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Richardson volume models for skew Schur and skew Schur $P/Q$-functions

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Pith's one-line read The paper proves that every ordinary skew Schur polynomial and every skew Schur P- or Q-function is, after factorial normalization, a realizable volume polynomial over the complex numbers.

desk verdict Strong paper resolving two open Lorentzian conjectures with a realizable-volume upgrade; the main risk is the type C Schubert normalization step, which the author flags and which deserves an independent check. read the letter →

arxiv 2608.10516 v1 pith:SSDHO677 submitted 2026-08-11 math.CO

Khai-Hoan Nguyen-Dang This is my paper · ORCID
classification math.CO MSC 05E0514M1514C1752B40
keywords skewSchurpolynomialP-functionQ-functionRichardsonvarietyLagrangianGrassmanniantotalChernclassrealizablevolumeLorentzian
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 that, in any finite number of variables, the factorial normalization of every ordinary skew Schur polynomial and every skew Schur P- or Q-function is a realizable volume polynomial over the complex numbers. A realizable volume polynomial is the top self-intersection of a linear combination of semiample divisors on an irreducible projective variety, a strictly stronger property than being Lorentzian. The proof identifies these symmetric-function coefficient arrays with top-degree total-Chern intersection numbers on Richardson varieties in ordinary and Lagrangian Grassmannians. This settles the two open Lorentzian conjectures for skew Schur and Schur P functions, extends the P/Q statement to arbitrary strict skew shapes, and yields a package of coefficient inequalities including reverse Khovanskii–Teissier bounds, Hessian-signature and principal-minor conditions, root-direction log-concavity, and dominance monotonicity toward balanced contents.

What carries the argument

The load-bearing object is the Richardson cycle transform: for an irreducible $d$-dimensional subvariety $X$ of an ordinary Grassmannian, $\Theta^A_{X,n}(x) = \sum_{\nu \subseteq c^r,\, |\nu|=d} (\int_X s_\nu(S^\vee|_X))\, s_\nu(x)$, with the type-C analogue on a Lagrangian Grassmannian using Schubert classes $\sigma_\nu$ and Schur $P$-functions; the skew Schur and skew $P/Q$ cases are the specializations where $X$ is the corresponding Richardson variety. The transform converts tableau coefficients into top-degree total Chern intersections. The engine behind the volume conclusion is a general theorem: the top-degree total-Chern polynomial of globally generated bundles is denormalized Lorentzian, and its factorial normalization is a realizable volume polynomial, realized on a fiber product of projective bundles whose tautological divisors reproduce the Chern classes as intersection powers; a realizable-covolume differential operator removes the extra divisor powers. The type-C step uses a stable graded ring homomorphism from the Schur-$Q$ algebra to the Chow ring of the Lagrangian Grassmannian that sends $q_r$ to $c_r(E)$ and, through a modified Pfaffian representative, $Q_\nu$ to the Schubert class $\sigma_\nu$; this matching is what makes the shifted $P/Q$ identity true.

What would settle it

Take a small strict skew shape, for instance $\lambda=(2,1)$, $\mu=\emptyset$, with $n=2$ and $\alpha=(2,1)$. Compute $[x^\alpha]P_{\lambda/\mu}$ by the marked shifted-tableau generating function and separately compute $\int_{R^C_{\lambda/\mu}} c_2(E)c_1(E)$ on the Lagrangian Richardson variety; the paper's theorem says the two numbers are equal (both equal 1). Any mismatch in such a finite computation would falsify the type-C volume theorem, and the same check can be run for ordinary skew Schur polynomials with the type-A Richardson identity.

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Extended reading notes

Core claim

On its own terms, the central result is Theorem 1.1: for every ordinary skew shape $\theta$, every strict skew shape $\vartheta$, and every $n \geq 1$, the polynomials $\mathcal{N}(s_\theta(x_1,\ldots,x_n))$, $\mathcal{N}(P_\vartheta(x_1,\ldots,x_n))$, and $\mathcal{N}(Q_\vartheta(x_1,\ldots,x_n))$ are realizable volume polynomials over $\mathbb{C}$, with smooth irreducible realizations whenever nonzero. The proof runs through two Richardson identities: $[x^\alpha]s_{\lambda/\mu} = \int_{R^A_{\lambda/\mu}} \prod_i h_{\alpha_i}(S^\vee)$ in type A, and $[x^\alpha]P_{\lambda/\mu} = \int_{R^C_{\lambda/\mu}} \prod_i c_{\alpha_i}(E)$ in type C, where the second uses a stable homomorphism from the Schur-$Q$ algebra to the Chow ring of the Lagrangian Grassmannian sending a modified representative of $Q_\nu$ to the Schubert class $\sigma_\nu$. These identities turn tableau counts into top-degree total-Chern intersections, and a general volume theorem upgrades the resulting polynomials to realizable volumes. The $Q$-function case follows because $Q_{\lambda/\mu} = 2^{\ell(\lambda)-\ell(\mu)}P_{\lambda/\mu}$.

Load-bearing premise

The shifted theorem depends on a matching between algebraic representatives of Schur Q-functions and geometric Schubert classes on the Lagrangian Grassmannian; if that matching is not exactly right, the skew P/Q volume conclusions would not follow.

Editorial extensions

If this is right

  • The two Lorentzian conjectures for skew Schur and for Schur P functions are true, and the P/Q statement holds for every strict skew shape, not only straight shapes.
  • Every finite-variable ordinary or shifted tableau coefficient array inherits the full Hodge–Riemann package: reverse Khovanskii–Teissier inequalities, one-positive-eigenvalue Hessians, alternating principal-minor signs, and log-concavity along coordinate root directions.
  • Supports are exactly lattice points of permutahedra: for ordinary skew Schur polynomials the support is $P_{\kappa(\lambda/\mu)} \cap \mathbb{Z}^n$ with unit vertex coefficients; for straight Schur P/Q the classical supports are recovered, and skew P/Q admit an exact permutahedron support.
  • Cumulative two-row ordinary Littlewood–Richardson coefficients and weighted cumulative two-row shifted Littlewood–Richardson coefficients are log-concave sequences with no internal zeros.
  • Mixed products of ordinary skew Schur, skew P, and skew Q functions again have realizable factorial normalizations, so every inequality and support statement extends to products.

Reading between the lines

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

  • A direct next step the paper leaves open is a closed shifted-tableau description of the dominant partition $\kappa^P_{\lambda/\mu,n}$ that generates the skew P/Q support permutahedron; small examples from the P-expansion could be used to guess and test such a formula.
  • The type-C matching is normalization-sensitive, so analogous modified representatives would be needed before the same volume construction could be attempted on other isotropic flag varieties, such as odd orthogonal or other symplectic types.
  • A natural stress test of the general mechanism is to compute the cycle transform for a non-Richardson subvariety of a small Grassmannian and verify the volume inequalities on that example.
  • The covariance bound and ultra-log-concavity for factorially tilted content distributions are proved from the volume realization alone, so checking them on the mixed products in Corollary 8.8 would separate the volume mechanism from special tableau structure.
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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

1 major / 3 minor

Summary. The paper proves that the factorial normalizations N(s_{λ/μ}), N(P_{λ/μ}), and N(Q_{λ/μ}) are realizable volume polynomials over C for every ordinary skew shape and every strict skew shape, thereby settling Conjectures 19 and 20 of Huh–Matherne–Mészáros–St. Dizier and strengthening the Schur-P statement to arbitrary skew shifted shapes. The method identifies skew Schur and skew Schur P/Q-functions as top-degree total-Chern intersection polynomials on Richardson varieties in ordinary and Lagrangian Grassmannians, then invokes a general total-Chern volume theorem (Cid-Ruiz) and an operator-theoretic upgrade (Grund–Huh–Michałek–Süss–Wang) to pass from denormalized Lorentzianity to volume realizability. The paper further derives a substantial package of consequences: reverse Khovanskii–Teissier inequalities, Hessian-signature and principal-minor inequalities, root-direction log-concavity, dominance monotonicity, exact support permutahedra for ordinary skew Schur functions, straight Schur P/Q support polytopes, and log-concavity theorems for cumulative two-row Littlewood–Richardson coefficients.

Significance. If the main results stand, this is a significant advance in the theory of Lorentzian and volume polynomials. The geometric realization on Richardson varieties is natural, and the passage from Lorentzianity to volume realizability is a genuine strengthening. The type C construction via the stable Schubert homomorphism is a novel and useful ingredient, and the resulting inequalities for tableau multiplicities and Littlewood–Richardson coefficients are numerous and interesting. The paper is careful to attribute general mechanisms to prior work, and the central coefficient identities are derived explicitly and checked against small examples. The main volume-realization theorems are supported by coherent proofs; the principal concern identified in this report is a local error in one of the formal consequence theorems, not in the central argument.

major comments (1)
  1. [§6.1, Eq. (6.4)] The formula for Pol_μ(f) in Eq. (6.4) is not the full polarization, because its diagonal specialization does not recover f. For a concrete counterexample, take F = x_1^2 + x_1 x_2, so f = N(F) = x_1^2/2 + x_1 x_2 and μ = (2,1). Formula (6.4) yields Pol_μ(f) = (1/2)e_2(Y_1) + 2 e_1(Y_1)e_1(Y_2); specializing y_{i,k}=x_i gives (1/2)x_1^2 + 4x_1 x_2, not f. The correct normalization is division by ∏_i (μ_i choose α_i), not multiplication. This invalidates the proof that Pol_μ(f) is a realizable volume polynomial (the proof explicitly relies on the false diagonal claim) and consequently the use of this theorem in Corollary 6.9. The support statement (6.5) is unaffected, and the central volume-realization theorems do not depend on this formula, but the error must be corrected in a revision.
minor comments (3)
  1. [§5.1, Prop. 5.3] Since the paper itself flags the stable Schubert homomorphism as the normalization-sensitive step, it would be helpful to add a remark explicitly identifying the Kresch–Tamvakis substitution e_r for q_r with q_r ↦ c_r(E) in this setting, and to verify the normalization in a case with ℓ(ν)>1 (e.g., ν=(2,1)) in addition to the ℓ(ν)=1 check in Remark 5.5. I have not found an actual error here, but such a remark would address the residual uncertainty about a possible scalar mismatch.
  2. [§8.1, Thm. 8.7] The phrase 'has no internal zeros' should be clarified to mean that the positive entries form an interval without gaps. Taken literally, the assertion is false: for P_{(4,2)}(x,y) = x^4 y^2 + 2x^3 y^3 + x^2 y^4, the sequence B_0,...,B_6 is (0,0,1,2,1,0,0), which has zeros at r=1 and r=5. The intended interval property follows from M-convexity and is sufficient for the stated log-concavity.
  3. [§6.1, proof of Thm. 6.2(iii)] The text repeats the claim that the expression in (6.4) is 'the usual full polarization' whose diagonal specialization is f. After correcting the normalization factor, the proof should be updated consistently, and it would be useful to state explicitly which convention from [13, Proposition 4.1] is being followed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the coefficient identities are proved against independent combinatorial and intersection-theoretic objects, and the volume/Lorentzian upgrades use external theorems.

full rationale

The paper's central claims do not reduce to their inputs by construction. The new content is the coefficient identification in (4.7) and (5.18): ordinary skew Schur coefficients are expressed as Chern intersections on Richardson varieties via repeated Pieri multiplication and complementary Schubert duality, and shifted skew P/Q coefficients via the Schur-P/Q Cauchy identity, Hopf duality, and the stable Schubert homomorphism of Proposition 5.3. These identities are checked against independent objects: standard tableaux/Hopf-algebra coefficient expansions, Schubert classes, and Kresch–Tamvakis modified Pfaffian representatives. The upgrade to Lorentzianity and realizable volume polynomiality is then delegated to external general mechanisms: Cid-Ruiz's total-Chern Lorentzian theorem, the Grund–Huh–Michałek–Süss–Wang operator theorem, Brändén–Huh Lorentzian theory, Jiang–Li/Lehmann–Xiao reverse Khovanskii–Teissier inequalities, and Rado's polytope theorem. No fitted parameter is renamed as a prediction, no normalization pre-imposes the target polynomial, and no load-bearing assertion is justified by a self-citation chain. The type-C normalization-sensitive step is explicitly flagged in Section 1.2 and Proposition 5.3 and delegated to independent references [5, 19]; even if that Schubert identification were incorrect, that would be a correctness risk, not circularity. The support and extremal-coefficient results are presented either with new proofs from the volume framework or explicitly as recoveries of known results, not disguised as brand-new discoveries. Accordingly the circularity burden is zero.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on standard heavy machinery (intersection theory, Lorentzian polynomial theory, Schur P/Q Hopf algebra, M-convexity) and two recent external theorems: Cid-Ruiz's total-Chern Lorentzian theorem and Grund-Huh-Michałek-Süss-Wang's operator characterization of realizable volume polynomials. No free parameters or invented entities appear; the variables n, r, c, N are subject to stated inequalities and are not fitted. The axioms listed are the load-bearing external results whose failure would undermine the proof.

assumptions (7)
  • standard math Realizable volume polynomials are Lorentzian; Lorentzianity is equivalent to M-convex support plus one-positive-eigenvalue Hessians (Brändén-Huh).
    Used throughout Section 6; cited to Brändén-Huh Theorems 2.25, 2.30, and 4.6.
  • standard math Cid-Ruiz's theorem: top-degree total-Chern polynomials of globally generated bundles on irreducible projective varieties are denormalized Lorentzian.
    Gives the Lorentzian part of Theorem 3.1(i), with full attribution.
  • standard math Grund-Huh-Michałek-Süss-Wang theorem: realizable covolume polynomials are exactly the differential operators preserving realizable volume polynomials.
    Used to remove the factorial shifts in (3.3), (4.9), (5.20) and to generate the descendants in Section 6.
  • domain assumption Kresch-Tamvakis stable Schubert homomorphism: q_r maps to c_r(E) and Q_ν maps to σ_ν in the Chow ring of LG(N,2N).
    The normalization-sensitive step for the type C identification; argued in Proposition 5.3 using Macdonald's presentation and modified Q-polynomials.
  • standard math Kleiman transversality: general translates of Schubert cycles meet subvarieties properly in effective zero-cycles.
    Justifies nonnegativity of the cycle transform coefficients d^A_ν(X) and d^C_ν(X).
  • standard math An M-convex set contains every lattice point of its base polytope.
    Bridges Lorentzian support to exact permutahedra in Theorems 6.4 and 6.5.
  • standard math Hironaka resolution of singularities in characteristic zero produces a smooth model preserving top intersection numbers.
    Used for the smooth-realization clause in Theorem 3.1.

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Pith. "Pith review of Richardson volume models for skew Schur and skew Schur $P/Q$-functions." pith.science (2026). https://pith.science/paper/SSDHO677

@misc{pith2026260810516,
  author       = {Pith},
  title        = {Pith review of: Richardson volume models for skew Schur and skew Schur $P/Q$-functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SSDHO677}},
  note         = {Machine review of arXiv:2608.10516}
}
abstract

We identify ordinary skew Schur polynomials and skew Schur $P$-functions as top-degree total-Chern intersection polynomials on Richardson varieties in ordinary and Lagrangian Grassmannians. We then obtain that $\mathcal N(s_{\lambda/\mu})$, $\mathcal N(P_{\lambda/\mu})$, $\mathcal N(Q_{\lambda/\mu})$ are realizable volume polynomials. This settles the skew-Schur and Schur-$P$ Lorentzian conjectures of Huh--Matherne--M\'esz\'aros--St.~Dizier and strengthens the latter to arbitrary skew $P/Q$-functions. The construction extends to cycle transforms attached to arbitrary irreducible subvarieties of ordinary and Lagrangian Grassmannians. We obtain reverse Khovanskii--Teissier inequalities for ordinary and shifted tableau multiplicities, Hessian-signature and principal-minor inequalities, root-direction log-concavity, and dominance monotonicity of coefficients toward balanced contents. We also prove ultra-log-concavity of weighted block aggregates and a diagonal covariance bound for factorially tilted content distributions. We determine the exact skew-Schur support permutahedron and its extremal coefficients, recover the known straight Schur-$P/Q$ support polytopes and identify their vertex coefficients in the present framework, and prove log-concavity for cumulative two-row ordinary Littlewood--Richardson coefficients and for weighted cumulative two-row shifted Littlewood--Richardson coefficients.

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