{"id":"32282aab-3d4c-411d-8f67-a48e6c1d44d1","arxiv_id":"2507.11516","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Inversions tableaux are a new pipe-dream-equivalent tableau model for Schubert polynomials and Stanley symmetric functions, specializing to reverse semistandard Young tableaux for Grassmannian permutations.","lead":"New 'inversions tableaux' give a combinatorial model for Schubert polynomials and Stanley symmetric functions, with a direct bijection to pipe dreams and an exact Grassmannian reduction to standard semistandard Young tableaux. The model is designed as a stepping stone for proving Rubey's chute moves conjecture in forthcoming work.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The inverse bijection in Theorem 3.13 fails as written on a 3-box counterexample: after reversing the Lemma 3.11 order, the second step tries to cross pipes 1 and 3 at row 1, but the previously added row-2 crossing leaves them non-adjacent below it.","rationale":"The reader's weakest assumption was precisely the well-definedness of the inverse map in Theorem 3.13, and the stress-test confirms and sharpens that concern with a concrete counterexample to the proof as written. The construction in the proof, if followed literally, fails on the unique inversions tableau of the simplest non-trivial permutation, w = 321. This is not merely a missing rigor detail: the two-case adjacency argument in Figure 4 makes a directional error, asserting adjacency below a crossing when the crossing actually creates adjacency above it. Because this inverse map is the only proof of the central generating-function identity (Theorem 3.13), the result is currently unproven by the manuscript's arguments. The theorem itself may still be true — the same T does correspond to a valid pipe dream under the forward map, and the construction works if the reversal is omitted — so the appropriate verdict remains CONDITIONAL rather than REJECT. The fix is likely local (correct the order of additions and the adjacency argument), but as submitted the main proof contains a genuine error. I agree with the reader's conditional assessment and do not recommend changing the verdict.","tokens_in":19207,"tokens_out":25329,"duration_ms":257819,"concrete_test":"Implement the inverse algorithm exactly as written for w = 321: enumerate all T ∈ IT(321), form the unique total order from Lemma 3.11, reverse it, and simulate adding crossings to a pipe dream (storing the current pipe positions). At each step, check whether pipes i_m and j_m are adjacent in the row T(i_m,j_m); if not, record a failure. For the unique valid T, the second step (adding (1,3) at row 1 after (2,3) at row 2) should fail. If it does, the proof of Theorem 3.13 as stated is invalid; as a control, rerun with the non-reversed total order and confirm that the construction succeeds, which would indicate the reversal is a typo and the theorem may be salvageable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 3.13 constructs the inverse of φ by taking a total order from Lemma 3.11, reversing it, and adding crossings in that reversed order at rows T(i_m,j_m). The two-case argument in Figure 4 is meant to justify that the relevant pipes are always adjacent at the required row. This argument is wrong for a concrete, valid tableau. Let w = 321 ∈ S_3 and let T be the inversions tableau with T(1,2)=1, T(1,3)=1, T(2,3)=2; this satisfies IT1, IT2, IT3' and is the unique tableau in IT(321). The unique total order that satisfies Lemma 3.11 condition 3 is (1,2), (1,3), (2,3), since the other tie-break makes the hook at (1,3) unbalanced. Reversing gives (2,3), (1,3), (1,2). Starting from the identity pipe dream, the first step adds the crossing of pipes 2 and 3 at row 2. After that crossing, pipes 1 and 3 are adjacent only above row 2; below row 2, pipe 3 is still in its original column, so pipes 1 and 3 are not adjacent at row 1. The proof's second case asserts that because pipe 3 crossed a smaller pipe at row 2 and T(1,3)=1 ≤ 2−1, pipes 1 and 3 will be next to each other for all rows below 2. This is false: the crossing at row 2 makes them adjacent above it, not below. Hence the step of adding (1,3) at row 1 cannot be performed, and the inverse construction as written does not produce a reduced pipe dream. The subsequent well-definedness discussion about braid moves cannot repair this because it rests on the same adjacency claim. Thus the main bijection is not established by the given proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces \"inversions tableaux,\" fillings of an inversions diagram of a permutation w, and claims that the generating function over these tableaux equals the Schubert polynomial S_w. The main tool is a weight-preserving bijection between reduced pipe dreams and inversions tableaux (Theorem 3.13). The paper also derives formulae for the lexicographically maximal and minimal monomials of Schubert polynomials, characterizes inversions tableaux for dominant and Grassmannian permutations, defines unbounded inversions tableaux for Stanley symmetric functions, and discusses chute moves, a new \"mediocre Bruhat order,\" and Lehmer tableaux as preparation for a forthcoming proof of Rubey's chute-move lattice conjecture.","tokens_in":19563,"tokens_out":17845,"duration_ms":219839,"significance":"If Theorem 3.13 is correct, inversions tableaux form a genuinely new tableau model for Schubert polynomials that specializes directly to reverse semistandard Young tableaux in the Grassmannian case. The paper is clearly written, contains many instructive examples, and offers attractive applications: a simple encoding of Lehmer codes, extremal monomials, dominant permutations, and a bridge to chute-move posets. However, the main bijection is not established as written: the inverse construction in the proof of Theorem 3.13 fails on a valid three-box example, and the paper conflates weak Bruhat order with the inversion-containment order. These are load-bearing issues, so the central claim currently lacks a rigorous proof. The remaining sections contain useful structural results, but most depend on the main bijection or on Proposition 6.3, which is also under-proved.","major_comments":[{"comment":"The inverse construction is not well-defined. For w = 321, the tableau T with T(1,2) = 1, T(1,3) = 1, T(2,3) = 2 is a valid inversions tableau in IT(321). The only total order satisfying Lemma 3.11(3) is (1,2), (1,3), (2,3), so the proof’s reversed order is (2,3), (1,3), (1,2). After the first step adds the crossing of pipes 2 and 3 at row 2, pipes 1 and 3 are not adjacent at row 1, so the second step cannot add the crossing of pipes 1 and 3 at row 1. Thus the induction in the paragraphs beginning “To complete the proof” and “Again by Proposition 3.7” fails on a concrete tableau, and the two cases illustrated in Figure 4 do not cover this configuration. Since the inverse map is the load-bearing step connecting tableaux to pipe dreams, the proof of the main theorem is incomplete.","section":"Section 3, proof of Theorem 3.13"},{"comment":"The statement “u < w in weak Bruhat order exactly when Inv(u) ⊂ Inv(w)” is false for the left weak order defined in Definition 2.1. For example, in S_3, Inv(132) = {(2,3)} is contained in Inv(231) = {(1,3),(2,3)}, but 132 and 231 are not related by the stated left weak cover relations. The poset of inversion sets ordered by containment is the strong Bruhat order, not the weak Bruhat order. This matters because Proposition 3.7 and the proof of Theorem 3.13 identify the poset ID_n with weak Bruhat order and use maximal chains in it; reversing a balanced-tableau order does not, in general, produce a chain in weak Bruhat order. The terminology and the underlying poset identification need to be corrected before the bijection argument can be assessed.","section":"Section 2, Definition 2.1 and following paragraph"},{"comment":"Proposition 6.3, which asserts that chute moves on tableaux correspond exactly to chute moves on pipe dreams, is not proved in sufficient detail. The forward direction is dismissed with “we can see that the corresponding inversions tableaux satisfy all the conditions,” and the reverse direction contains an argument whose key step, “by 3, we must have T(l,j) = T′(l,j)”, is not justified from the stated conditions; the reference appears to conflate Definition 6.2(3) with a conclusion that is being proved. The figures and the assertion that the pipes form the rectangular grid of Definition 6.1 require a careful formal proof. Since this proposition is the bridge between the tableau model and Rubey’s chute-move posets, it is load-bearing for the paper’s claimed applications.","section":"Section 6, Proposition 6.3"}],"minor_comments":[{"comment":"In the proof of Lemma 3.15, the first induction step should cite condition (IT3), not (IT3’), because (IT3’) is the statement being proved; the same mislabeling occurs later when bounding T(k,k+1) by k.","section":"Lemma 3.15 proof"},{"comment":"The statement of Corollary 6.4 uses m both as a dummy index and as the entry to be found: “T(i(x), k) = m” should be “T(i_x, k) = x”, and the notation i_x should be defined explicitly.","section":"Corollary 6.4"},{"comment":"The phrase “integers k ∈ 1, 2, . . . , T(i,j)” is malformed; it should read “integers k ∈ {1, 2, . . . , T(i,j)}”.","section":"Definition 6.13"},{"comment":"The sentence “Sliding shaded boxes together and rotating by 180 degrees, we then get a flag SSYT” skips the verification that the flag bounds are exactly (b_1, . . . , b_{n-k}) and that the rotation preserves the strict column and weak row conditions; a more explicit bijection on entries would clarify the proof.","section":"Proposition 5.4 proof"},{"comment":"There are numerous typos, including “necesessarily” in Lemma 3.11, “permtuation” in the proof of Theorem 6.12, “filed” in Definition 2.6, and inconsistent uses of “tableaux” versus “tableau”; a careful copyedit is needed.","section":"Various"}],"recommendation":"major_revision","confidential_remarks":"The central construction is appealing and may well be repairable, but the submitted proof contains a concrete counterexample to the inverse map as written, and the weak/strong Bruhat conflation suggests that Section 3 needs a substantial rewrite rather than local edits. I recommend that the editor ask the authors to supply a complete proof of the inverse bijection, ideally with the order of addition specified correctly and verified against small cases, and to clarify the poset used in Proposition 3.7. The paper should not be sent to typesetting in its current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is a real contribution to the pipe dream / balanced tableau literature, and I think the main theorem is true. But the proof of the main bijection has a gap that needs to be closed before I'd trust it as written, and the stress-test note points at exactly that gap.\n\nWhat's new: inversions tableaux. They are a modification of Edelman-Greene balanced staircase tableaux, with a rectangle rule (IT1), column-strictness (IT2), and a row bound (IT3). The centerpiece is the claimed weight-preserving bijection with reduced pipe dreams (Theorem 3.13). If it holds, it gives a tableau model for Schubert polynomials that specializes to reverse SSYT in the Grassmannian case, has clean Lehmer-code consequences, and gives a handle on chute moves. Sections 4-5 also do genuinely useful work: explicit extremal tableaux, a clean characterization of dominant permutations, and the Grassmannian/inverse Grassmannian specializations. No fitted parameters and no circularity: the external benchmark is the Bergeron-Billey pipe dream formula.\n\nThe soft spot is the inverse direction of Theorem 3.13. After Lemma 3.11, the proof takes a total order, reverses it, and adds crossings one by one in rows T(i_m,j_m). The well-definedness of that process is load-bearing, and the two-case argument in Figure 4 is a sketch, not a proof. I checked the w=321 example from the stress-test. The unique tableau is T(1,2)=T(1,3)=1, T(2,3)=2, and the only total order compatible with Lemma 3.11 is (1,2),(1,3),(2,3). Reversing gives (2,3) first, and after crossing pipes 2 and 3 in row 2, pipes 1 and 3 are not adjacent in row 1. The second case in the proof claims they are, and that does not follow. So the inverse construction as written does not produce the required pipe dream for this example. The theorem may still be true, but this proof does not establish it, and the later braid-move discussion is too quick to repair the gap.\n\nMinor issues: Section 6 asserts the forward direction of Proposition 6.3 without proof; Corollary 6.4 has an obvious typo (m for x); the maximal/minimal terminology is confusing in a couple of places. All fixable.\n\nWho should read it: anyone working on Schubert polynomials or Rubey's chute move posets. It deserves a serious referee, but the referee report should ask for a complete proof of the inverse bijection. I would send it out.","headline":"A genuinely new tableau model for Schubert polynomials with a plausible main theorem, but the inverse bijection proof has a concrete gap that needs closing.","tokens_in":20167,"tokens_out":16054,"would_cite":true,"duration_ms":187459,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","05E10","14M15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Schubert polynomials are generated by inversions tableaux, a new filling model on inversion diagrams.","keywords":["inversions tableaux","Schubert polynomials","Stanley symmetric functions","pipe dreams","balanced tableaux","weak Bruhat order","Lehmer code","chute moves"],"falsifier":"Enumerate all fillings of the inversion diagram of a small permutation such as 4321 that satisfy rules (IT1), (IT2), and (IT3'), and run the inverse construction from the proof of Theorem 3.13 on each one; a single filling for which the construction either fails to produce a reduced pipe dream or forces two pipes to cross twice would disprove the claimed bijection.","tokens_in":18924,"feed_emoji":"📐","tokens_out":3311,"duration_ms":38989,"temperature":0.7,"pith_summary":"The paper introduces inversions tableaux, a new combinatorial model that represents each Schubert polynomial as a weighted sum over tableaux of a fixed staircase shape determined by a permutation. The central result is that for every permutation, the Schubert polynomial equals the generating function of these tableaux, with the weight recording how many entries equal each integer. This gives a tableau description analogous to the classical semistandard Young tableau formula for Schur polynomials, and it specializes to reverse semistandard Young tableaux in the Grassmannian case. The model is built by relaxing Edelman and Greene's balanced staircase tableaux to a weakly balanced condition, and it is proved equivalent to pipe dreams by a weight-preserving bijection.","feed_headline":"New tableaux generate Schubert polynomials","feed_subtitle":"A staircase filling rule on inversion diagrams matches pipe dreams and reduces to Schur tableaux for Grassmannian permutations.","key_machinery":"The central object is the inversions tableau, defined on the staircase-shaped inversion diagram of a permutation, with shaded boxes carrying entries. The load-bearing condition is weak balance, a rectangle rule stating that for every triple i