Generic pipe dreams, lower-upper varieties, and Schwartz-MacPherson classes

Allen Knutson; Paul Zinn-Justin

REVIEW 3 major objections 5 minor 2 cited by

Generic pipe dreams, lower-upper varieties, and Schwartz-MacPherson classes

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves that the equivariant cohomology class of every lower-upper variety $E_w$ is, up to a power of $A+B$, the generic pipe dream polynomial $G_w$, a finite weighted sum over three-tile pipe dream grids.

desk verdict A genuinely useful unification of pipe dream formulas, likely correct, but the main equivariant-class theorem is presented as a sketch with a load-bearing proof gap. read the letter →

arxiv 2411.11208 v1 pith:NTXJP2BG submitted 2024-11-17 math.CO math.AG

Allen Knutson , Paul Zinn-Justin This is my paper · ORCID

classification math.COmath.AG MSC 14M1514N1514C17

keywords genericpipedreamslower-uppervarietiesequivariantcohomologydoubleSchubertpolynomialscommutingvarietySchwartz-MacPhersonclassesKazhdan-Lusztig

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

This paper aims to show that one combinatorial gadget, the generic pipe dream (GPD), controls the topology of the whole family of lower-upper varieties $E_w$. The central claim is Theorem 4: under the $(B_- \times B_+ \times (\mathbb{C}^\times)^2)$-action, the equivariant cohomology class of $E_w$ equals $(A+B)^{-n}G_w$, where $G_w$ is the sum, over all GPDs for $w$, of products of tile weights $A+x_i-y_j$, $B-x_i+y_j$, and $A+B$. If this is right, the same enumeration contains the classic and bumpless pipe dream formulas for double Schubert polynomials, the degree of the $n$th commuting variety as a sum of powers of 2, and Chern-Schwartz-MacPherson classes of Kazhdan-Lusztig varieties, matrix Schubert varieties, and double Bruhat cells. The reason to care is that an infinite geometric invariant becomes a finite, explicit combinatorial count.

What carries the argument

The central object is the generic pipe dream (GPD): an $n\times n$ square tiled by three tiles — a crossing, a bump, and a blank — with labeled pipes entering from the West and North according to the permutation $w$. Each tile contributes a linear equivariant weight, $A+x_i-y_j$ for a crossing, $B-x_i+y_j$ for a bump, and $A+B$ for a blank, and $G_w$ is the sum of products of these weights over all GPDs. The argument is carried by two mechanisms: a combinatorial one, in which $G_w$ is shown to obey the same divided-difference recursions that uniquely determine the equivariant class in the cited Brauer loop scheme work; and a geometric one, in which a degeneration of $E_w$ with flux variables on the grid forces every limiting component to lie in one of the complete intersections $F_\delta$, whose classes are the individual GPD terms.

What would settle it

Take $n=2$, $w$ the transposition, expand $(A+B)^{-2}G_w$ as a polynomial in $A,B,x_1,x_2,y_1,y_2$, and compare one coefficient, say that of $x_1^2$, with the same coefficient obtained by applying the divided-difference recursion the paper cites; if they differ, Theorem 4 is false. A geometric check would compute the initial ideal of the defining equations of $E_w$ for a small $n$ and look for any component not among the $F_\delta$.

Watch

Extended reading notes

Core claim

The paper's discovery is a closed formula for the equivariant cohomology class of each lower-upper variety $E_w = \{(X,Y) : XY \text{ lower triangular},\ YX \text{ upper triangular},\ \operatorname{diag}(XY) = w \cdot \operatorname{diag}(YX) \text{ nonrepeating}\}$ inside $(\mathrm{Mat}_{n\times n})^2$. With the scaling action of $(\mathbb{C}^\times)^2$ added to the $B_- \times B_+$ action, the class is $(A+B)^{-n}$ times the generic pipe dream polynomial $G_w$, which is the sum over $n\times n$ grids of crossing, bump, and blank tiles (with boundary labels determined by $w$) of the product of the three corresponding weights. The proof has two halves: one shows combinatorially that $G_w$ satisfies the same divided-difference recursions that uniquely determine the class; the other exhibits an equivariant degeneration of $E_w$ into complete intersections $F_\delta$, one per GPD, whose classes are exactly the individual terms of the formula. As the paper stresses, the leading terms of $G_w$ recover the classical and bumpless pipe dream formulas for double Schubert polynomials, and the same GPDs, with modified weights, compute CSM classes.

Load-bearing premise

The load-bearing premise is the paper's assertion, given as a sketch rather than a proof, that the generic pipe dream polynomials satisfy exactly the same divided-difference recursions that uniquely determine the equivariant classes, together with the unproved geometric step that the degeneration of $E_w$ has no components outside the complete intersections $F_\delta$.

Editorial extensions

If this is right

For every permutation $w$, the equivariant class of the lower-upper variety becomes an explicit finite sum over generic pipe dreams, so it can be computed by enumeration rather than by solving geometric recursions.
The leading terms of the pipe-dream sum reproduce the classical double Schubert polynomial formula and the bumpless pipe dream formula, making the generic pipe dream the common parent of both calculi.
The degree of the $n$th commuting variety equals a sum of powers of 2 indexed by generic pipe dreams for the identity permutation, giving an effective way to compute these degrees for large $n$.
The same modified pipe-dream weights compute the Chern-Schwartz-MacPherson classes of open Kazhdan-Lusztig varieties, matrix Schubert varieties, and double Bruhat cells, so all of these classes are governed by one combinatorial model.
The K-theoretic analogue replaces tile weights by K-theoretic ones and conjecturally provides K-classes of lower-upper varieties, including the commuting variety.

Reading between the lines

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

Inference: If Theorem 4 is correct, the lower-upper variety is a geometric realization of pipe-dream connectivity: its equivariant class depends only on the weighted set of pipe configurations, so any deformation of the variety that preserves that set has the same class.
Inference: The flux-degeneration argument that produces the complete intersections $F_\delta$ could plausibly be applied to other schemes defined by matrix product conditions, yielding pipe-dream-type enumerations for their classes; the paper does not state this.
Inference: Because the pipe-dream weights are linear in the equivariant parameters, the theorem suggests that all the Schubert and CSM formulas discussed here are specializations of one master polynomial; checking the conjectural K-theoretic pipe-dream formula for small $n$ would test that uniformity.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know about this one. It gives a single weighted sum over generic pipe dreams (GPDs) that recovers the classic pipe dream formula at the highest B-power and the bumpless pipe dream formula at the highest A-power. That is a real unification, and the paper is upfront about which terms survive the two limits. The reason it matters is Theorem 4: the (B_- × B_+ × (C^×)^2)-equivariant class of the lower-upper variety E_w is (A+B)^{-n} times the GPD polynomial. That theorem is only sketched, and the sketch has a load-bearing gap.

What is new and good: the GPD model itself; Theorem 1's exchange argument is clear and self-contained; the commuting-variety degree formula as a sum of powers of 2 is a nice byproduct, and they compute it up to n=16 and cite an independent proof from [GZJ22]; the CSM section produces closed formulas for B_-wB_+ and double Bruhat cells, with an explicit small example. The heavy self-citation to [KZJ14] and [GZJ22] is legitimate—those are the sources of the uniqueness theorem and the K-theoretic recursion, not padding.

The soft spot is Theorem 4's proof. The authors say the classes are uniquely determined by divided-difference recursions from their 2014 paper and that 'it is not hard to show' the GPD polynomials satisfy the same recursions, pointing to [GZJ22, prop. 6], a K-theoretic statement. The stress-test note is right: a K-theoretic recursion does not automatically specialize to the exact cohomological recursion with the right normalization, base case, and action of A and B. The paper gives none of those details. And Theorem 6 cannot backstop Theorem 4, since its conclusion explicitly assumes Theorem 4. So the central identification is likely true—the two leading-term limits and the independent degree formula give real support—but as written it is a conjecture-with-evidence, not a proof. That is a serious gap for a paper whose main theorem is Theorem 4. A referee should demand a complete proof of the recursion check or a precise pointer to where it appears.

This is for Schubert calculus people and anyone working on CSM classes of orbits; the reading group would enjoy it. It deserves peer review, but with the expectation of a revision that fills or precisely cites the missing computation. I would cite the GPD construction and the CSM formulas, with a caveat on Theorem 4.

Referee Report

3 major / 5 minor

Summary. The paper introduces a new family of combinatorial objects, 'generic pipe dreams' (GPDs), assembled from four tile types on an n x n grid, and defines polynomials G_w by assigning to each tile a weight A+x_i-y_j, B-x_i+y_j, or A+B. The main geometric claim is Theorem 4: the (B_- x B_+ x (C^×)^2)-equivariant cohomology class of the lower-upper variety E_w is (A+B)^{-n}G_w. The authors also prove leading-term statements recovering classic and bumpless pipe dream formulas for double Schubert polynomials (Theorem 1), derive a formula for the degree of the commuting variety as a sum of powers of 2, and apply GPDs to compute Segre-Schwartz-MacPherson classes of Kazhdan-Lusztig varieties, double Bruhat cells, and matrix Schubert varieties (Theorems 10, 13). A conjectural K-theoretic extension is discussed in Section 4.

Significance. If established, Theorem 4 provides a uniform and elegant weighted-enumeration formula for the equivariant classes of all lower-upper varieties, recovering two previously independent pipe dream models as leading-term specializations. The degree formula for the commuting variety is striking and algorithmically effective. The applications to SSM classes are potentially valuable and connect GPDs to Schubert calculus, cluster algebras, and characteristic class theory. The manuscript is honest about what is proved and what is conjectural, and it explicitly labels the two main proof sections as sketches. However, the paper's central result currently rests on an unverified assertion about divided-difference recursions, and the degeneration argument in Theorem 6 has significant gaps. The overall picture is plausible, but the manuscript is not yet at the level of completeness expected for a journal publication.

major comments (3)

[§2, Theorem 4] The proof of Theorem 4 consists of two premises: the equivariant classes of E_w are uniquely determined by divided-difference recursions from [KZJ14, §4], and the GPD polynomials G_w satisfy the same recursions. The second premise is asserted as 'not hard to show' with a pointer to [GZJ22, prop. 6], but no proof is given here. This is load-bearing: if G_w does not satisfy exactly the same recursions, with the same normalization and the same action of A and B, the identification of the class of E_w with (A+B)^{-n}G_w does not follow. The cited proposition is K-theoretic, and the specialization of its divided-difference operators and base cases to the cohomological setting of [KZJ14, §4] is not automatic. I am not claiming the assertion is false, only that it is unproved in this manuscript. The authors should supply a complete proof or a precise proposition-by-proposition specialization of [GZJ22, prop. 6].
[§2, Theorem 6] The degeneration argument is incomplete in exactly the places that matter. The step 'An easy calculation of its initial term leads to the equation X_ij(Φ_S - Φ_E) = 0' is not shown, and the subsequent flux-propagation step ('starting from the bottom/left and adding one square at a time, we conclude that fluxes Φ_e can only take the values 0, t_1, ..., t_n') is only asserted. The conclusion that each component of the degeneration of E_w lies in a single complete intersection F_δ is the crux of the geometric interpretation, but the proof does not justify why no other flux configurations survive. Moreover, the final comparison explicitly invokes Theorem 4, so Theorem 6 cannot serve as an independent verification of the class formula. Please provide the missing initial-term computation and a rigorous induction for the flux propagation, including a discussion of embedded components.
[§3.2, Theorem 13 proof] In the proof of Theorem 13, after erasing the pipes coming from the South, the text states that 'the weight of a blank is the sum of the weight of a bump and of that of a cross,' citing Eq. (4). With the weights displayed in Eq. (4), this identity would read 1 = (x-y) + (x-y+1), which is not an identity. If the intended modified weights carry signs, as hinted by the preceding sentence ('It differs from the original weight (1) by signs'), that sign convention must be written explicitly and the identity verified. As written, the CSM formula for B_- w B_+ is not justified, and this result is one of the applications announced in the abstract.

minor comments (5)

[§1, Eq. (1) and throughout] The manuscript relies heavily on custom tile diagrams that are not legible in the version I received; since the definition of GPD and the weights in Eqs. (1), (4), and the definition of F_δ in Theorem 6 all depend on these diagrams, please ensure they are typeset clearly in the final version.
[§1, Example 2] The GPDs for w = 2431 are displayed but not labelled; it would help the reader if the 'first two' and 'last' GPDs were identified explicitly, for instance by numbering the pictures.
[§3.2] There is a typo: 'a set of GDPs' should read 'a set of GPDs'.
[§3.1, Theorem 10] In formula (3), the index set I is used without an explicit definition; please state that I ⊆ {1, ..., |Q|} is the set of positions at which the subword R has letters, and clarify the notation ∏_{i∈I} Q_i as a product in the Weyl group.
[§3.2, Corollary 11] The phrase 'these GPDs have no blanks' is slightly confusing in light of Eq. (4), which assigns weight 1 to the 'otherwise' case; please make explicit whether blanks occur in this setting and, if not, why the 'otherwise' weight is needed.

Circularity Check

1 steps flagged · score 4.0 of 10

Theorem 4's equality rests on an unproved recursion check for G_w, delegated to the authors' own [KZJ14] and [GZJ22]; independent limits keep it from being circular by construction.

uniqueness imported from authors [Section 2, Theorem 4, sketch of proof (p. 3)]
"These cohomology classes are shown in [KZJ14, §4] to be uniquely determined by certain inductive divided difference formulæ. It is not hard to show, using Yang–Baxter type arguments, that the generic pipe dream polynomials satisfy the same inductive formulæ; see [GZJ22, prop. 6] for the K-theoretic version."

The central identification [E_w] = (A+B)^{-n} G_w is made to follow in one sentence: the class of E_w is forced by a uniqueness theorem cited to the authors' earlier [KZJ14, §4], and G_w is asserted to satisfy the same recursions, with the verification dispatched to a pointer to the authors' own [GZJ22, prop. 6]. Neither the divided-difference recursion for G_w nor the specialization of the K-theoretic statement to the needed cohomological operators is proved here. The load-bearing step is therefore an unverified claim plus a self-citation, not a self-contained derivation. It is not definitional circularity because G_w and E_w are independently defined and the theorem is anchored by leading-term limits and degree checks, but the derivation chain is not independent at its key step.

full rationale

The paper's main new result (Theorem 4) equates a geometric equivariant class with a combinatorially defined generic pipe dream polynomial. This is not an identity by construction: G_w is defined by summing over new combinatorial objects, while E_w comes from a scheme defined by matrix equations, and the paper gives independent evidence for the match—Theorem 1 recovers the classical and bumpless pipe dream formulas for double Schubert polynomials as leading-term limits, and Corollary 5 plus Example 7 give a computable degree formula that agrees with the independent [GZJ22, theorem 4]. Those anchors keep the central claim from collapsing into its own input. However, the proof of Theorem 4 has a real self-citation gap: the uniqueness of the equivariant classes is imported from the authors' [KZJ14], and the crucial fact that G_w satisfies those same divided-difference recursions is only asserted as 'not hard to show' with a pointer to the coauthor's K-theoretic [GZJ22, prop. 6]. No proof or explicit specialization is supplied in this paper, so the central equality rests on a load-bearing self-citation plus an unverified recursion check. Theorem 6 cannot fill this gap independently: its conclusion explicitly invokes Theorem 4 ('Because we already know equality of cohomology classes according to theorem 4') and its own 'with a bit more work' flux-propagation argument is only sketched. The paper is therefore not fully self-contained at its central step, and the claimed derivation leans on the authors' earlier work rather than on an independent computation. Since the theorem has substantial independent content and the missing recursion check is a concrete, checkable assertion rather than a definitional equivalence, the appropriate score is 4 rather than a higher 'reduces by construction' score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

The central claims rest on six cited or asserted premises rather than on fitted parameters; there are no free parameters because the A, B, x_i, y_j are formal equivariant variables. The most fragile inputs are the unproved recursion-verification and the flux-propagation step in the degeneration argument.

assumptions (6)

domain assumption The equivariant cohomology class of E_w is uniquely determined by the inductive divided difference formulas from [KZJ14, §4].
Theorem 4's proof starts with this cited uniqueness theorem; the GPD polynomials are then claimed to satisfy the same recursions.
domain assumption Generic pipe dream polynomials satisfy the same inductive recursions (Yang-Baxter type), as in [GZJ22, prop. 6] for K-theory.
This is the load-bearing 'not hard to show' step admitting the equality in Theorem 4; not proved in this paper.
ad hoc to paper In the Groebner degeneration of E, the initial ideal computation leads to X_ij(Phi_S - Phi_E)=0 and flux conservation at each square; the unproved propagation step ('adding one square at a time') forces fluxes to take pipe labels.
Theorem 6's containment of the degeneration in the union of F_delta rests on this; the uniqueness of flux propagation is asserted without details.
domain assumption The toric degeneration BS_Q to TV_Q is smooth and gives an equivariant diffeomorphism preserving cells (from [PP16]).
Used in Theorem 10 to transfer SSM computations from Bott-Samelson to the toric variety.
domain assumption Schuermann's transversality lemma for SSM classes [Sch17, lemma 8(2)] holds in the equivariant setting.
Used in Lemma 9 and Theorem 10 to multiply SSM classes of transverse intersections.
domain assumption The Kazhdan-Lusztig isomorphism of pairs (X_w^o ∩ U_v subset U_v) to (X_w^o ∩ X_v^o subset X_v^o) times X_v^o is T-equivariant.
Used in Corollary 11 to reduce the CSM class of an open Kazhdan-Lusztig variety to the GPD formula.

invented entities (1)

Generic pipe dreams (GPDs) independent evidence
purpose: New diagram model interpolating between classic and bumpless pipe dreams; used to compute equivariant cohomology classes of lower-upper varieties and CSM classes of matrix orbit closures.
The objects are new to this paper; their limits reproduce the known classic and bumpless pipe dream formulas for double Schubert polynomials, and the commuting-variety degree formula matches the independent proof in [GZJ22, theorem 4].

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Pith. "Pith review of Generic pipe dreams, lower-upper varieties, and Schwartz-MacPherson classes." pith.science (2026). https://pith.science/paper/NTXJP2BG

@misc{pith2026241111208,
  author       = {Pith},
  title        = {Pith review of: Generic pipe dreams, lower-upper varieties, and Schwartz-MacPherson classes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NTXJP2BG}},
  note         = {Machine review of arXiv:2411.11208}
}

abstract

We recall the lower-upper varieties from [Knutson '05] and give a formula for their equivariant cohomology classes, as a sum over generic pipe dreams. We recover as limits the classic and bumpless pipe dream formulae for double Schubert polynomials. As a byproduct, we obtain a formula for the degree of the $n$th commuting variety as a sum of powers of 2. Generic pipe dreams also appear in the Segre-Schwarz-MacPherson analogue of the AJS/Billey formula, and when computing the Chern-Schwarz-MacPherson class of the orbit $B_- w B_+ \subseteq Mat_{k\times n}$ or of a double Bruhat cell $B_-u B_+ \cap B_+ v B_-$.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Flag positroid pipe dreams
math.CO 2026-05 unverdicted novelty 7.0 of 10

Flag positroid pipe dreams are new diagrams in bijection with Bruhat intervals whose elbow counts are Richardson-cell dimensions and whose row rule characterizes nonnegatively representable elementary positroid quotients.
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.

Reference graph

Works this paper leans on

14 extracted references · 7 canonical work pages · cited by 2 Pith papers

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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David Anderson and William Fulton, Equivariant cohomology in algebraic geometry, Cambridge University Press, 2024, https://people.math.osu.edu/anderson.2804/ecag/index.html

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William Fulton, Flags, S chubert polynomials, degeneracy loci, and determinantal formulas , Duke Math. J. 65 (1992), no. 3, 381--420. 1154177

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Rebecca Goldin and Allen Knutson, Schubert structure operators and K_T(G/B) , S\'em. Lothar. Combin. 82B (2020), http://arxiv.org/abs/1909.05283 arXiv:1909.05283

work page Pith review arXiv 2020

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Alexandr Garbali and Paul Zinn-Justin, Shuffle algebras, lattice paths and the commuting scheme, Hypergeometry, Integrability and Lie Theory, 2022, Editors Erik Koelink, Stefan Kolb, Nicolai Reshetikhin and Bart Vlaar, http://arxiv.org/abs/2110.07155 arXiv:2110.07155

work page Pith review arXiv 2022

[5] [5]

Megumi Harada and Jihyeon Jessie Yang, N ewton-- O kounkov bodies of B ott-- S amelson varieties and G rossberg-- K arshon twisted cubes , 2015, http://arxiv.org/abs/1504.00982 arXiv:1504.00982

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Allen Knutson and Ezra Miller, Gr\"obner geometry of S chubert polynomials , Ann. of Math. (2) 161 (2005), no. 3, 1245--1318. MR2180402 (2006i:05177)

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Allen Knutson and Paul Zinn-Justin, The B rauer loop scheme and orbital varieties , J. Geom. Phys. 78 (2014), 80--110, http://arxiv.org/abs/1001.3335 arXiv:1001.3335

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