Pith. sign in

REVIEW 3 major objections 5 minor 4 references

Sums of Schubert structure constants with bounded Coxeter length

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that for each classical Lie family, the sum of Schubert structure constants over triples with Coxeter length $k$ is eventually a polynomial in $n$ with leading term $(2n)^k/k!$.

desk verdict New proof strategy and likely-true lead term, but the load-bearing count in Proposition 3.5 (and Lemma 3.4) is wrong as written and needs a serious revision. read the letter →

arxiv 2506.13684 v1 pith:XHPBAVLE submitted 2025-06-16 math.CO math.RT

Ada Stelzer This is my paper · ORCID
classification math.COmath.RT MSC 05E1414N1520F55
keywords SchubertstructureconstantsequivariantcohomologyDynkinsupportequivalenceWeylgroupsCoxeterlengthclassicalLietypespolynomialityGKMtheory
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's goal is to show that the sums $\gamma_k(n)$ of Schubert structure constants over triples $u,v,w$ with $\ell(w)=k$ are eventually polynomial in $n$, with the single leading term $(2n)^k/k!$, for each of the four classical Lie families (types A, B, C, and D). The method is to group Weyl-group elements $w$ by an equivalence relation: two elements are equivalent when their reduced words draw on isomorphic Dynkin subdiagrams and the isomorphism carries one element to the other. Stability of Schubert structure constants under Dynkin-diagram embeddings makes the inner sum over $u,v$ depend only on the class, so $\gamma_k(n)$ becomes a finite sum of a class constant times a class count $N_w(n)$. The leading class is the one supported on $k$ isolated $A_1$ nodes, which corresponds to products of $k$ commuting simple reflections; its structure-constant sum is $2^k$ and its count contributes $n^k/k!$. If correct, this answers the request for a conceptual proof and fixes the exact asymptotic size of bounded-length Schubert sums.

What carries the argument

The central object is the Dynkin-support equivalence $w \sim w'$: two Weyl-group elements are equivalent when their reduced words involve induced subdiagrams of the relevant classical Dynkin diagram that are isomorphic, with the isomorphism sending $w$ to $w'$. The load-bearing mechanism is the equivariant restriction formula, which expresses the restriction of an equivariant Schubert class at a fixed point as a sum over reduced subwords, together with the stability property that follows from it: an embedding of Dynkin diagrams $\iota: \Delta \to \Delta'$ forces the equivariant structure constants $C^w_{u,v}$ and $C^{\iota(w)}_{\iota(u),\iota(v)}$ to agree up to relabelling. These tools let the paper replace a sum over all of $W_n$, which grows with $n$, by a finite sum over equivalence classes represented inside $W_{2k}$, reducing the $n$-dependence to a polynomial class count $N_w(n)$.

What would settle it

In type A, fix $k$ and take $w$ to be a product of $k$ pairwise non-adjacent simple reflections. The elements equivalent to $w$ in $W_n$ are exactly the ways to choose $k$ non-adjacent nodes from the $n$-node Dynkin diagram, so the exact class count is $\binom{n-k+1}{k}$, with leading term $n^k/k!$. Checking whether Proposition 3.5 reproduces this count, rather than a $k!$-multiple of it, for any fixed $k$ and large $n$ settles the proof's central counting step.

Watch

Extended reading notes

Core claim

The paper's central claim, stated as Theorem 1.2, is that for fixed $k$ and any one of the families $\{SL_{n+1}\}$, $\{SO_{2n+1}\}$, $\{SP_{2n}\}$, $\{SO_{2n}\}$, the number $\gamma_k(n) = \sum_{u,v,w \in W_n, \ell(w)=k} c^w_{u,v}$ is, for all sufficiently large $n$, a polynomial in $n$ whose leading term is $(2n)^k/k!$. The proof splits $\gamma_k(n)$ by the Dynkin-support equivalence relation, obtaining $\gamma_k(n) = \sum_{[w], w \in W_{2k}, \ell(w)=k} (\sum_{u,v \le w} c^w_{u,v}) N_w(n)$, where $N_w(n)$ counts elements equivalent to $w$ in $W_n$. Because the inner structure-constant sum is stable under Dynkin embeddings, the only $n$-dependence is in $N_w(n)$; because $N_w(n)$ is eventually polynomial of degree equal to the number of type-A components of the support, the largest degree occurs when the support is $A_1^k$, where the inner sum equals $2^k$ and the count has leading term $n^k/k!$. The paper concludes that the leading term is therefore $(2n)^k/k!$.

Load-bearing premise

Everything rests on the count of Weyl-group elements equivalent to a given element $w$ under the Dynkin-support relation: if that count is wrong by a multiplicative factor, the claimed leading term is wrong by the same factor.

Editorial extensions

If this is right

  • For each fixed $k$ and each classical type, $\gamma_k(n)$ is eventually a degree-$k$ polynomial in $n$, so bounded-length structure-constant sums have a stable, type-independent asymptotic shape.
  • The leading term $(2n)^k/k!$ is the same for the $SL$, $SO$, and $Sp$ families, so the growth rate of these sums does not distinguish Lie type at top order.
  • The leading contribution comes entirely from elements $w$ whose Dynkin support is $A_1^k$, i.e. products of $k$ commuting simple reflections; every other support contributes at most degree $k-1$.
  • Because $\gamma_k(n)$ is a finite linear combination of fixed structure-constant sums from $W_{2k}$ with polynomial coefficients $N_w(n)$, the full polynomial can in principle be computed once the finite $W_{2k}$ data and the counts are known.

Reading between the lines

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

  • A natural extension not pursued in the paper is to track the first subleading term of $\gamma_k(n)$; the degree-$k-1$ coefficient should be computable from classes whose Dynkin support has $k-1$ components or one non-$A$ component, and it may be the first place where Lie type shows up.
  • The same Dynkin-support grouping should apply to sums weighted by $\ell(u)$ or $\ell(v)$, or to equivariant structure constants at specializations, since the stability property used here is equivariant; one can test whether the leading $n$-dependence remains $(2n)^k/k!$.
  • The explicit thresholds in Theorem 3.6 (roughly $n \ge k$ in the classical types) make the formula computationally checkable: for each small $k$, direct computation of $\gamma_k(n)$ in types A, B, C, D for several $n$ above the threshold would either confirm the polynomial or expose the discrepancy.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. Fix one of the four classical families of complex Lie groups, with Weyl groups of types A, B, C, and D. The paper studies γ_k(n), the sum of all Schubert structure constants c^w_{u,v} with ℓ(w)=k, and proves that for fixed k and all sufficiently large n, γ_k(n) is a polynomial in n with leading term (2n)^k/k!. The proof introduces an equivalence relation on Weyl group elements by isomorphism of Dynkin supports, then uses the equivariant restriction formula of Andersen–Jantzen–Soergel and a stability theorem of Robichaux–Yadav–Yong to show that all elements in one equivalence class contribute the same sum of structure constants. This reduces γ_k(n) to a finite sum over classes represented in W_{2k}, weighted by the class sizes N_w(n). Proposition 3.5 estimates these class sizes, Lemma 3.7 evaluates the sum of structure constants for the class A1^k, and the lead term is extracted from that class.

Significance. The proposed method is attractive and, if correct, would be a substantial improvement over the signed-puzzle proof of Pak–Robichaux: it gives the leading term for all classical types and cleanly separates the n-dependent counting from finitely many Schubert computations. The external tools used, including GKM localization and the AJS equivariant restriction formula, are standard and are applied in a plausible way; I see no circularity and no fitted parameters. The main obstacle is the counting in Proposition 3.5, which is currently incorrect; however, the small cases k=1 and k=2 indicate that the stated leading term is plausible, so the theorem may well survive a correct count. For these reasons the paper merits a major revision rather than rejection.

major comments (3)
  1. [Proposition 3.5] The displayed identity N_w(n) = N'_w(n)·∏ a_w(j)! is incorrect, and it is also inconsistent with the lead term stated in the same proposition. If N'_w(n) has leading term n^p, then the displayed product would have leading term n^p·∏ a_w(j)!, not n^p/∏ a_w(j)!. Concretely, for w = s_1...s_k with Δ(w) = A1^k, the k simple reflections commute, so each induced subgraph gives exactly one element equivalent to w; the true count is approximately n^k/k!, whereas the displayed product gives approximately n^k. For w = s_1s_2 with Δ(w) = A_2, the Dynkin automorphism sends w to s_2s_1, so each edge support contributes two equivalent elements, while ∏ a_w(j)! = 1. The correct multiplicity is the orbit size |Aut(Δ(w))|/|Stab(w)|, after first passing from labelled embeddings to induced subgraphs. Since Proposition 3.5 is the counting input used in Theorem 3.6, this error is load-bearing.
  2. [Theorem 3.6 and proof of Theorem 1.2] Theorem 3.6 uses N_w(n) as the size of the equivalence class [w]∩W_n, and the final lead-term extraction uses the leading coefficient of N_w(n) for the class A1^k. Because Proposition 3.5's N_w(n) is not the true class size, neither statement is established as written. For example, for k=2 in type A, the true class A1^2 has approximately n^2/2 elements, not n^2 as the product formula gives; combined with Lemma 3.7's factor 4, this would change the coefficient of n^2 unless the counting is repaired. The intended value (2n)^2/2 = 2n^2 is consistent with the true count, so I expect the theorem remains true, but the proof must be reworked after fixing Proposition 3.5.
  3. [Lemma 3.4 and Proposition 3.5] The counting convention in Lemma 3.4 should be stated explicitly. For Δ = A1^c, the lemma's formula is (k+1-c)!/(k+1-2c)! ≈ k^c, which counts labelled embeddings, one for each labelling of the c copies of A1. However, N'_w(n) is defined as the number of induced subgraphs isomorphic to Δ(w), which is an unlabeled count. The factor ∏ a_w(j)! in Proposition 3.5 is meant to mediate between these conventions, but it is applied in the wrong direction and it does not account for automorphisms of Δ(w) that do not come from permuting equal components, such as the automorphism of A_2. Please clarify the labelling convention and replace the identity with the correct orbit-size formula.
minor comments (5)
  1. [Throughout] The word 'polynomality' should be 'polynomiality' throughout the manuscript.
  2. [Definition 3.1] The statement that Δ(w) does not depend on the choice of reduced word I should be made part of the definition, not asserted afterward.
  3. [Definition 3.2] In the display following Definition 3.2, the index j is used both for the ambient dimension and for the component type, which makes the definition of a_w(j) unnecessarily confusing.
  4. [Example 2.3] The computation in Example 2.3 would be easier to check if the two subwords J of I were written out explicitly before evaluating the restriction formula.
  5. [Proposition 3.5, Figure 2] For the exceptional D_n embeddings, the text lists the relevant shapes but does not explain why these are the only exceptional shapes; a sentence of justification would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is self-contained and rests on external foundation results; the disputed counting lemma is a correctness concern, not a circular one.

full rationale

The paper's central claim, Theorem 1.2, is derived from Theorem 3.6, Proposition 3.5, and Lemma 3.7. Theorem 3.6 is a regrouping of the sum defining gamma_k(n) by the equivalence relation w~w', and the needed invariance of the inner sum of Schubert structure constants under ~ is supplied by Proposition 2.5, cited to Robichaux-Yadav-Yong [4, Theorem 2.1], which itself rests on the Andersen-Jantzen-Soergel equivariant restriction formula and GKM localization. None of these cited inputs is the target theorem, and none is authored by Stelzer. Proposition 3.5 is a separate combinatorial enumeration of equivalent Weyl group elements; although its displayed identity N_w(n) = N'_w(n) * product_j(a_w(j)!) appears internally inconsistent with its stated lead term n^p / product_j(a_w(j)!), and may miscount orbit sizes, a false or unsupported counting lemma is a correctness or rigor issue, not circularity: it does not assume Theorem 1.2, nor does it rename a fitted parameter as a prediction. Lemma 3.7 computes the minimal-support structure-constant sum by a direct factorization argument from Proposition 2.6, which is proved within the paper from the external AJS formula. No step reduces Theorem 1.2 to an input equivalent to itself; no fitted parameter is called a prediction; and no load-bearing self-citation occurs, since [3] is the theorem being reproved but is not used as a premise in the proof. Therefore the circularity score is 0.

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

No free parameters are introduced. The proof imports standard Lie-theoretic results. The new combinatorial objects are Dynkin support and the equivalence relation, which are definitions rather than independent entities. The fragile counting identity in Proposition 3.5 is classified as an ad hoc assumption because the proof does not correctly derive the automorphism orbit sizes.

assumptions (6)
  • standard math Andersen-Jantzen-Soergel equivariant restriction formula
    Used in Section 2 to compute equivariant restrictions and derive support and stability properties; taken as a theorem from [1].
  • standard math GKM localization injection
    Identifies equivariant cohomology with functions on the finite fixed point set; from [2].
  • domain assumption Dynkin diagram embedding invariance of Schubert structure constants
    Allows grouping structure constants by Dynkin support equivalence; cited from [4].
  • standard math Parabolic factorization of equivariant structure constants
    Proved in the paper using the restriction formula and Bruhat support; needed for Lemma 3.7 and for sums over products of simple reflections.
  • domain assumption Any Dynkin support with at most k vertices embeds into the type-compatible diagram on 2k vertices
    Used in Lemma 3.3 to represent every equivalence class by an element of W_{2k}; a combinatorial fact about Dynkin diagrams.
  • ad hoc to paper Polynomiality and degree of equivalence class counts
    Proposition 3.5 asserts that N_w(n) is eventually polynomial with degree equal to the number of type-A components. The conclusion is plausible, but the proof contains an incorrect identity; the lead term for the maximal class A1^k is correct.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Sums of Schubert structure constants with bounded Coxeter length." pith.science (2026). https://pith.science/paper/XHPBAVLE

@misc{pith2026250613684,
  author       = {Pith},
  title        = {Pith review of: Sums of Schubert structure constants with bounded Coxeter length},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XHPBAVLE}},
  note         = {Machine review of arXiv:2506.13684}
}
abstract

Pak-Robichaux recently introduced a signed puzzle rule for Schubert structure constants, which they use to show that sums $\gamma_k(n)$ of these constants with a bounded number of inversions are polynomial. We give a different, conceptual proof of their theorem. Our argument computes the lead term of $\gamma_k(n)$ and extends to all classical Lie types.

Figures

Figures reproduced from arXiv: 2506.13684 by the authors.

Figure 1
Figure 1. An embedding of ∆ = A1 × A1 × A2 into A7. Proposition 3.5. For w ∈ Wm for some m, let Nw(n) denote the number of elements v ∈ Wn such that w ∼ v. Let n ′ denote the number of vertices in ∆(w), and p the number of type-A components in ∆(w). Then for n ≥    n ′ − 1 in type A, n ′ in types B and C, n ′ + 1 in type D, Nw(n) is a polynomial with lead term LT(Nw(n)) = n p Qm j=1(aw(j)!). Proof. Let N′ w(n) denote the… view at source ↗
Figure 2
Figure 2. An “exceptional” embedding of ∆ = A1 × A2 × A3 into D8. Theorem 3.6. Fix k and a choice of classical Lie type. Then γk(n) is the finite sum γk(n) = X [w] w∈W2k,ℓ(w)=k X u,v≤w c w u,v! Nw(n). In particular, γk(n) is a polynomial for n ≥    k − 1 in type A, k in types B and C, k + 1 in type D. Proof. Recall that by definition, γk(n) = X u,v,w∈Wn ℓ(w)=k c w u,v. By Lemma 2.4, we may restrict the sum to u, v ≤ w. B… view at source ↗

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

4 extracted references · 3 canonical work pages

  1. [1]

    Representations of quantum groups at a pth root of unity and of semisimple groups in characteristic p: independence of p , Astrisque No

    Andersen, Henning; Jantzen, Jens; Soergel, Wolfgang. Representations of quantum groups at a pth root of unity and of semisimple groups in characteristic p: independence of p , Astrisque No. 220 (1994), 321 pp

    work page 1994

  2. [2]

    Equivariant cohomology, Koszul duality, and the localization theorem, Invent

    Goresky, Mark; Kottwitz, Robert; MacPherson, Robert. Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), no. 1, 25–83

    work page 1998

  3. [3]

    Signed puzzles for Schubert coefficients, preprint, 2025

    Pak, Igor; Robichaux, Colleen. Signed puzzles for Schubert coefficients, preprint, 2025. arXiv:2504.17734

    arXiv 2025

  4. [4]

    The A·B·C·Ds of Schubert calculus, S´ em

    Robichaux, Colleen; Yadav, Harshit; Yong, Alexander. The A·B·C·Ds of Schubert calculus, S´ em. Lothar. Combin. 85 ([2020–2021]), Art. B85a, 12 pp. Dept. of Mathematics, U. Illinois at Urbana-Champaign, Urbana, IL 61801, USA Email address : astelzer@illinois.edu

    work page 2020

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.