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arxiv: 2511.02649 · v2 · submitted 2025-11-04 · 🧮 math.CO

A geometric and generating function approach to plethysm

\'Alvaro Guti\'errez , Rosa Orellana , Franco Saliola , Anne Schilling , Mike Zabrocki This is my paper

Pith reviewed 2026-05-18 00:55 UTC · model grok-4.3

classification 🧮 math.CO
keywords plethysm coefficientsSchur functionsgenerating functionsrational functionsEhrhart theoryMacMahon's analysissymmetric functionslinear recursions
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The pith

When the length of the partition λ is bounded, the bivariate generating function for plethysm coefficients is a rational function.

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

The paper shows that a bivariate generating function summing plethysm coefficients a_{μ[ν]}^λ over partitions λ with bounded length is rational. This rationality is established using MacMahon's combinatory analysis as a key step. For the case where the length is at most 2, an explicit geometric algorithm is provided based on q-Ehrhart theory of half-open polytopes. A reader might care because this offers a new way to compute or understand these coefficients that appear in the expansion of plethysm of Schur functions, which is central to symmetric function theory and representation theory of symmetric groups. The work also derives linear recursions for the coefficients in the SL_2 case and a reciprocity theorem.

Core claim

We study a bivariate generating function of plethysm coefficients when λ has bounded length. We show that this generating function is rational. A key step is MacMahon's combinatory analysis. When the bound on the length is 2 we give an explicit geometric algorithm to compute it using q-Ehrhart theory. We give evidence that the generating function is the quantum Ehrhart series of a union of half-open polytopes and show that it satisfies a reciprocity theorem reminiscent of Ehrhart reciprocity. Furthermore, we give a set of linear recursions that completely describe the SL_2-plethysm coefficients.

What carries the argument

The realization of the bivariate generating function as the quantum Ehrhart series of a union of half-open polytopes, whose rationality is deduced from MacMahon's combinatory analysis.

Load-bearing premise

Restricting to partitions λ of bounded length allows the bivariate generating function to be realized as the quantum Ehrhart series of a union of half-open polytopes.

What would settle it

A computation for a small bound on length showing that the generating function is not rational, or that the reciprocity theorem fails for the plethysm coefficients.

Figures

Figures reproduced from arXiv: 2511.02649 by \'Alvaro Guti\'errez, Anne Schilling, Franco Saliola, Mike Zabrocki, Rosa Orellana.

Figure 1
Figure 1. Figure 1: Let w = 3. On the left, the integral points of the cube 9□ divided into its six chambers. On the right, a generic slice by a plane perpendicular to (1, 1, 1), overlaid with the braid hyperplane arrangement. Originally, the RSK algorithm gives a bijection between words of length w in the alphabet {0, 1, . . . , h} and pairs of tableaux in S λ⊢w SSYTh+1(λ)×SYT(λ), see for example [Sag01, Sta99]. Now consider… view at source ↗
Figure 2
Figure 2. Figure 2: On the left, the Chfine(−) faces of □ for w = 3; in the middle, the Ch(−) chambers; on the right, the Chcoar(−) subdivision. Proof. The chamber Ch(π) is the union of the faces Chfine(γ) for all γ such that π γ = π. In particular, note that these set partitions are exactly those such that α γ refines Des(π). Hence ipeCh(π) (h; x0, . . . , xh) = X γ : πγ=π ipeChfine(γ) (h; x0, . . . , xh) = X γ : πγ=π Mαγ (x… view at source ↗
Figure 3
Figure 3. Figure 3: A generic slice of [0, 1]4 projected to a 3-dimensional sphere. Illustrated is the front part of the sphere, containing 12 chambers, indexed by permutations of S4. Each color represents a different coarse chamber Chcoar(−). Note also that since mapping π to π −1 interchanges the insertion and recording tableaux under RSK, we can write Chcoar(Q) = [ π∈Sw RSK(π)1=Q Ch(π). (2.2) Proposition 2.12. Let Q ∈ SYT(… view at source ↗
Figure 4
Figure 4. Figure 4: Let P˜ = [−1/2, 1/2]3 . The PTq operator applied to qEhru P˜(z, q) with u = (1, . . . , 1) is the q-Ehrhart series of the displayed sub-polytope of P˜, with respect to the grading u. The second conceptual way of understanding the PTα operator is through the lens of quantum calculus. The relationship between QEhrµ(z, q) and Aµ(z, q) is evident from the following formulas: QEhrµ(z, q) = X k⩾1,h⩾0 a [k] µ[h] … view at source ↗
read the original abstract

Plethysm coefficients $\mathsf{a}_{\mu[\nu]}^\lambda$ are the structure coefficients of the plethysm of Schur functions $s_\mu[s_\nu] = \sum_{\lambda} \mathsf{a}_{\mu[\nu]}^\lambda s_\lambda$. We study a bivariate generating function of plethysm coefficients when $\lambda$ has bounded length. We show that this generating function is rational. A key step is MacMahon's combinatory analysis. When the bound on the length is $2$ we give an explicit geometric algorithm to compute it using $q$-Ehrhart theory. We give evidence that the generating function is the quantum Ehrhart series of a union of half-open polytopes and show that it satisfies a reciprocity theorem reminiscent of Ehrhart reciprocity. Furthermore, we give a set of linear recursions that completely describe the $\mathrm{SL}_2$-plethysm coefficients.

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

3 major / 2 minor

Summary. The paper studies the bivariate generating function G(x,y) for plethysm coefficients a_{μ[ν]}^λ of Schur functions when the length of λ is bounded by a fixed k. It claims to prove that G is rational, with MacMahon's combinatory analysis as the key step. For the special case k=2, it supplies an explicit geometric algorithm via q-Ehrhart theory, presents evidence that G coincides with the quantum Ehrhart series of a union of half-open polytopes, proves a reciprocity theorem analogous to Ehrhart reciprocity, and derives a complete set of linear recursions for the SL_2-plethysm coefficients.

Significance. If the rationality proof and the geometric constructions hold, the work would furnish a new geometric and recursive framework for computing and understanding plethysm coefficients, linking them to Ehrhart theory and MacMahon's classical results. The explicit algorithm for length 2 and the reciprocity theorem are potentially useful for explicit calculations and structural insights in algebraic combinatorics.

major comments (3)
  1. [generating function and rationality section] The rationality claim for the bivariate generating function rests on reinterpreting the bounded-length plethysm coefficients via MacMahon's combinatory analysis, yet the manuscript does not exhibit the explicit change of variables, bijection, or combinatorial reinterpretation that converts the sum over a_{μ[ν]}^λ into a MacMahon-type series whose rationality is already known. This mapping is load-bearing for the central claim (see the paragraph introducing G(x,y) and the subsequent application of MacMahon's analysis).
  2. [length-2 geometric algorithm] In the length-2 case, the geometric algorithm is said to realize the generating function as the quantum Ehrhart series of a union of half-open polytopes, but the manuscript does not provide the explicit construction of these polytopes from the plethysm data or verify that the q-Ehrhart series is indeed rational by direct appeal to known results without additional assumptions. This construction is central to the explicit algorithm (see the section on the bound on the length equal to 2).
  3. [recursions for SL_2-plethysm coefficients] The linear recursions are asserted to completely describe the SL_2-plethysm coefficients, but the manuscript supplies no verification that the recursion set is minimal or that it generates all coefficients without external input; small explicit examples or a comparison with known tables would be needed to confirm completeness (see the section deriving the recursions).
minor comments (2)
  1. [introduction and definitions] The notation for the bivariate generating function G(x,y) should include an explicit summation formula with the precise exponent of y to avoid ambiguity.
  2. [length-2 geometric algorithm] Figure captions for any polytopal diagrams in the length-2 section could be expanded to indicate the precise half-open regions and the q-weighting.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below, clarifying the existing arguments where possible and indicating revisions that will strengthen the exposition without altering the core claims.

read point-by-point responses

  1. Referee: [generating function and rationality section] The rationality claim for the bivariate generating function rests on reinterpreting the bounded-length plethysm coefficients via MacMahon's combinatory analysis, yet the manuscript does not exhibit the explicit change of variables, bijection, or combinatorial reinterpretation that converts the sum over a_{μ[ν]}^λ into a MacMahon-type series whose rationality is already known. This mapping is load-bearing for the central claim (see the paragraph introducing G(x,y) and the subsequent application of MacMahon's analysis).

    Authors: The manuscript introduces G(x,y) and applies MacMahon's analysis to establish rationality, but we agree that the explicit change of variables and bijection deserve a more detailed presentation to make the argument fully self-contained. In the revised version we will insert a dedicated paragraph (or short subsection) immediately after the definition of G(x,y) that spells out the precise change of variables, the combinatorial reinterpretation of the sum, and the direct appeal to the known rationality of the resulting MacMahon-type series. revision: yes

  2. Referee: [length-2 geometric algorithm] In the length-2 case, the geometric algorithm is said to realize the generating function as the quantum Ehrhart series of a union of half-open polytopes, but the manuscript does not provide the explicit construction of these polytopes from the plethysm data or verify that the q-Ehrhart series is indeed rational by direct appeal to known results without additional assumptions. This construction is central to the explicit algorithm (see the section on the bound on the length equal to 2).

    Authors: The manuscript states that the generating function coincides with the quantum Ehrhart series of a union of half-open polytopes and invokes known rationality results for such series. We acknowledge that an explicit, step-by-step construction of the polytopes directly from the length-2 plethysm data is not written out in full detail. In the revision we will add this construction (including the precise half-open polytopes and the verification that their q-Ehrhart series is rational by direct appeal to existing theorems on q-Ehrhart theory) in the section treating the length-2 case. revision: yes

  3. Referee: [recursions for SL_2-plethysm coefficients] The linear recursions are asserted to completely describe the SL_2-plethysm coefficients, but the manuscript supplies no verification that the recursion set is minimal or that it generates all coefficients without external input; small explicit examples or a comparison with known tables would be needed to confirm completeness (see the section deriving the recursions).

    Authors: The manuscript derives a set of linear recursions from the geometric and generating-function framework and asserts that they completely describe the SL_2-plethysm coefficients. To address the request for verification, the revised version will include a short subsection with small explicit examples (low-weight cases) together with direct comparisons against known tables of plethysm coefficients, thereby confirming that the recursions generate all coefficients and illustrating their completeness. revision: yes

Circularity Check

0 steps flagged

No circularity: rationality follows from external MacMahon analysis and q-Ehrhart theory applied to bounded-length plethysm

full rationale

The paper's central claim—that the bivariate generating function for plethysm coefficients with bounded length(λ) is rational—rests on reinterpreting the sum via MacMahon's combinatory analysis (an external, classical result on generating functions for plane partitions and related objects) and, for the length-2 case, on an explicit geometric algorithm using q-Ehrhart theory of half-open polytopes. These are independent external benchmarks whose rationality proofs predate the paper and do not depend on its plethysm coefficients. The abstract and described results further supply linear recursions and a reciprocity theorem derived from the geometric model, without any self-definitional loop, fitted-input renaming, or load-bearing self-citation. The derivation chain therefore remains self-contained against external combinatorial tools.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard combinatorial background rather than new free parameters or invented entities; MacMahon's analysis and q-Ehrhart theory are treated as established domain tools.

axioms (2)
  • domain assumption MacMahon's combinatory analysis applies directly to the bivariate generating function of bounded-length plethysm coefficients
    Cited as key step in the rationality proof.
  • domain assumption The length-2 case admits a description as quantum Ehrhart series of a union of half-open polytopes
    Required for the explicit geometric algorithm and reciprocity theorem.

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Reference graph

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