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Changing Bases with Pipe Dream Combinatorics

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Pith's one-line read This paper gives explicit diagrammatic rules for changing between the Grothendieck and Schubert polynomial bases, including a back stable expansion.

desk verdict 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. read the letter →

arxiv 2506.07306 v1 pith:PQQ2MUP5 submitted 2025-06-08 math.CO

Anna Weigandt This is my paper
classification math.CO MSC 05E05
keywords SchubertpolynomialsGrothendieckbumplesspipedreamsbackstablecalculusco-transitionBruhatorderK-theoryofflagvarieties
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [§7.3, Lemma 7.8 and Theorem 1.9] 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.
  2. [§3.1, Lemmas 3.5, 4.7, 4.11, and 7.7] 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.
  3. [§7.2, Lemma 7.7] 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.
minor comments (4)
  1. [§3.1, (3.1)–(3.4)] 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.
  2. [§4.4, Lemma 4.9] 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.
  3. [§6.3, Theorem 6.7 and Example 6.14] 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.
  4. [§4.2, Lemma 4.7 proof] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the BPD change-of-basis formulas transfer the known Lenart/Lascoux pipe-dream expansions through the external Gao-Huang bijection. The only flagged issue is a black-box external commutation lemma, which is a correctness risk, not a circular reduction.

full rationale

The derivation chain is not circular in any of the enumerated senses. Theorems 1.5 and 1.6 are quoted as the known results of Lenart and Lascoux, and the paper gives independent proofs of them from Knutson's co-transition recurrences and the standard pipe-dream monomial formulas (Theorems 4.3, Propositions 5.6 and 5.7); neither input assumes the target expansions. The transfer to BPDs is coefficient comparison: in the proof of Theorem 1.1 the coefficient a_{w,v} in G_w = sum (-1)^{...} a_{w,v} S_v is computed first by Theorem 1.5 as a count of reduced co-pipe dreams and then by Theorem 1.9 as the same count of reduced co-BPDs. Theorem 1.2 is identical for the Schubert-to-Grothendieck coefficients; no parameter is fitted and no quantity is defined in terms of the output. Theorem 1.9 itself is an induction whose two ingredients are Lemma 7.7, proved in the paper from the BPD insertion algorithm, and Lemma 7.8, which is quoted as a special case of [GH23, Theorem 4.5], an external result. One genuine concern is that the printed Lemma 7.8 has the hypothesis 'B 7-> B' by insertion into column j' and concludes kappa(P' - {(i,j)}) = B, whereas the proof of Theorem 1.9 invokes it in the converse direction to conclude that B' is obtained from B by column insertion from P' = P union {(i,j)}, B' = kappa(P'). This is a verification gap concerning an external theorem, not a circular definition. The back stable Theorem 8.1 is derived by specializing the finite Theorem 1.1 and comparing coefficients in a back stable basis, with no use of the target expansion as an input. The only self-citations, e.g. [Wei21, Lemma 4.6] for a structural fact about BPDs and the BPD monomial formula, are prior published results that do not assume the change-of-basis theorems; they are normal citations rather than circular loads. Overall, the core new claims have independent content, and the paper is self-contained against external benchmarks once [GH23, Theorem 4.5] and [Hua23a] are accepted.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper is transparent about depending on several external theorems from the literature. No new free parameters are introduced, and the co-pipe dreams and co-BPDs are combinatorial objects, not new physical entities.

assumptions (5)
  • domain assumption Knutson's co-transition recurrences for pipe dreams (Proposition 5.6)
    Used as the foundation for the new proofs of Lenart and Lascoux's formulas; the paper does not prove this theorem, only sketches details.
  • domain assumption Gao-Huang canonical bijection between pipe dreams and BPDs, including the commutation property in [GH23, Theorem 4.5]
    This external theorem is essential to transfer results from pipe dreams to BPDs (Lemma 7.8, Theorem 1.9).
  • domain assumption Huang's column insertion theorem on BPDs [Hua23a, Theorem 5]
    Used in section 7 to analyze how insertion modifies the co-permutation.
  • domain assumption The linear independence of back stable Schubert polynomials [LLS21, Theorem 3.5]
    Used in the proof of Theorem 8.1 to identify coefficients.
  • standard math Standard divided difference calculus identities, e.g. N_i = N_i x_{i+1} + N_i
    Used in the proof of Theorem 1.6; standard in Schubert calculus.

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Pith. "Pith review of Changing Bases with Pipe Dream Combinatorics." pith.science (2026). https://pith.science/paper/PQQ2MUP5

@misc{pith2026250607306,
  author       = {Pith},
  title        = {Pith review of: Changing Bases with Pipe Dream Combinatorics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PQQ2MUP5}},
  note         = {Machine review of arXiv:2506.07306}
}
read the original abstract

Lascoux and Sch\"utzenberger introduced Schubert and Grothendieck polynomials to study the cohomology and K-theory of the complete flag variety. We present explicit combinatorial rules for expressing Grothendieck polynomials in the basis of Schubert polynomials, and vice versa, using the bumpless pipe dreams (BPDs) of Lam, Lee, and Shimozono. A key advantage of BPDs is that they are naturally back stable, which allows us to give a combinatorial formula for expanding back stable Grothendieck polynomials in terms of back stable Schubert polynomials. We also provide pipe dream interpretations for the rules originally given by Lenart (Grothendieck to Schubert) and Lascoux (Schubert to Grothendieck), which were previously formulated in terms of binary triangular arrays. We give new proofs of these results, relying on Knutson's co-transition recurrences. As a consequence, we obtain a formula for expanding Grothendieck polynomials into Schubert polynomials using chains in Bruhat order. The key connection between the pipe dream and BPD change of basis formulas is the canonical bijection of Gao and Huang. We show that co-permutations are preserved by this map.

Figures

Figures reproduced from arXiv: 2506.07306 by the authors.

Figure 1
Figure 1. Pictured on the left is a ladder move and on the right a K-theoretic ladder move. Remark 4.10. BPDs are in direct bijection with states the six-vertex model. As explained in [Wei21], one can obtain a BPD B from a state of the six-vertex model by drawing pipe segments corresponding to the arrows that point left and down. To obtain the corresponding co-BPD Bˇ, instead select the arrows that point left and up. Lemma 4.… view at source ↗
Figure 2
Figure 2. Visualization of the replacements in Lemma 5.8, when k = 4. A portion of the graph of w is on the left, and the graph of wU,i is on the right, with U = {i1, i2, i3, i4}. Note that no dots appear in the interior of the pictured partition shapes. We note that the first part Proposition 5.7 is a special case of [Len03, Theorem 3.1]. The second part can be derived from Monk’s rule. 5.4. Proof of combinatorial co-transit… view at source ↗
Figure 3
Figure 3. The graph G3 is pictured on the left. G3 is on the right. 123 213 132 231 312 321 (1,1) (2,1) (1,2) (2,1) (1,1) (1,1) [PITH_FULL_IMAGE:figures/full_fig_p033_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: The graph R3. (2) Sw = x n−1 1 x n−2 2 · · · xn−1 X p∈pathsGn (w,w0) 1 wt(p) . (3) G (1) w = X p∈pathsRn (w,w0) p is reduced Sw0·δ(word(p)). (4) Sw = X p∈pathsGn (w,w0) Gw0·δ(word(p)). Before proceeding, we give an example of Theorem 6.7 [PITH_FULL_IMAGE:figures/full_…

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  1. Hybrid pipe dreams for the lower-upper scheme

    math.CO 2025-09 conditional novelty 7.0 of 10

    Hybrid generic pipe dreams give a hybridization-independent equivariant formula for the classes of lower-upper varieties, proved via Yang-Baxter equations and a degeneration into complete intersections.

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