$g$-vectors and $DT$-$F$-polynomials for Grassmannians

Bernhard Keller; Sarjick Bakshi

arxiv: 2410.01037 · v4 · pith:OA4KZC3Vnew · submitted 2024-10-01 · 🧮 math.RT · math.AG

g-vectors and DT-F-polynomials for Grassmannians

Sarjick Bakshi , Bernhard Keller This is my paper

Pith reviewed 2026-05-23 19:54 UTC · model grok-4.3

classification 🧮 math.RT math.AG

keywords Grassmanniancluster algebrag-vectorF-polynomialDonaldson-Thomas transformationFrobenius categoryPlücker coordinateYoung diagram

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The pith

The g-vectors of Plücker coordinates with respect to the triangular initial seed are determined via Frobenius categorification, and the F-polynomials for the Donaldson-Thomas transformation are expressed in terms of 3-dimensional Young 3D

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

The paper reviews Hom-infinite Frobenius categorification of cluster algebras with coefficients and applies it to Jensen-King-Su's categorification of the Grassmannian. This produces explicit g-vectors for the Plücker coordinates relative to the triangular seed. It also yields expressions for the F-polynomials attached to the Donaldson-Thomas transformation that are indexed by three-dimensional Young diagrams. The work supplies a new proof of a theorem of Daping Weng. A reader would care because g-vectors and F-polynomials govern the mutation behavior and positivity properties of the cluster algebra on the Grassmannian.

Core claim

Using the Hom-infinite Frobenius categorification of cluster algebras with coefficients, applied to Jensen-King-Su's Frobenius categorification of the Grassmannian, the g-vectors of the Plücker coordinates are determined with respect to the triangular initial seed, and the F-polynomials associated with the Donaldson-Thomas transformation are expressed in terms of 3-dimensional Young diagrams, providing a new proof for a theorem of Daping Weng.

What carries the argument

Hom-infinite Frobenius categorification of cluster algebras with coefficients, which converts the g-vectors and F-polynomials into data coming from the stable category of the Grassmannian categorification.

Load-bearing premise

Jensen-King-Su's Frobenius categorification of the Grassmannian is valid and the Hom-infinite categorification applies directly to produce the stated g-vectors and F-polynomials.

What would settle it

Direct computation of the g-vector of a specific Plücker coordinate such as p_{1,2,3} in Gr(3,6) from the exchange matrix of the triangular seed, compared against the value predicted by the categorification.

Figures

Figures reproduced from arXiv: 2410.01037 by Bernhard Keller, Sarjick Bakshi.

**Figure 1.** Figure 1: Example of a module LI For I P Ipk, nq we can associate to each YI the Jensen–King–Su module LI as follows. Let Y T I denote the transpose of the Young diagram YI . Rotate it by 3π{4 in the counterclockwise direction. We identify the upper rim of this rotated diagram with the upper rim of the JKS diagram of the module LI associated with I, cf [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗

read the original abstract

We review $\mathrm{Hom}$-infinite Frobenius categorification of cluster algebras with coefficients and use it to give two applications of Jensen--King--Su's Frobenius categorification of the Grassmannian: 1) we determine the $g$-vectors of the Pl\"ucker coordinates with respect to the triangular initial seed and 2) we express the $F$-polynomials associated with the Donaldson--Thomas transformation in terms of $3$-dimensional Young diagrams thus providing a new proof for a theorem of Daping Weng.

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.

Desk Editor's Note private letter to a colleague

The paper computes explicit g-vectors for Plücker coordinates and expresses DT F-polynomials via 3D Young diagrams by applying the Hom-infinite categorification to the Jensen-King-Su Grassmannian category, but the transfer of all hypotheses from the general theorem needs explicit verification.

read the letter

The main takeaway is that Bakshi and Keller use the Jensen-King-Su Frobenius category to produce concrete g-vectors for the Plücker coordinates relative to the triangular seed, and they give a 3D Young diagram formula for the F-polynomials tied to the Donaldson-Thomas transformation. This also yields a new proof of Weng's theorem on those polynomials. The results are presented as direct applications of the reviewed Hom-infinite framework for cluster algebras with coefficients. The combinatorial descriptions are the clear output that readers in this area can use right away. The paper does a service by spelling out how the general method produces these specific objects without extra machinery. The soft spot is the applicability step itself. The central claims rest on the general Hom-infinite theorem holding verbatim once the Jensen-King-Su category is plugged in. The manuscript reviews the framework, yet the description gives no sign of a point-by-point check that the Hom-infinite property, the existence of enough projectives and injectives, and the precise seed compatibility all transfer without adjustment. If any of those conditions requires extra argument, the g-vector and F-polynomial claims lose their justification. No other internal contradictions or fitting issues show up in what is described. This paper is for people already working on categorifications of Grassmannian cluster algebras or on Hom-infinite methods more broadly. A reader who knows the Jensen-King-Su and the general theorem papers will extract the formulas quickly; others will need the background. It deserves a serious referee because the claims are specific, the method is reproducible in principle, and the only real question is whether the hypothesis transfer is handled cleanly. I would send it to review.

Referee Report

1 major / 0 minor

Summary. The manuscript reviews Hom-infinite Frobenius categorification of cluster algebras with coefficients and applies it to Jensen--King--Su's Frobenius categorification of the Grassmannian. It determines the g-vectors of the Plücker coordinates with respect to the triangular initial seed and expresses the F-polynomials associated with the Donaldson--Thomas transformation in terms of 3-dimensional Young diagrams, providing a new proof for a theorem of Daping Weng.

Significance. If the transfer of hypotheses is rigorously justified, the results supply explicit combinatorial formulas for g-vectors and DT F-polynomials on Grassmannians via categorification, together with an alternative proof of Weng's theorem. This strengthens links between Frobenius categories and cluster invariants and may enable further combinatorial applications.

major comments (1)

[Review of general framework and applications] The central claims rest on invoking the general Hom-infinite Frobenius categorification theorem on the Jensen--King--Su category. The manuscript reviews the general framework but does not contain an explicit checklist confirming each hypothesis (Hom-infinite property, existence of enough projectives/injectives, compatibility of the Frobenius structure with the cluster algebra with coefficients, and the precise form of the initial seed) holds verbatim for the Grassmannian case. This verification is load-bearing for the stated g-vectors and the identification of F-polynomials with 3D diagrams.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive report and for identifying a point that will improve the clarity of our application of the general framework. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Review of general framework and applications] The central claims rest on invoking the general Hom-infinite Frobenius categorification theorem on the Jensen--King--Su category. The manuscript reviews the general framework but does not contain an explicit checklist confirming each hypothesis (Hom-infinite property, existence of enough projectives/injectives, compatibility of the Frobenius structure with the cluster algebra with coefficients, and the precise form of the initial seed) holds verbatim for the Grassmannian case. This verification is load-bearing for the stated g-vectors and the identification of F-polynomials with 3D diagrams.

Authors: We agree that an explicit checklist would strengthen the exposition and make the transfer of hypotheses fully transparent. In the revised manuscript we will add a new subsection (placed immediately after the review of the general framework) that systematically verifies each listed hypothesis for the Jensen--King--Su category, citing the relevant statements and proofs from Jensen--King--Su together with the precise form of the triangular initial seed used in our applications. revision: yes

Circularity Check

0 steps flagged

Application of reviewed general categorification theorems to Grassmannian yields independent g-vector and F-polynomial results

full rationale

The paper reviews the Hom-infinite Frobenius categorification framework and invokes Jensen--King--Su's separate categorification of the Grassmannian to derive the g-vectors of Plücker coordinates and the expression of DT F-polynomials via 3D Young diagrams. These steps constitute applications of external theorems rather than self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the claimed results to their inputs by construction. No equations or definitions in the provided abstract or context exhibit the enumerated circularity patterns; the derivation chain remains self-contained against the cited external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger records the explicit dependencies stated there. The central claims rest on the correctness of two prior categorification results.

axioms (2)

domain assumption Jensen--King--Su's Frobenius categorification of the Grassmannian holds and can be used to compute the stated invariants
The two applications are direct consequences of this categorification.
domain assumption The reviewed Hom-infinite Frobenius categorification of cluster algebras with coefficients is valid
The paper reviews and then applies this framework to obtain the results.

pith-pipeline@v0.9.0 · 5615 in / 1445 out tokens · 32191 ms · 2026-05-23T19:54:33.124468+00:00 · methodology

discussion (0)

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ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

we express the F-polynomials associated with the Donaldson–Thomas transformation in terms of 3-dimensional Young diagrams

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Works this paper leans on

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