{"id":"d0140e2c-298e-4f0b-a686-1922e61bb57c","arxiv_id":"2506.07306","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New combinatorial formulas, via bumpless pipe dreams and pipe dreams, express Grothendieck polynomials in the Schubert basis and Schubert polynomials in the Grothendieck basis, and extend to the back stable setting.","lead":"Grothendieck polynomials can be expanded into Schubert polynomials, and vice versa, using tilings called bumpless pipe dreams and pipe dreams. This paper gives new diagram-based formulas for these expansions, including the first formula for the back stable version.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 7.8 is stated in the wrong direction for the use made of it in Theorem 1.9: from κ(P′)=B′ alone one cannot conclude that B′ is the column-insertion of B unless the commutation property of [GH23, Thm 4.5] is assumed.","rationale":"The paper's new BPD formulas are plausible and the worked examples are internally consistent, including the non-multiplicity-free checks in Example 7.12 and the back-stable example in Section 8.5. The chain model in Section 6.3 gives independent-looking evidence for the pipe-dream formulas. The load-bearing issue is the transfer from pipe dreams to BPDs. The reader's weakest assumption already identifies Lemma 7.8 and [GH23, Theorem 4.5] as the external black box; I agree with that concern and sharpen it: as printed, Lemma 7.8 is not merely unproved but logically reversed relative to Theorem 1.9's use of it. The proof step needs the forward commutation statement, which is only available by interpreting [GH23, Thm 4.5] in a way that is not spelled out. This does not force rejection, because the missing statement is likely true and could be supplied from the Gao–Huang paper or proved directly; but until it is verified, the central BPD change-of-basis formulas and the back-stable expansion are conditional on an unverified external compatibility result. I therefore keep the reader's CONDITIONAL verdict rather than moving to ACCEPT or REJECT.","tokens_in":40630,"tokens_out":10665,"duration_ms":115586,"concrete_test":"Check the exact statement of [GH23, Theorem 4.5] in the column-weight-preserving version: does it assert κ(P∪{(i,j)}) = col-ins_j(κ(P)) for j = α(δ(P)), for all P, including non-reduced pipe dreams? Then verify this commutation computationally on a small case such as w = 2143: enumerate P ∈ Pipes(2143), compute κ(P) using the canonical bijection, form P′ = P∪{(1,1)}, compute κ(P′), and compare with the column-insertion algorithm of Section 7.2 applied to κ(P). If all pairs agree, restate Lemma 7.8 in the forward direction and the proof of Theorem 1.9 is repaired; if any pair disagrees, Theorems 1.1, 1.2, and 8.1 are not established as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 1.9, the author sets P′ = P ∪ {(i,j)} and B′ = κ(P′), then says “by Lemma 7.8, B′ is obtained from B by column insertion.” Lemma 7.8 as printed, however, has the hypothesis “B 7→ B′ by insertion into column j of B” and concludes κ(P′−{(i,j)}) = B. That is the inverse direction, not the direction used. The step actually needed is the commutation κ(P∪{(i,j)}) = ins_j(κ(P)) for j = α(w), or at least the converse of Lemma 7.8. The paper cites [GH23, Theorem 4.5] for this, but neither the statement nor the proof of that commutation is reproduced. Since Theorem 1.9 is the bridge converting the fully proved pipe-dream formulas (Theorems 1.5 and 1.6) into the BPD formulas (Theorems 1.1 and 1.2), and since Theorem 8.1 inherits the same dependence, the central claim rests on an unverified external compatibility statement. There is a further subtlety: Theorem 1.1 involves pipe dreams whose co-pipe dream is reduced but whose pipe dream need not itself be reduced, and Theorem 1.2 involves reduced pipe dreams; the printed Lemma 7.8 does not make clear which reducedness conventions are covered. If [GH23, Thm 4.5] is reduced-only or is not exactly the commutation asserted here, the transfer argument for non-reduced terms fails.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops combinatorial change-of-basis formulas between Grothendieck and Schubert polynomials. Theorems 1.5 and 1.6 reformulate Lenart's and Lascoux's pipe-dream-style rules and give new proofs via Knutson's co-transition recurrences; Theorem 6.7 adds a chain model in Bruhat order. The main new claims are Theorems 1.1 and 1.2, which state analogous formulas in terms of bumpless pipe dreams and their co-objects, and Theorem 8.1, which uses these to expand back stable Grothendieck polynomials into back stable Schubert polynomials. The bridge from the pipe-dream results to the BPD results is Theorem 1.9, asserting that the Gao--Huang canonical bijection preserves co-permutations. The paper contains many worked examples and detailed, largely self-contained arguments, with the Gao--Huang theorem [GH23] used as an external input.","tokens_in":40979,"tokens_out":17435,"duration_ms":167230,"significance":"If correct, the BPD formulas are a genuinely useful contribution: BPDs are naturally back stable, and Theorem 8.1 provides the first combinatorial expansion of back stable Grothendieck polynomials into back stable Schubert polynomials. The paper also gives new proofs of the Lenart and Lascoux pipe-dream rules, and the chain-based description in Theorem 6.7 is a nice byproduct. The exposition is generally careful, the examples are instructive, and the paper builds on established machinery rather than introducing ad hoc assumptions. However, two load-bearing gaps prevent the main theorems from being fully established as written: the direction of Lemma 7.8 is used in Theorem 1.9 in a way that does not follow from its statement, and the planar-history sign conventions in Lemma 3.5 are applied with the opposite direction in Lemmas 4.7, 4.11, and 7.7. These issues appear local and repairable, but until they are fixed the central transfer to BPDs is not proven.","major_comments":[{"comment":"Lemma 7.8 is stated with hypothesis “B → B′ by insertion into column j of B” and conclusion κ(P′−{(i,j)}) = B. In the proof of Theorem 1.9, after setting P′ = P ∪ {(i,j)} and B′ = κ(P′), the text says “by Lemma 7.8, B′ is obtained from B by column insertion.” This is the converse of the printed lemma, not its statement. The needed property is κ(P∪{(i,j)}) = ins_j(κ(P)), or at least the converse of Lemma 7.8, which is neither stated nor proved. Theorem 1.9 is the only bridge from the fully proved pipe-dream formulas (Theorems 1.5 and 1.6) to the BPD formulas (Theorems 1.1 and 1.2), and Theorem 8.1 inherits the same dependence. This is a load-bearing gap.","section":"§7.3, Lemma 7.8 and Theorem 1.9"},{"comment":"There is a sign conflict between Lemma 3.5 and its uses. For NW/SE planar histories, Lemma 3.5(2) says that δ(P)(b) > δ(P)(a) if and only if the pipes exiting in rows a and b do not cross. Consequently, a descent at k, i.e. δ(k) > δ(k+1), implies that the pipes exiting in rows k and k+1 do cross. Lemma 4.7 and Lemma 4.11 use a descent of the co-object to infer the opposite, namely that those pipes do not cross. Similarly, the final subcase of Lemma 7.7 uses crossing of pipes in a co-BPD to infer an ascent at i, whereas Lemma 3.5(2) gives a descent. Thus the descent-set containments and the key computation δ(ˇB) = δ(ˇB′)□τ_i are not established with the conventions as printed. Either Lemma 3.5's direction, the reading conventions for co-objects, or the affected proofs must be corrected; Theorems 1.1, 1.2, and their back-stable version depend on these lemmas.","section":"§3.1, Lemmas 3.5, 4.7, 4.11, and 7.7"},{"comment":"The proof of Lemma 7.7 is an extensive case analysis that is presented largely through diagrams rather than coordinate specifications. In particular, the claims that the interior replacements “consist of repeated applications” of the displayed local moves, and that each displayed move preserves the co-permutation, are asserted rather than demonstrated. The classification of boundary configurations also depends on the same sign convention discussed above. Because Lemma 7.7 is essential to the proof of Theorem 1.9, the argument would need to be written in a more formal, checkable form before the central claim can be regarded as verified.","section":"§7.2, Lemma 7.7"}],"minor_comments":[{"comment":"The local label-propagation rules for the four orientations are conveyed only by pictures; since Lemmas 3.5–3.7 are purely combinatorial statements about these rules, an explicit textual or coordinate description of the label propagation would greatly help the reader verify the claimed equivalences.","section":"§3.1, (3.1)–(3.4)"},{"comment":"The tile-by-tile replacement defining the bijection from BPDs to co-BPDs is described only by images; a coordinate description of each of the six replacements would make the well-definedness and invertibility of the map checkable rather than immediate from the pictures.","section":"§4.4, Lemma 4.9"},{"comment":"In the path model, the definition of word(p) and the direction in which the word is read should be stated more explicitly, since Theorem 6.7(3) applies w0·δ(word(p)) and the example’s path lists are otherwise easy to misread.","section":"§6.3, Theorem 6.7 and Example 6.14"},{"comment":"The step from “the tile at position (k,1) is not a cross in P” to “the tile at (k+1,1) is a cross in ˇP” is not immediate; a short diagram or coordinate argument would clarify the geometry of the associated co-pipe dream.","section":"§4.2, Lemma 4.7 proof"}],"recommendation":"major_revision","confidential_remarks":"The paper is potentially correct and the new formulas are appealing, but the proof of the central transfer result Theorem 1.9 is not complete as written. The key point to check in a revision is the exact relation between Lemma 7.8 and [GH23, Theorem 4.5]: the converse direction is needed, and it should be stated and proved or cited with full precision. The sign inconsistency around Lemma 3.5 is concerning but seems local; I would want a corrected proof before accepting. The back-stable application in Theorem 8.1 is contingent on the same fix."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the paper delivers a genuinely new and useful back stable Grothendieck-to-Schubert expansion (Theorem 8.1), and the BPD versions of the change-of-basis formulas (Theorems 1.1 and 1.2) are natural and likely correct. Second, as written there are two presentation bugs that block certification: Lemma 3.5 has a sign condition that is wrong for SW planar histories, and the proof of Theorem 1.9 uses Lemma 7.8 in the reverse of the direction stated. Both look fixable, but the paper needs a revision before the main theorems are airtight.\n\nWhat's new and good: the pipe dream proofs of Lenart and Lascoux via co-transition are clean, the chain formulation in Theorem 6.7 is a nice byproduct, and the co-permutation preservation result (Theorem 1.9) is a genuinely interesting structural statement about the Gao–Huang bijection. The back stable expansion is the standout; it gives the first combinatorial formula of its kind and will likely be cited.\n\nSoft spots. Lemma 3.5 groups NE and SW planar histories into one statement, but for SW the crossing-to-order relation is opposite: a crossing produces a descent, not an ascent. The proofs of Lemmas 4.7, 4.11, and 7.7 use the correct version, so this is probably a typo, but as printed the lemma is false and those proofs are formally invalid. More seriously, the proof of Theorem 1.9 says \"by Lemma 7.8, B′ is obtained from B by column insertion,\" but Lemma 7.8 assumes insertion and concludes the pipe dream equality. The needed direction is the forward commutation κ(P∪{(i,j)}) = ins_j(κ(P)), which is cited to [GH23, Theorem 4.5] but not reproduced. The reducedness issue compounds this: κ is described as a bijection on reduced pipe dreams, yet Theorem 1.1 needs a comparison of sets containing non-reduced pipe dreams with reduced co-pipe dreams (Example 1.7 shows such pipe dreams exist). The paper does not explain how κ covers those cases.\n\nOverall, the mathematical core is probably sound—the worked examples are consistent and the structure is coherent—but the formal dependencies need to be cleaned up. This is a paper for Schubert calculus specialists; the back stable formula is the reason to care. I would send it to peer review and ask for a major revision that fixes Lemma 3.5, states the exact commutation from [GH23], and clarifies the reducedness conventions.","headline":"Genuinely new BPD and back stable formulas, but a sign error in Lemma 3.5 and a gap in Theorem 1.9 need fixing before the central claims are airtight.","tokens_in":41539,"tokens_out":24468,"would_cite":true,"duration_ms":242228,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper gives explicit diagrammatic rules for changing between the Grothendieck and Schubert polynomial bases, including a back stable expansion.","keywords":["Schubert polynomials","Grothendieck polynomials","bumpless pipe dreams","pipe dreams","back stable Schubert calculus","co-transition","Bruhat order","K-theory of flag varieties"],"falsifier":"For a fixed small permutation such as $w=2143$, enumerate every reduced pipe dream and every reduced BPD with the same column weights, apply the canonical bijection, and compare $\\delta(\\check P)$ with $\\delta(\\check B)$; any mismatch would refute the transfer theorem. Independently, expand both sides of the Grothendieck-to-Schubert identity for $w=13452$ at monomial level: the formula predicts the coefficient of $S_{23451}$ is $-3$, so a different value would refute the expansion.","tokens_in":40420,"feed_emoji":"🧩","tokens_out":8802,"duration_ms":88731,"temperature":0.7,"pith_summary":"The paper proves explicit combinatorial change-of-basis rules between the two standard polynomial bases of Schubert calculus: every Grothendieck polynomial expands as an alternating sum of Schubert polynomials, and every Schubert polynomial expands as a positive sum of Grothendieck polynomials, with each term labelled by a bumpless pipe dream (BPD) and its associated co-BPD. Because BPDs are naturally back stable, the author also obtains the first combinatorial formula expanding back stable Grothendieck polynomials in the back stable Schubert polynomial basis. The same machinery gives new proofs of the older change-of-basis rules stated in terms of binary triangular arrays, and rephrases them as sums over pipe dreams and over chains in Bruhat order. These formulas answer a basic basis-change question in the cohomology and K-theory of complete flag varieties, where Schubert and Grothendieck polynomials represent geometric classes.","feed_headline":"Bumpless pipe dreams expand Grothendieck into Schubert polynomials","feed_subtitle":"Each term is read off a BPD diagram and its co-diagram, including for back stable polynomials.","key_machinery":"The objects that carry the argument are four diagram families: pipe dreams, co-pipe dreams, bumpless pipe dreams (BPDs), and co-BPDs, together with the canonical column-weight-preserving bijection between pipe dreams and BPDs. A pipe dream is a tiling of an $n \\times n$ grid whose crossing set encodes a permutation; a BPD is the analogous northeast-oriented tiling whose pipes run from bottom to right. The co-objects are obtained by fixed local tile replacements, and each carries a co-permutation $\\delta(\\cdot)$ obtained by tracing pipes through the diagram. The co-transition recurrences on pipe dreams, which bijectively add an addable cell to a diagram and split the result according to a set $\\Phi_i(w)$ of permutations, supply the recursive engine for the proofs, and the canonical bijection transfers the resulting identities from pipe dreams to BPDs.","core_discovery":"The central claim is a pair of identities. For a permutation $w$, the Grothendieck polynomial expands as a signed sum over reduced co-BPDs attached to BPDs of $w$: $G_w = \\sum_{B \\in \\mathrm{BPD}(w), \\check B \\text{ reduced}} (-1)^{\\ell(\\delta(\\check B))-\\ell(w)} S_{\\delta(\\check B)}$. In the reverse direction, $S_w = \\sum_{B \\in \\mathrm{BPD}(w)} G_{\\delta(\\check B)}$, a positive but not generally multiplicity-free sum. The proof transfers the analogous pipe-dream identities through a canonical bijection between pipe dreams and BPDs, and the key theorem is that this bijection preserves the co-permutation associated to each diagram, so that $\\delta(\\check P)=\\delta(\\check B)$. A doubly infinite version of the first identity gives the expansion of back stable Grothendieck polynomials into back stable Schubert polynomials, which is described as the first such combinatorial formula.","pith_inferences":["If the co-permutation preservation property extends to the marked or K-theoretic version of the canonical bijection mentioned in the paper, the same transfer would give combinatorial proofs of further K-theoretic change-of-basis identities.","The chain formulation suggests a route to dynamic-programming algorithms for structure constants or monomial-support questions, since paths in Bruhat order can be counted without first listing all pipe dreams.","Specializing the back stable expansion to partition-shaped permutations should reproduce known stable Grothendieck-to-Schur expansions, offering a direct numerical check of the new back stable formula."],"forward_implications":["The Grothendieck-to-Schubert expansion can be read directly from reduced co-BPDs: each contributes $S_{\\delta(\\check B)}$ with sign $(-1)^{\\ell(\\delta(\\check B))-\\ell(w)}$, and the underlying monomial expansion is cancellation-free.","The Schubert-to-Grothendieck expansion is a positive sum over all co-BPDs of BPDs of $w$, although it need not be cancellation-free at the monomial level.","Back stable Grothendieck polynomials admit the same BPD-style expansion into back stable Schubert polynomials, indexed by doubly infinite reduced co-BPDs; this is the first such combinatorial formula.","The pipe-dream versions of both change-of-basis rules are re-proved from co-transition recurrences and can be generated recursively by chains in Bruhat order, giving a smaller search space than listing all pipe dreams."],"supporting_citations":[{"why":"Supplies the canonical bijection between pipe dreams and BPDs whose commutation with column insertion carries the proof of the BPD transfer theorem.","marker":"[GH23]"},{"why":"Provides the co-transition recurrences on pipe dreams that drive the new proofs and the chain model.","marker":"[Knu22]"},{"why":"Gives the original Grothendieck-to-Schubert expansion, stated in binary triangular arrays, which the paper re-proves and transfers to BPDs.","marker":"[Len99]"},{"why":"Gives the original Schubert-to-Grothendieck expansion, stated in binary triangular arrays, which the paper re-proves and transfers to BPDs.","marker":"[Las04]"},{"why":"Introduces BPDs and the back stable Schubert polynomial basis that form the setting of the back stable theorem.","marker":"[LLS21]"},{"why":"Defines back stable Grothendieck polynomials, whose back stable Schubert expansion is the paper's final theorem.","marker":"[LLS23]"},{"why":"Provides the BPD formula for Grothendieck polynomials and the droop-move framework used throughout the BPD arguments.","marker":"[Wei21]"},{"why":"Gives the column-insertion algorithm on BPDs used to prove that co-permutations are preserved by the canonical bijection.","marker":"[Hua23a]"}],"fun_headline_variants":["Back stable Grothendieck to Schubert via BPDs","BPDs give explicit Grothendieck-Schubert expansions","First BPD rule for back stable Grothendieck-Schubert","BPDs expand Grothendieck into Schubert with back stability","BPDs: new combinatorial rules for Grothendieck and Schubert"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The transfer of the pipe-dream formulas to BPDs depends on the assertion, taken from a cited theorem rather than proved in this paper, that the canonical bijection between the two diagram families respects the recursive insertion step used to build the diagrams.","fun_headline_variants_meta":{"raw":{"variants":["Back stable Grothendieck to Schubert via BPDs","BPDs give explicit Grothendieck-Schubert expansions","First BPD rule for back stable Grothendieck-Schubert","BPDs expand Grothendieck into Schubert with back stability","BPDs: new combinatorial rules for Grothendieck and Schubert"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000782,"raw_usage":{"total_tokens":3470,"prompt_tokens":977,"completion_tokens":2493,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":593,"completion_tokens_details":{"reasoning_tokens":2398}},"tokens_in":593,"tokens_out":2493,"duration_ms":19534,"temperature":1.0,"reasoning_tokens":2398,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:37:45.356881+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a fixed small permutation such as $w=2143$, enumerate every reduced pipe dream and every reduced BPD with the same column weights, apply the canonical bijection, and compare $\\delta(\\check P)$ with $\\delta(\\check B)$; any mismatch would refute the transfer theorem. Independently, expand both sides of the Grothendieck-to-Schubert identity for $w=13452$ at monomial level: the formula predicts the coefficient of $S_{23451}$ is $-3$, so a different value would refute the expansion.","supporting_citations":[],"review_version":1}