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arxiv: 2606.08570 · v1 · pith:YHEMVT75new · submitted 2026-06-07 · 🧮 math.AG · math.RT· math.SG

Newton--Okounkov bodies of partial flag varieties via cluster algebras

Yunhyung Cho , Myungho Kim , Yoosik Kim , Euiyong Park This is my paper

Pith reviewed 2026-06-27 18:10 UTC · model grok-4.3

classification 🧮 math.AG math.RTmath.SG
keywords Newton-Okounkov bodiescluster algebraspartial flag varietiesSchubert varietiesLagrangian toriunipotent cellssymplectic geometry
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The pith

Cluster algebras on unipotent cells construct Newton-Okounkov polytopes for Schubert varieties in partial flag varieties, yielding infinitely many nonequivalent ones when the algebra is infinite type.

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

The paper shows how to build Newton-Okounkov polytopes for Schubert varieties inside partial flag varieties by leveraging the cluster algebra on a unipotent cell. This method works for arbitrary type. In cases where the cluster algebra is infinite type and the flag variety is simply laced, any very ample line bundle gives rise to a family of polytopes that includes infinitely many distinct ones up to integral affine transformations. As a result, the construction produces infinitely many distinct monotone Lagrangian tori in these varieties.

Core claim

The authors establish that the cluster algebra structure on the unipotent cell defines a valuation on the coordinate ring whose Newton-Okounkov body matches the one associated to the Schubert variety and the homogeneous line bundle. When the cluster algebra is of infinite type, this produces infinitely many pairwise nonequivalent polytopes up to integral affine transformation for any very ample bundle on simply laced partial flag varieties.

What carries the argument

The cluster algebra structure on the unipotent cell, used to define a valuation whose Newton-Okounkov body coincides with the one from the Schubert variety and given line bundle.

If this is right

  • The family of Newton-Okounkov polytopes contains infinitely many pairwise nonequivalent ones up to integral affine transformation.
  • Infinitely many distinct monotone Lagrangian tori arise in simply laced partial flag varieties.
  • The construction applies uniformly to Schubert varieties in partial flag varieties of arbitrary type.
  • Each very ample homogeneous line bundle produces its own infinite family of distinct polytopes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may extend to other varieties equipped with cluster structures, generating similar infinite families of polytopes.
  • Different polytopes in the family could encode distinct symplectic capacities or other geometric invariants.
  • One could check whether the resulting tori are Hamiltonian isotopic by comparing their Newton-Okounkov bodies directly.

Load-bearing premise

The cluster algebra structure on the unipotent cell can be used to define a valuation whose associated Newton-Okounkov body coincides with the one coming from the Schubert variety and the given line bundle.

What would settle it

A computation for a specific infinite-type cluster algebra and simply laced partial flag variety showing only finitely many distinct polytopes up to integral affine transformation would disprove the claim of infinitude.

Figures

Figures reproduced from arXiv: 2606.08570 by Euiyong Park, MyungHo Kim, Yoosik Kim, Yunhyung Cho.

Figure 1
Figure 1. Figure 1: The marked quivers (Q, L) and (Q′ , L ′ ). We now explain how to associate a marked extended exchange matrix to a parabolic subgroup of a semisimple algebraic group over C. Let I denote the index set for the simple roots of g and let w0 be the longest element of W. For a subset K ⊆ I, consider the corresponding parabolic subgroup P K and the quotient WK ≃ W/WK where WK is the parabolic Weyl group. Let wK b… view at source ↗
Figure 2
Figure 2. Figure 2: Diamond shape subquiver Using Lemma 7.6, we first deal with the affine types. Lemma 7.7 (see Lemma 6.5 in [CKKP25]). For p, q > 0, let Q′ be the quiver of type Aep,q with all vertices unfrozen. Let Q be the quiver obtained from Q′ by adding a single frozen vertex f and one arrow from f to w0 as in [PITH_FULL_IMAGE:figures/full_fig_p036_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Quivers Q and Q♢ in Lemma 7.7 Lemma 7.8. Let Q′ be the quiver of type Ee6 with all vertices unfrozen. Let Q be the quiver obtained from Q′ by adding one frozen vertex f and one single arrow from 6 to f as in [PITH_FULL_IMAGE:figures/full_fig_p037_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Then the following hold: (1) the quiver Q′ is of finite mutation type and (2) the quiver Q is of infinite mutation type. Proof. Apply the following sequence µ of mutations to Q (6, 1, 8, 4, 3, 5, 7, 6, 7, 3, 2, 4, 1, 5, 8, 3, 7, 5, 2). Let Q♢ := µ(Q). By Lemma 7.6 (b2 > 0), Q is of infinite mutation type. □ [PITH_FULL_IMAGE:figures/full_fig_p037_5.png] view at source ↗
Figure 5
Figure 5. Figure 5: Quivers Q and Q♢ in Lemma 7.9 Lemma 7.10. Let Q′ be the quiver of type Ee7 with all vertices unfrozen. Let Q be the quiver obtained from Q′ by adding one frozen vertex f and one single arrow from f to 7 as in [PITH_FULL_IMAGE:figures/full_fig_p038_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Then the following hold: (1) the quiver Q′ is of finite mutation type and (2) the quiver Q is of infinite mutation type. Proof. Apply the following sequence µ of mutations to Q (8, 9, 5, 1, 3, 4, 5, 2, 6, 5, 1, 3, 2, 3, 6, 7, 4, 8, 5, 9, 6, 3, 5) Let Q♢ := µ(Q). By Lemma 7.6, Q is of infinite mutation type. □ Next, we deal with the extended affine types. Lemma 7.12 (Section 5 in [FT24]). Suppose that Q′ be… view at source ↗
Figure 7
Figure 7. Figure 7: Quivers Q and Q♢ in Lemma 7.11 Lemma 7.13. Let Q′ be the quiver of type E (1,1) 7 with all vertices unfrozen. Let Q be the quiver obtained from Q′ by adding one frozen vertex f and one single arrow from 8 to f as in [PITH_FULL_IMAGE:figures/full_fig_p039_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Quivers Q and Q♢ in Lemma 7.13 Lemma 7.14. Let Q′ be the quiver of type E (1,1) 8 with all vertices unfrozen. Let Q be the quiver obtained from Q′ by adding one frozen vertex f and one single arrow from 8 to f as in [PITH_FULL_IMAGE:figures/full_fig_p039_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Quivers Q and Q♢ in Lemma 7.14 Example 7.15. Consider a classical group G of type A5 and let K = {1, 3, 5}. Let P = P K be the corresponding parabolic subgroup. Then G/P ≃ Fℓ(2, 4; 6), which has dimension m = 12. The Weyl group W of G is isomorphic to the symmetric group S6. The minimal length representative of the coset w0WK is given by wK =  1 2 3 4 5 6 5 6 3 4 1 2  . A reduced expression of wK is s4s3… view at source ↗
Figure 10
Figure 10. Figure 10: The marked quiver for A5 type with K = {1, 3, 5} Let V ′ = {1, 2, 3, 7, 9} be a subset of the vertex set V of QP and consider the induced subquiver Q′ = QP |V′. By applying the mutation sequence (2, 3, 1), we see that Q′ is of finite mutation type Ae3,2. Now take V = V ′ ∪ {12} and consider the induced subquiver Q = QP |V. By Lemma 7.7, the quiver Q is of infinite mutation type. Remark 7.16. In Example 7.… view at source ↗
Figure 11
Figure 11. Figure 11: The marked quiver for D4 type with K = {1, 3, 4} Let V ′ = Vuf. Applying the mutation at the nodes (4, 1, 6, 2, 7, 5), the quiver QP contains a subquiver Q′ = QP |V′ of finite mutation type Ee6. Now set V = V ′ ∪ 10 and consider the induced subquiver Q = QP |V. By Lemma 7.8, the quiver Q is of infinite mutation type [PITH_FULL_IMAGE:figures/full_fig_p041_11.png] view at source ↗
read the original abstract

We construct Newton--Okounkov polytopes of Schubert varieties in partial flag varieties of arbitrary type using the cluster structure on a unipotent cell. When the governing cluster algebra is of infinite type, we prove that for any very ample homogeneous line bundle over a simply laced partial flag variety, the resulting family of Newton--Okounkov polytopes contains infinitely many pairwise nonequivalent polytopes up to integral affine transformation. As an application to symplectic geometry, we construct infinitely many distinct monotone Lagrangian tori in a broad class of simply laced partial flag varieties.

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

2 major / 2 minor

Summary. The manuscript constructs Newton--Okounkov polytopes of Schubert varieties in partial flag varieties of arbitrary type by using the cluster algebra structure on a unipotent cell. When the governing cluster algebra is of infinite type, it proves that for any very ample homogeneous line bundle over a simply laced partial flag variety the resulting family contains infinitely many pairwise nonequivalent polytopes up to integral affine transformation. As an application it constructs infinitely many distinct monotone Lagrangian tori in a broad class of such varieties.

Significance. If the central identification between the cluster-algebra valuation and the geometric Newton--Okounkov body holds, the work supplies a new combinatorial route to these polytopes and establishes their infinitude in infinite-type cases, with direct consequences for the existence of many monotone Lagrangian tori. The cluster-algebra approach is a genuine addition to the existing literature on Newton--Okounkov bodies.

major comments (2)
  1. [Abstract; construction section (likely §3)] The abstract and the construction section present the cluster-algebra valuation on the unipotent cell as producing the Newton--Okounkov bodies of the Schubert varieties, yet no explicit argument is given that this valuation coincides with (or yields an integral-affine equivalent to) the standard valuation coming from the very ample line bundle on the homogeneous coordinate ring of the Schubert variety. This equality is load-bearing for both the title and the infinitude claim.
  2. [Infinitude theorem (likely §5)] The infinitude theorem (likely §5) is stated for the family of Newton--Okounkov polytopes obtained from the cluster construction; without the identification in the previous point, the result applies only to an auxiliary family rather than to the geometric bodies asserted in the main theorem and the symplectic application.
minor comments (2)
  1. [Introduction] The introduction would benefit from a short paragraph recalling the definition of a Newton--Okounkov body and the precise sense in which two polytopes are considered equivalent up to integral affine transformation.
  2. [Introduction] A few sentences comparing the new construction with earlier combinatorial descriptions of Newton--Okounkov bodies for flag varieties (e.g., via Gelfand--Tsetlin or other polytopes) would help situate the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for an explicit identification between the cluster-algebra valuation and the standard Newton-Okounkov valuation. We agree that this step is essential for the claims in the title, main theorems, and symplectic application, and we will strengthen the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract; construction section (likely §3)] The abstract and the construction section present the cluster-algebra valuation on the unipotent cell as producing the Newton--Okounkov bodies of the Schubert varieties, yet no explicit argument is given that this valuation coincides with (or yields an integral-affine equivalent to) the standard valuation coming from the very ample line bundle on the homogeneous coordinate ring of the Schubert variety. This equality is load-bearing for both the title and the infinitude claim.

    Authors: We acknowledge that the original manuscript does not contain a direct, self-contained argument establishing that the valuation defined via the cluster algebra on the unipotent cell coincides with (or is integral-affine equivalent to) the standard valuation induced by a very ample homogeneous line bundle on the homogeneous coordinate ring of the Schubert variety. In the revision we will add a new subsection in the construction section (approximately §3) containing a lemma that compares the two valuations on a set of generators of the ring. The argument will use the fact that the cluster variables generate the coordinate ring as an algebra and that the cluster valuation is compatible with the grading and filtration coming from the line bundle; this will show the resulting polytopes are the geometric Newton-Okounkov bodies. revision: yes

  2. Referee: [Infinitude theorem (likely §5)] The infinitude theorem (likely §5) is stated for the family of Newton--Okounkov polytopes obtained from the cluster construction; without the identification in the previous point, the result applies only to an auxiliary family rather than to the geometric bodies asserted in the main theorem and the symplectic application.

    Authors: We agree that, without the identification, the infinitude statement applies only to the auxiliary cluster family. Once the explicit identification is added as described above, the infinitude theorem will apply directly to the geometric Newton-Okounkov bodies. In the revision we will restate the main theorem and the symplectic application to make this dependence explicit and to confirm that the family in question consists of the geometric bodies. revision: yes

Circularity Check

0 steps flagged

No circularity: construction uses external cluster data without self-referential reduction

full rationale

The abstract describes a construction of Newton-Okounkov polytopes via the cluster structure on a unipotent cell, followed by a proof of infinitude of nonequivalent polytopes for infinite-type cluster algebras on simply-laced partial flag varieties. No equations, fitted parameters, or self-citations are visible that would reduce the claimed polytopes or the infinitude statement to a definition or input by construction. The derivation chain relies on external cluster-algebra structures rather than redefining the target bodies in terms of themselves. This is the most common honest non-finding for papers whose central objects are defined from independent algebraic data.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the construction is described at the level of existence via cluster algebras.

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