometric applications of Schur S- and Q-polynomials","cited_arxiv_id":null,"evidence_quote":"Establishes that Schur P- and Q-functions represent the Schubert classes of the maximal orthogonal and Lagrangian Grassmannians, providing the algebraic setting for the conjecture."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Develops shifted tableaux and ballot rules for the classical shifted Littlewood-Richardson coefficients, the base theory on which the new edge-labeled generalization is built."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides Fomin's growth diagram formulation of jeu de taquin, adapted here to e-partitions to prove commutativity of the ring $R_n$."},{"cited_title":"Monical, Polynomials in algebraic combinatorics , Ph.D thesis, University of Illinois at Urbana- Champaign, 2018","cited_arxiv_id":null,"evidence_quote":"Formulates the equivariant Horn-type conjecture for orthogonal Grassmannians and reports computational checks for small staircases, framing the open problem that the new rule targets."}],"review_version":1}