Equivariant cohomology, Schubert calculus, and edge labeled tableaux

Alexander Yong; Colleen Robichaux; Harshit Yadav

REVIEW 3 major objections 6 minor 75 references

Equivariant cohomology, Schubert calculus, and edge labeled tableaux

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Pith's one-line read This paper proposes a conjectural Littlewood-Richardson rule for the equivariant Schubert calculus of isotropic Grassmannians, built from shifted edge labeled tableaux and the Anderson-Fulton ring.

desk verdict New shifted edge labeled tableaux give a plausible conjectural LR rule for isotropic Grassmannians, but the central ring isomorphism is open and depends on a rectification-order convention with no geometric justification. read the letter →

arxiv 1908.11224 v1 pith:KF3R5RRN submitted 2019-08-29 math.CO math.AG

Colleen Robichaux , Harshit Yadav , Alexander Yong This is my paper · ORCID

classification math.COmath.AG MSC 05E1014M1505E05

keywords shiftededgelabeledtableauxequivariantSchubertcalculusisotropicGrassmanniansjeudetaquinLittlewood-RichardsonruleAnderson-Fultonringgrowthdiagramscommutative

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 is partly a survey of edge labeled tableaux as a model for equivariant Schubert calculus of Grassmannians, and partly an announcement of a shifted analogue aimed at maximal orthogonal and Lagrangian Grassmannians. It introduces shifted edge labeled tableaux, in which edge labels are allowed only on the southern edges of diagonal boxes, and defines a product on formal basis elements $[\lambda]$ by counting tableaux of shape $\nu/\lambda$ that row-rectify to a fixed superstandard tableau of shape $\mu$. The central conjecture is that this product makes the ring $R_n$ isomorphic to the Anderson-Fulton ring $P$, whose structure constants specialize to equivariant Schubert structure constants of the Lagrangian Grassmannian. If true, this supplies the missing combinatorial Littlewood-Richardson rule for the equivariant cohomology of the isotropic Grassmannians and opens the same circle of questions (nonvanishing, saturation, complexity) that has been answered for ordinary Grassmannians. The paper proves commutativity of $R_n$ by growth diagrams and verifies the isomorphism in all cases $n \le 4$ and many $n = 5$ cases.

What carries the argument

The key machinery is the shifted edge labeled tableau together with its shifted jeu de taquin slides: the usual two-box slides (J1) and (J2) are supplemented by diagonal-edge moves (J3') and (J4') that move a box label past an edge-label set or replace the smallest edge label. Rectification is fixed to row order, and the count of tableaux rectifying to the superstandard tableau $S_\mu$ produces the structure constants. A growth-diagram formalism based on e-partitions (partitions with edge-label multiplicities recorded on diagonal edges) is then used to prove that the resulting product is commutative, by reflecting the diagram across its antidiagonal.

What would settle it

For any strict partitions $\lambda,\mu,\nu \subseteq \rho_5$ not among the cases already checked, compute $d^\nu_{\lambda,\mu}$ by the row-rectification count and compare it with the Anderson-Fulton coefficient $D^\nu_{\lambda,\mu}$ obtained from the Pfaffian formula in Section 10; a single disagreement would disprove Conjecture 10.1.

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Extended reading notes

Core claim

The central discovery is a new combinatorial object, the shifted edge labeled tableau: a filling of a shifted skew shape in which every label appears once, rows and columns increase, and the southern edge of each diagonal box may carry a subset of labels. Counting such tableaux of shape $\nu/\lambda$ that rectify to a fixed superstandard tableau under a row-by-row jeu de taquin order defines coefficients $d^\nu_{\lambda,\mu}$, packaged as $D^\nu_{\lambda,\mu} = 2^{L(\nu;\lambda,\mu)-\Delta(\nu;\lambda,\mu)} z^{\Delta(\nu;\lambda,\mu)} d^\nu_{\lambda,\mu}$. The formal span of symbols $[\lambda]$ with product $[\lambda] \star [\mu] = \sum_\nu D^\nu_{\lambda,\mu}[\nu]$ forms a ring $R_n$ that the authors conjecture is isomorphic to the Anderson-Fulton ring $P$ with basis $\sigma_\lambda$. Under that isomorphism, the structure constants become the equivariant Schubert structure constants of the Lagrangian Grassmannian after specializing $\alpha_1 = z$ and $\alpha_i = 0$ for $i > 1$. Evidence includes a growth-diagram proof of commutativity, matching special cases $d^\lambda_{\lambda,(p)} = \binom{\ell(\lambda)}{p}2^{p-1}$ and $d^{\rho_n}_{\rho_n,\rho_n} = 2^{\binom{n}{2}}$, and exhaustive verification for $n \le 4$.

Load-bearing premise

The product coefficients are defined using a fixed row-by-row order of sliding moves, but the final tableau can depend on that order once edge labels are present; if the correct geometric rule needs a different order, the conjectured ring would not be the Anderson-Fulton ring.

Editorial extensions

If this is right

If Conjecture 10.1 holds, the coefficients $D^\nu_{\lambda,\mu}$ give an explicit Littlewood-Richardson rule for the equivariant Schubert structure constants of the Lagrangian Grassmannian after the specialization $\alpha_1 = z$, $\alpha_i = 0$ for $i > 1$.
The conjectured isomorphism implies Conjecture 9.2: every $D^\nu_{\lambda,\mu}$ is an integer polynomial in $z$, with nonnegative coefficients inherited from equivariant positivity.
Commutativity of $R_n$ is already established, and the Pieri-type cases match the Anderson-Fulton ring, so the remaining gap to the full isomorphism is associativity together with a general comparison of structure constants.
A proof of associativity combined with the Pieri rule would complete the standard associativity-argument route to the isomorphism, mirroring the classical proof of the Littlewood-Richardson rule.

Reading between the lines

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

The dependence of rectification on row order, highlighted by Example 9.1, is the most delicate point in the construction: if the geometrically correct jeu de taquin uses a different order, the entire conjectural dictionary would shift, and Example 9.9 shows the order sensitivity is real.
The equality $d^{\rho_n}_{\rho_n,\rho_n} = 2^{\binom{n}{2}}$ invites a bijective explanation: staircase-shaped shifted edge labeled tableaux rectifying to the superstandard tableau may correspond to labeled graphs on $n$ vertices, Aztec diamond tilings, or Gelfand-Tsetlin patterns, exposing recursive structure that could help with the general conjecture.
Associativity of $R_n$ is a finite check for each fixed $n$ because $R_n$ is finitely generated as a $\mathbb{Z}[z]$-module; verifying it for $n = 5$ computationally would be a concrete intermediate test that does not require the geometric comparison.
If the conjecture holds, the specialization $\alpha_1 = z$ suggests that shifted edge labeled tableaux may also serve as a combinatorial model for a one-dimensional torus action on the Lagrangian Grassmannian, potentially giving a GKM-style proof independent of the Anderson-Fulton algebra.

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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

Colleague, this is a survey-and-announcement chapter from Yong's school. The genuinely new thing is the shifted edge labeled tableau model and the ring R_n built from it. Conjecture 10.1 identifies R_n with the Anderson-Fulton ring, which would give the long-sought equivariant Littlewood-Richardson rule for isotropic Grassmannians. That is a substantial research program, and the paper gives concrete evidence: the ring is proved commutative via a Fomin growth-diagram argument, the Pieri-type and maximal-case coefficients are computed in closed form, and there is a new proof of the O/L structure constant relation. The small cases are checked up to n=4 and many n=5 cases. The writing is good, and the survey parts are a useful entry point.

The soft spot is the one the stress-test flags. The definition of d^ν_{λ,μ} in Section 9.2 uses row rectification order, and shEqRect is not independent of that order when edge labels are present. Example 9.1 shows the two orders give different rectification target shapes, and Example 9.9 gives different counts (20 vs 16) for the same triple. The paper simply declares row rectification to be the rule. There is no argument that this convention is the one that matches the Anderson-Fulton ring. The only bridge to the geometry is equation (34), attributed to a private communication, so the numerics cannot distinguish between conventions beyond small cases. If the correct geometric rule requires a different order, the conjecture fails. That is a genuine load-bearing concern, and the authors are upfront about it, but they don't resolve it.

What is proved is real. The commutativity proof is nontrivial, the special cases are proven, and the O/L theorem (Theorem 8.2) is a solid contribution. The paper is honest about what is conjectural: Conjecture 9.2 (integrality), Conjecture 9.10 (associativity), and Conjecture 10.1 (isomorphism) are all clearly marked.

Proportionately: this is an announcement, not a complete theory. The soft spots are inherent to that genre. I would like to see a more serious attempt to justify the rectification order, or at least to check whether column order can ever match the AF ring. That would either strengthen or kill the conjecture, and it is the first thing I would ask a future version to address.

Who is this for? Anyone working on Schubert calculus, especially equivariant and isotropic cases. It deserves a serious referee, not a desk reject. I would recommend peer review with the expectation of a revised version. I'd also cite it if I work in this area, primarily as the source of the shifted edge labeled model and the conjecture.

Referee Report

3 major / 6 minor

Summary. This paper is a survey-announcement on equivariant Schubert calculus and edge labeled tableaux. It reviews the Thomas–Yong edge labeled jeu de taquin rule for equivariant structure constants of Grassmannians, the associated ballot rule, and applications to nonvanishing, saturation, Horn inequalities, and computational complexity. The new contribution is a shifted analogue of edge labeled tableaux: Section 9 defines shifted edge labeled tableaux with labels on diagonal edges, a jeu de taquin slide, and a rectification using a fixed “row rectification order.” It then defines d^ν_{λ,μ} as the number of such tableaux of shape ν/λ with |μ| labels that rectify to the superstandard tableau S_μ, and D^ν_{λ,μ} = 2^{ℓ(λ)+ℓ(μ)-ℓ(ν)-(|λ|+|μ|-|ν|)} z^{|λ|+|μ|-|ν|} d^ν_{λ,μ}. The authors prove commutativity of the resulting Z[z]-algebra R_n (Theorem 9.5), conjecture associativity (Conjecture 9.10), and conjecture that R_n is isomorphic to the Anderson–Fulton ring P with [λ]↦σ_λ (Conjecture 10.1). They verify special cases λ=(p) and λ=μ=ρ_n (Theorems 10.3 and 10.6) and report exhaustive checks for n≤4. The connection to equivariant Schubert calculus of the Lagrangian Grassmannian is made through identity (34) attributed to Anderson–Fulton private communication.

Significance. If true, Conjecture 10.1 would supply the first explicit Littlewood–Richardson-type rule for the equivariant Schubert structure constants of isotropic Grassmannians, a problem highlighted in Section 8. The paper’s proved results—commutativity of R_n, Theorem 8.2 relating the O and L structure constants, and the exact Pieri and top-class checks—provide nontrivial supporting evidence, and the survey portions are useful and clearly organized. The paper is also transparent about which statements are conjectural, and the numerical evidence is explicitly documented for small cases. However, the significance is conditional on an unresolved identification with the Anderson–Fulton ring and on the geometric correctness of the chosen rectification convention.

major comments (3)

[9.1–9.2, Examples 9.1 and 9.9] The structure constants d^ν_{λ,μ} and hence the ring R_n depend on the choice of row rectification order, as the paper itself shows in Example 9.1, and Example 9.9 records different counts (20 versus 16) under column rectification. Since the Anderson–Fulton structure constants are canonical, a proposed Littlewood–Richardson rule should be independent of such an arbitrary convention, or the paper should supply a geometric argument that row rectification is the correct convention. As written, the central conjecture 10.1 is tied to one fixed choice, and no evidence is given that this choice is forced by the geometry rather than by convenience.
[10.1, Eq. (34)] The identity L^ν_{λ,μ}(α1↦z, α2↦0, ..., αn↦0) = D^ν_{λ,μ} is attributed to a private communication with Anderson and Fulton. This identity is load-bearing: it is used in Proposition 10.2 and in Lemma 10.5 to connect the tableaux counts to the equivariant geometry, and it converts Conjecture 10.1 into a statement about Schubert calculus. A private communication is not a verifiable reference. The paper should either provide a proof or a detailed derivation of (34), or clearly restate the affected results as conditional on an unpublished identity.
[10.2, Claim 10.9 and Theorem 10.6] The proof of Theorem 10.6, which gives the check d^{ρ_n}_{ρ_n,ρ_n} = 2^{\binom{n}{2}} = d^{ρ_n}_{ρ_n,ρ_n}, rests on Claim 10.9, whose proof is deferred with the statement “The proof of this claim is lengthy and will appear elsewhere.” As a result, the claimed verification of Conjecture 10.1 in a nontrivial maximal case is conditional. The theorem should be reworded as conditional, or the proof of Claim 10.9 should be included or made available in a cited preprint.

minor comments (6)

[References] References [65] and [67] are identical; one should be removed or the two entries should be distinguished.
[7.3, displayed expansion of P_{(2,1)}] In the expansion of P_{(2,1)}(x_1,x_2,x_3), the monomial x_1x_2^2 appears twice; the first occurrence should presumably be x_1^2x_2.
[9.1, definition of diagonal edge] The sentence “This has one diagonal box but three diagonal edges” is confusing; the definition of a diagonal edge should be stated precisely and illustrated, since a single diagonal box would normally have one southern edge.
[Example 9.9] The counts 20 and 16 are stated without explanation; a short description of the computation or a table of the relevant tableaux would make the order-dependence check reproducible.
[9.1, row rectification order] The row rectification order is described informally in Example 9.1 as “choosing the southmost inner corner”; Section 9.2 should give a precise algorithm for this order, especially because the definition of shEqRect depends on it.
[10.1, Eq. (34)] The variables α_i in Eq. (34) are not reintroduced at the start of Section 10.1; the substitution should be stated explicitly, including the role of the indeterminate z.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction: the shifted edge-labeled tableau counts are defined independently and compared with the external Anderson–Fulton ring; self-citations and an unproved private-communication bridge create verifiability caveats but no by-construction circularity.

full rationale

The central derivation is not circular. The quantities d^ν_{λ,μ} are defined in Section 9.2 as cardinalities of shifted edge-labeled tableaux rectifying to S_μ under a fixed row-rectification convention; D^ν_{λ,μ} is a rescaling by 2^{L(ν;λ,μ)−Δ(ν;λ,μ)} z^{Δ(ν;λ,μ)}, and Conjecture 10.1 asserts equality with the independently defined Anderson–Fulton structure constants in the ring P introduced in Section 10.1. No parameter is fitted from the target ring to define d, and the conjecture is checked against external data (n≤4 and many n=5 cases). The commutativity proof for R_n (Theorem 9.5) is self-contained via growth diagrams and antidiagonal reflection. Several caveats exist but are not circularity: (i) shEqRect depends on rectification order (Example 9.1), and Example 9.9 shows column rectification gives 20 vs 16, so the row-order choice is a fragility rather than a forced construction; (ii) equation (34), connecting the Anderson–Fulton ring to equivariant Schubert calculus, is attributed to a private communication and is unproved, and Lemma 10.5 depends on it; (iii) survey material cites [66] by Thomas and the third author for the equivariant Grassmannian rule, but the new shifted model is not derived from that rule; (iv) Theorem 8.2 cites [55] by the same authors, although a proof is supplied in the text. These are verification and support-strength concerns, not reductions of the central conjecture to its own definitions.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

The central conjecture rests on the external identification (34), the row rectification convention, and standard Schubert calculus background. The paper is transparent about most of these.

assumptions (3)

domain assumption The specialization L^ν_{λ,μ}(α1→z, α2→0, ...) equals the Anderson-Fulton structure constant D^ν_{λ,μ} (Equation (34)), stated on private communication.
This identification is the bridge between the combinatorial ring R_n and equivariant Schubert calculus; it is not proved in the paper.
ad hoc to paper Rectification of shifted edge labeled tableaux is defined using row rectification order, even though shEqRect depends on the order (Example 9.1).
The ring structure constants d^ν depend on this convention; a different order gives different counts (Example 9.9).
standard math Standard results of equivariant cohomology and Schubert calculus (Graham positivity, GKM theory, Ikeda-Naruse formulas) are assumed.
These are background results cited and used throughout the survey sections.

invented entities (1)

Ring R_n of shifted edge labeled tableaux independent evidence
purpose: Provides a conjectural combinatorial model for the Anderson-Fulton ring and for equivariant Schubert calculus of isotropic Grassmannians.
The conjectured isomorphism to the independently defined Anderson-Fulton ring gives an external falsifiable check; small cases are verified.

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Pith. "Pith review of Equivariant cohomology, Schubert calculus, and edge labeled tableaux." pith.science (2026). https://pith.science/paper/KF3R5RRN

@misc{pith2026190811224,
  author       = {Pith},
  title        = {Pith review of: Equivariant cohomology, Schubert calculus, and edge labeled tableaux},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KF3R5RRN}},
  note         = {Machine review of arXiv:1908.11224}
}

read the original abstract

This chapter concerns edge labeled Young tableaux, introduced by H. Thomas and the third author. It is used to model equivariant Schubert calculus of Grassmannians. We survey results, problems, conjectures, together with their influences from combinatorics, algebraic and symplectic geometry, linear algebra, and computational complexity. We report on a new shifted analogue of edge labeled tableaux. Conjecturally, this gives a Littlewood-Richardson rule for the structure constants of the D. Anderson-W. Fulton ring, which is related to the equivariant cohomology of isotropic Grassmannians.

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