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Graham positivity of triple Schubert calculus

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The coefficients expressing a product of two double Schubert polynomials in three sets of variables are shown to be nonnegative combinations of $t_i - y_j$.

desk verdict Proves two conjectures via a plausible refined Graham positivity; main caveat is an unverified delegation to Anderson-Fulton. read the letter →

arxiv 2506.09421 v1 pith:TTQ3OASF submitted 2025-06-11 math.CO math.AG

Yibo Gao , Rui Xiong This is my paper · ORCID
classification math.COmath.AG MSC 05E0514M1514N15
keywords doubleSchubertpolynomialsGrahampositivityequivariantcohomologycalculusskewdivideddifferenceoperatorsflagvarietiesSamuel'sconjectureKirillov's
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 Samuel's conjecture that, in the expansion of the product of two double Schubert polynomials $S_u(x;y)S_v(x;t)$ as a combination of single double Schubert polynomials $S_w(x;t)$, every coefficient $c^w_{u,v}(y,t)$ is a polynomial with nonnegative integer coefficients in the differences $t_i - y_j$. The proof works by establishing a refined version of Graham's positivity theorem: an effective cycle in a flag variety that is invariant under the subgroup $B^-(w)$ has an equivariant cohomology class expressible as a combination of effective cycle classes with coefficients built from the negatives of the inversion roots of $w$. This refined statement is then applied to a geometric realization of $c^w_{u,v}(y,t)$ as the equivariant class of an intersection of two Schubert varieties. As a direct corollary, the paper settles Kirillov's conjecture that skew divided difference operators applied to Schubert polynomials yield polynomials with nonnegative coefficients.

What carries the argument

The load-bearing object is the refined Graham positivity theorem (Theorem 2.3), which asserts that $B^-(w)$-invariant effective cycles have equivariant classes lying in the cone generated by the negatives of the inversion roots of $w$: $[Y]_T \in \sum_i \mathbb{N}[-\alpha]_{\alpha \in I(w)} \cdot [Z_i]_T$. The proof is an induction on the length of $w$; the inductive step uses Anderson–Fulton's Proposition 19.4.4 on the subgroup pair $B^-(w s_i) \subset B^-(w)$ with character $\chi = -w\alpha_i$ to split the class of a $B^-(w s_i)$-invariant cycle into effective pieces. The machinery also includes a new geometric interpretation (Section 2.3) of $c^w_{u,v}(y,t)$ as the equivariant class of the intersection $\tau B^-uB/B \cap B^-vB/B$ expanded in the Schubert basis, where the special permutation $\tau$ has inversion set $\{y_j - t_i\}$; this is what converts the abstract positivity into the concrete statement in $t_i - y_j$.

What would settle it

For a small triple such as $u=v=w=s_1$ in $S_3$, expand $S_{s_1}(x;y)S_{s_1}(x;t)$ in the basis $S_w(x;t)$ and check whether every coefficient $c^w_{u,v}(y,t)$ is a nonnegative integer polynomial in $t_i - y_j$; any negative coefficient would falsify Theorem 1.1.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the refined Graham positivity theorem (Theorem 2.3) holds: for a $B^-(w)$-invariant effective cycle $Y$ on a non-singular variety $X$ with a $B^-$ action, $[Y]_T$ lies in the semiring generated by the negative inversion roots of $w$, with $B^-$-invariant effective cycles as coefficients: $[Y]_T \in \sum_i \mathbb{N}[-\alpha]_{\alpha \in I(w)} \cdot [Z_i]_T$ in $H^*_T(X)$. This strengthens Graham's original positivity theorem, which allows all positive roots, by restricting the cone of allowed characters to those of the form $-\alpha$ for $\alpha \in I(w)$. Applying this to the intersection $\tau B^-uB/B \cap B^-vB/B$ realizes the coefficients $c^w_{u,v}(y,t)$ as the coefficients of this class in the Schubert basis, and the inversion set of $\tau$ is exactly $\{y_j - t_i\}$, giving the required positivity. Setting $y=0$ then yields Kirillov's conjecture.

Load-bearing premise

The load-bearing premise is that the subgroup pair $B^-(w s_i) \subset B^-(w)$ satisfies the hypotheses of Anderson–Fulton's Proposition 19.4.4 with character $\chi = -w\alpha_i$, a condition the text asserts with a page reference but does not verify.

Editorial extensions

If this is right

  • Samuel's conjecture holds: for every $u,v,w \in S_\infty$, the coefficient $c^w_{u,v}(y,t)$ is a nonnegative integer polynomial in the differences $t_i - y_j$.
  • Kirillov's conjecture holds: for every $u,v,w$, the skew divided difference operator $\partial_{w/v}$ applied to $S_u(x)$ produces a polynomial with nonnegative coefficients.
  • The refined Graham positivity theorem (Theorem 2.3) is valid for any $B^-(w)$-invariant effective cycle, generalizing Graham's theorem and giving a geometric explanation of the positivity in Billey's formula.
  • The coefficients $c^w_{u,v}(y,t)$ are realized geometrically as the Schubert-expansion coefficients of the equivariant class of the intersection $\tau B^-uB/B \cap B^-vB/B$, which ties the positivity ring to the inversion set of $\tau$.

Reading between the lines

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

  • The refined positivity theorem is stated for arbitrary reductive groups, so the same argument may yield analogues of Samuel's and Kirillov's conjectures in other Lie types once the appropriate equivariant classes are identified.
  • If the Anderson–Fulton condition is verified explicitly, the induction in Theorem 2.3 could be made algorithmic, potentially producing a positive combinatorial formula for $c^w_{u,v}(y,t)$ rather than a purity statement.
  • The conjecture for double Grothendieck polynomials (Conjecture 2.7) suggests that the positivity should survive $K$-theoretic deformation; a $K$-theoretic analogue of the refined Graham theorem would be the natural tool.
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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

2 major / 4 minor

Summary. The paper proves Samuel's conjecture on Graham positivity of the expansion coefficients of products of double Schubert polynomials in three sets of variables (Theorem 1.1), and as a corollary Kirillov's conjecture on positivity of skew divided difference operators applied to Schubert polynomials (Corollary 1.2). The proof introduces a refined Graham positivity theorem (Theorem 2.3) for B^-(w)-invariant effective cycles and gives a geometric interpretation of the coefficients as an intersection of translated Schubert varieties (Lemma 2.5, proof of Theorem 1.1). The paper is concise and delegates substantial technical steps to Anderson and Fulton's book [1].

Significance. If the proof is completed, the paper resolves two previously open conjectures and establishes a refined positivity theorem that strengthens Graham's theorem. The geometric explanation of Billey's formula is a nice byproduct, and the K-theoretic conjecture at the end suggests further work. The overall strategy is plausible and the paper is clearly written, but two load-bearing technical steps—the application of [1, Prop. 19.4.4] and the proof of Lemma 2.5—are not adequately justified and must be repaired before the claims can be accepted.

major comments (2)
  1. [Section 2.1, proof of Theorem 2.3] The induction step asserts that the pair B^-(w s_i) ⊂ B^-(w) satisfies the hypotheses of [1, Proposition 19.4.4] with character χ = -w α_i, citing only [1, p. 384]. Lemma 2.2 establishes only that N^-(w s_i) is a normal subgroup of N^-(w). The cited proposition likely requires additional conditions (for instance, that the quotient is a one-parameter unipotent group with a specified T-action and that a normal-bundle condition holds); none of these are verified in the text. Because this proposition is the only tool that produces the decomposition [Y]_T = [Z_1]_T + χ[Z_2]_T, the induction is not established. The authors must state the hypotheses of [1, Prop. 19.4.4] explicitly and verify them for the pair B^-(w s_i) ⊂ B^-(w).
  2. [Section 2.3, Lemma 2.5] The proof of Lemma 2.5 is incorrect as written. The claim 'Since v∈S_n, s_i v > v for n≤i<2n, and thus B^-vB/B is invariant under s_i' is false: left multiplication by s_i sends B^-vB/B to B^- s_i v B/B, a different Schubert cell, and right multiplication by s_i sends it to B^- v s_i B/B, also different because v s_i ≠ v. Consequently the conclusion B^-vB/B = u_0 B^-vB/B is unjustified. This lemma is load-bearing for the geometric formula in the proof of Theorem 1.1; the authors need to supply a valid proof of properness and generic transversality of the intersection, or give a precise reference with a clear explanation of how it applies.
minor comments (4)
  1. [Section 2.1, proof of Theorem 2.3] The reference to 'Theorem 2.2' should be 'Lemma 2.2'.
  2. [Section 2.3, proof of Theorem 1.1] The reference to 'Theorem 2.5' should be 'Lemma 2.5'.
  3. [Section 2.3, proof of Theorem 1.1] After Lemma 2.5, the equality [τ B^-uB/B ∩ B^-vB/B]_T = Σ_w c^w_{u,v}(y,t) [B^-wB/B]_T is asserted in one sentence; the authors should explicitly state that the intersection cycle represents the equivariant product of the two classes and that the coefficients match the polynomial expansion (1).
  4. [Section 2.1, Corollary 2.4] The phrase 'finite many' should be 'finitely many'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main theorem is proved via an independent refined Graham positivity theorem applied to a geometric intersection; target conjectures are external and not assumed.

full rationale

The paper proves two external conjectures (Samuel's Conjecture 1.1 and Kirillov's Conjecture 1.2) using a strengthened version of Graham's positivity theorem (Theorem 2.3). Theorem 2.3 is proved by induction on ℓ(w) using Anderson–Fulton, Proposition 19.4.4, an external textbook result; it does not assume the conjectures it is meant to prove. The application to triple Schubert calculus proceeds by identifying the coefficient c^w_{u,v}(y,t) with the equivariant class of the transverse intersection τB^-uB/B ∩ B^-vB/B (via Lemma 2.5, delegated to Anderson–Fulton), and then applying Theorem 2.3/2.4 to this intersection, which is N^-(τ)-invariant. Corollary 1.2 uses the known identity ∂_{w/v}S_u(x;y) = c^w_{u,v}(y,x), cited to Samuel [21] and to Fan–Guo–Xiong [6]; this identity is a prior established formula, not a restatement of the conjecture. No fitted parameter is renamed as a prediction, no self-citation is used to justify the central premise, and no equation in the paper reduces by definition to another. The only substantive concern is whether the hypotheses of Anderson–Fulton, Proposition 19.4.4 are indeed satisfied by the pair B^-(ws_i) ⊂ B^-(w) with χ = -wα_i; the paper gives only a page reference and does not verify the hypotheses in full. That is a potential correctness or rigor gap, not circularity, and it does not affect the circularity score.

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

The proof depends on standard external machinery: Borel's presentation of equivariant cohomology, Anderson-Fulton's textbook propositions (Prop. 19.4.4 and Section 19.3), the known identity relating skew divided differences to Schubert structure constants, and basic facts about B^- -stable subvarieties of flag varieties. No free parameters are fitted to data, and no new speculative entities are introduced; the subgroups B^-(w) are natural definitions.

assumptions (5)
  • standard math Anderson-Fulton Proposition 19.4.4, applied to the pair B^-(w s_i) ⊂ B^-(w) with χ=-wα_i
    Used in the induction step of Theorem 2.3 to split [Y]_T into [Z_1]_T + χ[Z_2]_T; the paper cites [1, p. 384] without verifying the hypotheses.
  • standard math Borel presentation: double Schubert polynomials S_π(x;t) represent the equivariant classes [B^-πB/B]_T
    Invoked in Section 2.3 via [1, Theorem 10.6.4] to translate the geometric intersection into the polynomial expansion (1).
  • standard math Identity ∂_{w/v}S_u(x;y)=c^w_{u,v}(y,x)
    Used in the proof of Corollary 1.2 to reduce Kirillov's conjecture to Theorem 1.1; cited from [21, Section 2 p.5] and [6, Proposition 6.4].
  • domain assumption A B^- -invariant effective cycle in G/B expands as a nonnegative integer combination of Schubert classes [B^-wB/B]_T
    Corollary 2.4 relies on the finiteness of B^- -orbits and the fact that any B^- -stable subvariety is a union of B^- -orbit closures.
  • standard math Richardson variety intersections w_0 X_u ∩ X_v are proper and reduced at generic points
    Lemma 2.5 delegates the properness/transversality of τ B^-uB/B ∩ B^-vB/B to [1, Section 19.3] after conjugating by u_0.

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Pith. "Pith review of Graham positivity of triple Schubert calculus." pith.science (2026). https://pith.science/paper/TTQ3OASF

@misc{pith2026250609421,
  author       = {Pith},
  title        = {Pith review of: Graham positivity of triple Schubert calculus},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TTQ3OASF}},
  note         = {Machine review of arXiv:2506.09421}
}
read the original abstract

We prove Samuel's conjecture on certain Graham positivity of the expansion coefficient of two double Schubert polynomials in three sets of variables by establishing a refined version of Graham's positivity theorem. As a corollary, we prove Kirillov's conjecture on the positivity of skew divided difference operators applied to Schubert polynomials.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Equivariant Schubert Calculus for Inverse Grassmannian Permutations

    math.CO 2026-07 conditional novelty 7.0 of 10

    An equivariant product rule: double Schubert polynomials indexed by inverse Grassmannian permutations expand with structure constants given by double Schubert polynomials in two disjoint sets of variables.

Reference graph

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