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arxiv: 2604.27107 · v1 · submitted 2026-04-29 · 🧮 math.CO · cs.DM

Stretched Schubert coefficients are eventually quasi-polynomial

Igor Pak , Zachary Slonim This is my paper

Pith reviewed 2026-05-07 08:15 UTC · model grok-4.3

classification 🧮 math.CO cs.DM
keywords Schubert coefficientsstretched coefficientsquasi-polynomialspipe dreamsEhrhart theoryKirillov's conjecturesaturation conjecturecombinatorial positivity
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The pith

Stretched Schubert coefficients become eventually quasi-polynomial for fixed permutations.

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

The paper establishes that for any fixed permutations u, v, w, the function counting stretched Schubert coefficients f(N) equals c to the power N*w of N*u and N*v is eventually quasi-polynomial in the positive integer N. This means that for all sufficiently large N the value is given by a polynomial whose coefficients depend periodically on N. A reader would care because the result immediately implies that the ordinary generating function summing f(N) over all N is a rational function, which settles Kirillov's 2004 conjecture. The argument proceeds by rewriting the coefficients, via the pipe-dream model, as an alternating sum of lattice-point counts in certain polytopes that scale linearly with N.

Core claim

For permutations u, v, w in the symmetric group, the stretched Schubert coefficient f_{u,v,w}(N) := c_{N*u, N*v}^{N*w} is eventually quasi-polynomial. The proof expresses these coefficients as alternating sums of the numbers of integer points in polytopes coming from the pipe-dream model; these polytopes scale in a controlled way, so Ehrhart theory supplies the quasi-polynomial form.

What carries the argument

Pipe-dream polytopes whose lattice-point counts, under linear stretching by N, yield the Schubert coefficients via an alternating sum to which Ehrhart theory applies.

If this is right

  • The ordinary generating function summing f_{u,v,w}(N) over N is a rational function.
  • New infinite families of counterexamples to the saturation conjecture for Schubert coefficients exist.
  • Algorithms that evaluate or interpolate the quasi-polynomial can compute stretched coefficients for large N without enumerating all pipe dreams.
  • The same polytope construction yields effective bounds on the eventual period and degree of the quasi-polynomial.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same stretching-plus-Ehrhart technique may apply to other families of coefficients defined by alternating sums over combinatorial models, such as Littlewood-Richardson or Kronecker coefficients.
  • Rationality of the generating function supplies a new source of asymptotic formulas for the growth rate of ordinary Schubert coefficients.
  • The counterexamples to saturation suggest that positivity conjectures for Schubert coefficients may require stronger conditions than merely non-negative stretching.
  • Computational packages that compute Ehrhart quasi-polynomials of pipe-dream polytopes could be used to tabulate large tables of stretched coefficients automatically.

Load-bearing premise

The polytopes arising in the pipe-dream representation of Schubert coefficients scale linearly with the stretch factor N so that standard Ehrhart theory directly governs their lattice-point enumerators.

What would settle it

An explicit triple of permutations u, v, w together with a concrete computation of the first several dozen values of f(N) that fails to match any quasi-polynomial of the predicted period and degree.

Figures

Figures reproduced from arXiv: 2604.27107 by Igor Pak, Zachary Slonim.

Figure 2.1
Figure 2.1. Figure 2.1: Elbow tile (left) and cross tile (right) Example 2.1. Let u = 14862357 ∈ S8 and let D := {(1, 3),(1, 5),(2, 1),(2, 3),(2, 5),(2, 6),(3, 1),(3, 2),(3, 3),(4, 1)} ⊆ ∆8. Then D can be visualized as the pipe dream on the left of view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Pipe dream D for u with pipes drawn in (left) and simplified version with only dots and crosses (right) As shown on the right of view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: Ladder move Formally, let Li,j (D) := D + (i − m, j + 1) − (i, j), where the following conditions are satisfied (a) (i, j) ∈ D and (i, j + 1) ∈/ D, (b) (i − m, j),(i − m, j + 1) ∈/ D for some 0 < m < i, and (c) (i − k, j),(i − k, j + 1) ∈ D for all 1 ≤ k < m. Similarly, an inverse ladder move L −1 i−m,j+1 is a transformation of the type shown going right to left in view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: Bottom pipe dream Dbot(u) for u = 14862357 ∈ S8 . For a pipe dream D ∈ PD(u), denote by L(D) ⊆ PD(u) the set of pipe dreams for u that can be obtained from D by a sequence of ladder moves. We will need the following key combinatorial observation: Proposition 2.5 (Bergeron–Billey [BB93, Thm. 3.7(c)]). Let u ∈ Sn. Then L(Dbot(u)) = PD(u) view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Applying the algorithm of Lemma 3.4 to obtain Dbot(u) from D by a sequence of inverse ladder moves Thus, using the ordering of Example 3.1, we can see that x := φu(D) = (0, 0, 2, 1, 0, 0, 1, 1, 0, 0, 0) ∈ Φ4, where the nonzero entries correspond to x(2,1) = 2, x(2,2) = 1, x(1,1,2) = 1, x(1,1) = 1. We can then recover D by starting with Dbot(u) and performing the following ladder moves as determined by re… view at source ↗
read the original abstract

For a permutation $u\in S_n$, let $N\ast u\in S_{Nn}$ be the permutation with scaled Lehmer code. For given $u,v,w\in S_n$ and integer $N$, the stretched Schubert coefficients are defined as $f_{u,v,w}(N):=c_{N*u,N*v}^{N*w}$. Our main result is that the function $f_{u,v,w}(N)$ is eventually quasi-polynomial. This proves Kirillov's conjecture (2004), that the generating function for the sequence $\{f_{u,v,w}(N)\}$ is rational. For the proof, we use combinatorics of pipe dreams to show that Schubert coefficients are given as an alternating sum of the numbers of integer points in certain polytopes. These polytopes behave nicely under stretching, and we use Ehrhart theory to obtain the result. As a consequence of the proof, we also present new counterexamples to the saturation conjecture for Schubert coefficients, and give computational applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper claims that for fixed u, v, w in S_n the stretched Schubert coefficient f_{u,v,w}(N) := c_{N*u, N*v}^{N*w}, where N*u denotes the permutation whose Lehmer code is scaled by N, is eventually quasi-polynomial in N. The proof expresses Schubert coefficients via the pipe-dream model as an alternating sum of lattice-point counts in certain polytopes; these polytopes are shown to admit facet descriptions that scale linearly with the stretching parameter N. Ehrhart theory then implies that each lattice-point enumerator is a quasi-polynomial in N, and the alternating sum of finitely many such quasi-polynomials (with N-independent signs and number of summands) remains quasi-polynomial. This establishes rationality of the ordinary generating function, proving Kirillov's 2004 conjecture. The manuscript also derives new counterexamples to the saturation conjecture for Schubert coefficients and outlines computational applications.

Significance. Resolving Kirillov's conjecture is a substantial contribution to Schubert calculus and algebraic combinatorics. The argument combines the pipe-dream formula with Ehrhart theory in a direct manner that avoids new ad-hoc parameters and relies on independently established results, thereby keeping circularity risk low. The additional counterexamples to saturation and the computational consequences are concrete bonuses. The central modeling step—that the relevant polytopes scale appropriately under stretching—is the only non-routine ingredient, and the manuscript appears to establish the necessary niceness properties.

minor comments (3)
  1. [Introduction] The introduction would be strengthened by a short concrete example: take a small permutation u in S_3, compute N*u for N=2 and N=3, and display the corresponding Schubert coefficient values for small N to illustrate the stretching operation and the quasi-polynomial behavior.
  2. [§4] In the section applying Ehrhart theory, explicitly record the degree and period of the quasi-polynomials that arise from the pipe-dream polytopes; this would make the eventual quasi-polynomial statement more quantitative and facilitate comparison with known degree bounds for Schubert coefficients.
  3. [§6] The computational applications paragraph would benefit from a brief complexity discussion or pseudocode for extracting the quasi-polynomial from the polytope description; without it the claim of 'computational applications' remains somewhat vague.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our manuscript, which correctly identifies the main result on the eventual quasi-polynomiality of stretched Schubert coefficients, the proof strategy combining pipe dreams with Ehrhart theory, and the additional contributions regarding saturation counterexamples and computational applications. We are pleased that the referee views the resolution of Kirillov's conjecture as a substantial contribution and notes the low risk of circularity in the argument. The recommendation for minor revision is noted; since the report contains no specific major comments or requested changes, we interpret this as an invitation to incorporate any minor editorial or presentational improvements in the revised version.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper derives eventual quasi-polynomiality of f_{u,v,w}(N) by modeling the stretched Schubert coefficients via the pipe-dream formula as an alternating sum of lattice-point counts in polytopes whose facet inequalities scale linearly with the stretching parameter N. Standard Ehrhart theory for rational polytopes then supplies the quasi-polynomiality of each count (with N-independent signs and number of summands), so the alternating sum remains quasi-polynomial and its generating function is rational. Both the pipe-dream expression for Schubert coefficients and Ehrhart quasi-polynomials are independently established external results; the only non-routine step is the combinatorial verification that the relevant polytopes remain rational and scale linearly under stretching, which is proved directly rather than by self-definition, fitted parameters, or load-bearing self-citation. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof applies two standard external results without introducing fitted parameters or new entities.

axioms (2)
  • standard math Ehrhart theory: the number of integer points in a rational polytope dilated by N is a quasi-polynomial in N
    Invoked to conclude quasi-polynomiality once the coefficient is written as an alternating sum of lattice-point counts.
  • domain assumption Schubert coefficients admit a pipe-dream formula that expresses them as alternating sums of lattice points in explicitly described polytopes
    Central modeling step that allows the stretching argument to be transferred to the polytopes.

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