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Saturation property fails for Schubert coefficients

T0 review · 2 major / 3 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Kirillov's conjecture that Schubert coefficients satisfy saturation is false; a large family of triples has coefficient 1 before scaling and 0 after.

desk verdict The main theorem is false as stated: the proof's Lehmer-code identity is wrong, and the stress-test's explicit counterexample holds up. read the letter →

arxiv 2601.04182 v2 pith:AYRLE257 submitted 2026-01-07 math.CO

Igor Pak , Colleen Robichaux This is my paper
classification math.CO MSC 05E0505E1014M15
keywords SchubertcoefficientssaturationpropertyKirillov'sconjectureLehmercodeRothediagramMonk'sruleindicatortableauxvanishingproblem
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 disproves Kirillov's 2004 conjecture that Schubert coefficients obey a saturation property under scaling of permutation codes, analogous to the Littlewood–Richardson saturation theorem. The authors construct infinite families of permutations u,v,w where the Schubert coefficient c^w_{u,v}=1 but the scaled coefficient c^{N*w}_{N*u,N*v}=0 for every N>1. This shows the 'easy' direction of saturation already fails for Schubert coefficients, even for permutations with at most two descents. They also disprove an analogous saturation property under a different scaling, 'bit scaling', using a simpler dimension-count argument. The failure matters because saturation underlies the only known poly-time approach to deciding Schubert coefficient vanishing.

What carries the argument

The central objects are Lehmer codes (inversion-index vectors) of permutations, the associated Rothe diagrams, and the operation of code scaling N*w, which multiplies every code entry by N. The proof combines Monk's rule (which identifies Schubert coefficients for simple transpositions with a sum over cover relations) for the positive part, and the St. Dizier–Yong vanishing lemma (empty set of indicator tableaux implies coefficient zero) for the negative part. The key counting step compares the number of i-entries forced into row j of the scaled diagram with the number of available columns, yielding an inequality that contradicts N>1 and j-i≥2.

What would settle it

Compute the Schubert coefficient for the smallest instance (u=2143, w=4123, v=(1,2), N=2) by expanding the product of the corresponding Schubert polynomials; the theorem predicts 0. If the expansion gives a positive coefficient, the main claim collapses.

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Extended reading notes

Core claim

For any permutation u and a cover u ⋖ w = u·t_{ij} with i<j and u(i)<u(j)-1, setting v to the simple transposition (i,i+1) gives c^w_{u,v}=1 by Monk's rule, while code scaling all three permutations by any N>1 forces c^{N*w}_{N*u,N*v}=0. The vanishing is proved via the St. Dizier–Yong indicator tableau condition: any filling of the union of Rothe diagrams of N*u and N*v with the prescribed multiplicities would violate column-strictness and row-placement constraints, so no tableau exists and therefore the coefficient vanishes. The same phenomenon—coefficient equal to 1 before scaling, zero after—is established for bit scaling, with the scaled vanishing following from a mismatch in lengths (di

Load-bearing premise

The proof depends on an exact count of how many squares in row j of the scaled diagram lie directly below squares of row i; this count is asserted with a one-sentence justification but not proved as a separate lemma.

Editorial extensions

If this is right

  • Kirillov's Conjecture 1.1 is false, so saturation for Schubert coefficients under code scaling cannot be used as an approach to Schubert vanishing in polynomial time via linear programming.
  • The failure occurs already for permutations with at most two descents, a class where alternative combinatorial interpretations of Schubert coefficients exist; those interpretations do not rescue a saturation property.
  • Bit-scaling saturation also fails, and does so by a much simpler length mismatch, so any saturation-type conjecture for Schubert coefficients must be formulated with care about the scaling operation.
  • The counterexample family is explicit and infinite (all n≥4, all N>1), so the failure is robust, not a sporadic phenomenon.
  • The paper leaves open whether Schubert coefficient vanishing is decidable in polynomial time; only the saturation-based route is blocked.

Reading between the lines

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

  • The construction suggests that code scaling of permutations is not a geometric 'dilation' of the underlying variety in a way that preserves positivity; if it were, the coefficient would remain positive.
  • The St. Dizier–Yong indicator tableau condition is sufficient but not necessary for vanishing; this gap is exactly where the saturation failure enters, and might be explored to find an even larger family of counterexamples.
  • A natural testable extension: check whether the same counterexample family works for other scaling operations that multiply codes by different matrices; the proof structure suggests the inequality is delicate.
  • The dimension-count contradiction for bit scaling is so simple that it might generalize to any scaling that preserves the descent set and increases lengths in a nonlinear way.
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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 / 3 minor

Summary. The paper claims to disprove Kirillov's saturation conjecture for Schubert coefficients. Its main result, Theorem 1.2, asserts that for a Bruhat cover u ⋖ w = u t_ij with j - i ≥ 2 and u(i) < u(j) - 1, and for v = s_i, one has c^w_{u,v} = 1 but c^{N*w}_{N*u,N*v} = 0 for all N > 1. The proof uses Monk's rule for the positive part and a St. Dizier–Yong indicator-tableau emptiness criterion for the vanishing part, via a computation of Rothe diagrams under code scaling. A second family of counterexamples is given for a separately defined bit-scaling operation (Theorem 4.4).

Significance. If correct, Theorem 1.2 would be a clean and important negative answer to a twenty-year-old conjecture, with consequences for the complexity of Schubert vanishing. The paper is clearly written, surveys the relevant literature, and the bit-scaling construction is an interesting independent contribution. However, the central code-scaling theorem is false: the Rothe-diagram update formula in §3.3 is wrong, and an explicit instance satisfying the theorem's hypotheses admits a valid indicator tableau. The main claim of the paper therefore does not hold as stated.

major comments (2)
  1. [§3.3, displayed D(w) and code formulas] The claimed formula D(w) = D(u) - {(j,c): c∈[u(i)+1,u(j)-1]} ∪ {(i,c): c∈[u(i),u(j)-1]} is false. Take u = 31254 ∈ S_5, i = 3, j = 5. Then u ⋖ w = 31452 and u(3) = 2 < 3 = u(5)-1, so the hypotheses of Theorem 1.2 hold. Direct computation gives code(u) = (2,0,0,1,0) and code(w) = (2,0,1,1,0), whereas the displayed formula predicts code(w) = (2,0,2,1,-1), which is impossible. Consequently equations (3.1), (3.3), and (3.4) are not valid identities, and the contradiction 0 ≥ (N-1)(j-i-1) is not obtained from true statements.
  2. [§3.3, Theorem 1.2] The theorem is false as stated. For the same u = 31254, i = 3, j = 5, and N = 2, one computes 2*u = (5,1,2,6,3,4,7,8,9,10), 2*v = (1,2,5,3,4,6,7,8,9,10), and 2*w = (5,1,4,6,2,3,7,8,9,10). Then D(2*u) = {(1,1),(1,2),(1,3),(1,4),(4,3),(4,4)} and D(2*v) = {(3,3),(3,4)}. Fill row 1 of D(2*u) with 1s, row 4 of D(2*u) with 4s, and the shifted row-3 boxes of D(2*v) with 3s. The content is (4,0,2,2) = code(2*w), every column strictly increases, and each entry m lies in a row r ≥ m. Thus Tab^{2*w}_{2*u,2*v} is nonempty. Under the standard if-and-only-if form of [SY22, Thm B] cited in Lemma 3.2, this contradicts the asserted vanishing c^{2*w}_{2*u,2*v} = 0; at minimum, the claimed emptiness is false.
minor comments (3)
  1. [§3.2] The horizontal shift of D(v) is not specified when the permutations are replaced by their code scalings. The proof of Example 3.3 depends on this choice; please state whether the shift is by n, by N n, or by some other amount.
  2. [Lemma 3.2] Lemma 3.2 states only the implication Tab = ∅ ⇒ c = 0. If the cited St. Dizier–Yong theorem is an equivalence, the converse should be stated explicitly, since the proof of Theorem 1.2 relies on the nonemptiness of Tab as a certificate of positivity.
  3. [Example 3.3] The ASCII diagram for the indicator tableau is difficult to parse. Labeling rows and columns, or giving the coordinates of the boxes, would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the disproof is a self-contained calculation using external theorems; self-citations are contextual only.

full rationale

The central claim—that Kirillov's saturation conjecture fails—is derived from Monk's rule for c^w_{u,v}=1 and from the St. Dizier–Yong vanishing condition (Lemma 3.2) for c^{N*w}_{N*u,N*v}=0. The indicator-tableau machinery is imported from [ARY21] and [SY22]; the latter is an external published theorem, not an unverified self-citation, and neither source assumes the conjecture being disproved. The code-scaling and Rothe-diagram computations in §3.3 are algebraic derivations from definitions, not fitted parameters called predictions. The only self-citations ([ARY21], [PR25], [RYY22]) are contextual or motivational and do not carry the proof. The alleged false column-count identity/code formula flagged by the skeptic is a possible correctness gap, not a circularity: if the identity is wrong, the proof is invalid, but it is not an input configured to force the conclusion. Hence the paper is self-contained against external benchmarks, and the appropriate circularity score is 0–2; I assign 1 to acknowledge the minor self-citations without treating them as load-bearing.

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

The central claim rests on standard structure theory of Schubert polynomials and on two external theorems (Monk's rule, St. Dizier–Yong). The only internal ingredient that is asserted rather than fully derived is the column-count identity behind (3.3). No parameters are fitted and no new entities are introduced.

assumptions (5)
  • standard math Schubert polynomials form a basis and the structure constants c^w_{u,v} are nonnegative integers (eq. 1.3)
    Definition of Schubert coefficients.
  • standard math Monk's rule (Proposition 2.1)
    Gives c^w_{u,v}=1 when v is a simple transposition.
  • domain assumption St. Dizier–Yong vanishing condition (Lemma 3.2)
    External theorem: absence of indicator tableaux implies coefficient zero.
  • standard math Dimension condition (Proposition 2.2)
    Homogeneity of Schubert polynomials; used in bit-scaling proof.
  • domain assumption Column-count identity for Rothe diagrams under code scaling (§3.3)
    Terse one-sentence justification; load-bearing for the contradiction.

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Pith. "Pith review of Saturation property fails for Schubert coefficients." pith.science (2026). https://pith.science/paper/AYRLE257

@misc{pith2026260104182,
  author       = {Pith},
  title        = {Pith review of: Saturation property fails for Schubert coefficients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AYRLE257}},
  note         = {Machine review of arXiv:2601.04182}
}
read the original abstract

The saturation property for Littlewood--Richardson coefficients was established by Knutson and Tao in 1999. In 2004, Kirillov conjectured that the saturation property extends to Schubert coefficients. We disprove this conjecture in a strong form, by showing that it fails for a large family of instances. We also discuss computational complexity implications.

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Forward citations

Cited by 1 Pith paper

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

  1. Stretched Schubert coefficients are eventually quasi-polynomial

    math.CO 2026-04 unverdicted novelty 8.0 of 10

    Stretched Schubert coefficients f_{u,v,w}(N) are eventually quasi-polynomial, proving Kirillov's conjecture that their generating function is rational.

Reference graph

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