Quivers with potentials for Grassmannian cluster algebras

Jie Zhang; Wen Chang

REVIEW 5 major objections 6 minor 30 references

Quivers with potentials for Grassmannian cluster algebras

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Pith's one-line read For a Grassmannian cluster algebra, the quiver alone determines the potential, and this uniqueness identifies the symmetry groups of its category and cluster algebra.

desk verdict Useful and partly new uniqueness result for Grassmannian cluster algebra QPs, but the proof of the key theorem has a strict-vs-equality inequality gap that must be fixed. read the letter →

arxiv 1908.10103 v2 pith:24A2SPAW submitted 2019-08-27 math.RT

Wen Chang , Jie Zhang This is my paper · ORCID

classification math.RT MSC 13F6016G2014M15

keywords quiverswithpotentialsGrassmannianclusteralgebrasPostnikovdiagramsrigidityJacobiangeneralizedcategoriesautomorphismgroupsmutation

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

Associated to any Grassmannian $\mathbb{C}[\mathrm{Gr}(k,n)]$ there is a natural quiver with potential, built from a Postnikov diagram $D$: the quiver is dual to the diagram, and the potential is the alternating sum of the fundamental cycles corresponding to its oriented regions. The paper proves that mutating this quiver with potential matches, up to right-equivalence, the geometric exchange operation on $D$, so the quiver with potential is independent of the diagram chosen and deserves to be called the quiver with potential of the Grassmannian cluster algebra. It is shown to be rigid (no mutation ever creates a 2-cycle) and Jacobian-finite, and to be the unique non-degenerate, hence unique rigid, quiver with potential on its underlying quiver. From this uniqueness, any quiver in the mutation class determines its potential up to right-equivalence, which in turn implies that the auto-equivalence group of the generalized cluster category $\mathcal{C}_{(Q,W)}$ is isomorphic to the cluster automorphism group of the associated Grassmannian cluster algebra with trivial coefficients.

What carries the argument

The load-bearing object is the Postnikov diagram $D$ together with its dual quiver $Q(D)$: vertices are the alternating oriented regions into which the strands cut the disk, and arrows record crossings of strands. The potential $W(D)$ is the signed sum of the fundamental cycles of $Q(D)$, one for each oriented region, taken clockwise with positive sign and anticlockwise with negative sign; the iced version adds frozen boundary arrows. The decisive mechanism is the compatibility theorem: applying the quiver-with-potential mutation at the vertex corresponding to an alternating quadrilateral cell $R$ gives, after reduction and up to right-equivalence, exactly the quiver with potential of the geometrically exchanged diagram $\mu_R(D)$. Uniqueness is then produced by the general uniqueness criterion from [17], which says a QP satisfying a finiteness condition, a Jacobian-ideal condition on long cycles, and a perturbation condition is the unique non-degenerate QP on its quiver; this criterion is checked directly on the initial diagram $D_{\mathrm{ini}}$.

What would settle it

Exhibit two reduced $(k,n)$-Postnikov diagrams for the same pair $(k,n)$ that are not connected by geometric exchanges, which would make the QP in Definition 2.5 ill-defined. Alternatively, find a quiver $Q$ in the Grassmannian mutation class and a non-degenerate potential $W'$ on $Q$ that is not right-equivalent to $W$ or $W^{\mathrm{op}}$; even a single 2-cycle appearing after an iterated mutation of $(Q,W)$ would disprove rigidity.

Watch

Extended reading notes

Core claim

The central claim is that the Grassmannian cluster algebra $\mathbb{C}[\mathrm{Gr}(k,n)]$ has a canonical quiver with potential $(Q,W)$, well-defined up to mutation and right-equivalence, and that $(Q,W)$ is rigid, Jacobi-finite, and unique: every non-degenerate potential on the same quiver $Q$ is right-equivalent to $W$, and every non-degenerate potential on the opposite quiver $Q^{\mathrm{op}}$ is right-equivalent to $W^{\mathrm{op}}$. The proof constructs $W(D)$ from a Postnikov diagram as the signed sum of fundamental cycles, proves mutation compatibility with geometric exchange, and then applies a uniqueness criterion provided in [17] to the initial diagram. The consequence the authors draw is that, within the mutation class of a Grassmannian cluster algebra, the quiver determines the potential, so the cluster automorphism group of the algebra coincides with the auto-equivalence group of its generalized cluster category.

Load-bearing premise

Everything rests on the assumption that any two reduced $(k,n)$-Postnikov diagrams are connected by geometric exchanges, so the mutation class of the quiver with potential does not depend on which diagram one starts from; the proof of the local compatibility theorem also presents one configuration in detail and asserts the remaining configurations are similar.

Editorial extensions

If this is right

For any two reduced Postnikov diagrams representing the same $(k,n)$, the associated quivers with potentials are mutation-equivalent, so the QP of a Grassmannian cluster algebra is a well-defined invariant rather than an artifact of a diagram choice.
The QP $(Q,W)$ is rigid and non-degenerate: no iterated mutation produces a 2-cycle, and the Jacobian algebra $P(Q,W)$ is finite-dimensional.
If $Q'$ is a quiver in the mutation class and $Q'\cong Q$, then any QP $(Q',W')$ over $Q'$ is right-equivalent to $(Q,W)$; if $Q'\cong Q^{\mathrm{op}}$, it is right-equivalent to $(Q^{\mathrm{op}},W^{\mathrm{op}})$.
The generalized cluster category $\mathcal{C}_{(Q,W)}$ has the same symmetry group as the cluster algebra: $\mathrm{Aut}_T(\mathcal{C}_{(Q,W)}) \cong \mathrm{Aut}(\mathcal{A}_{(Q,W)})$ for trivial coefficients.
The authors conjecture the same quiver-determines-potential principle for all mutation classes of non-degenerate QPs, which would give the category-algebra isomorphism for every generalized cluster category; this paper establishes that case for Grassmannians.

Reading between the lines

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

The same uniqueness route should work for any cluster algebra whose quivers arise from dimer models with a signed fundamental-cycle potential, including the dimer-model, unipotent-group, and double-Bruhat-cell settings the authors list, so quiver-determines-potential may hold for a broad class of Jacobian-finite QPs beyond Grassmannians.
Because the iced QP is neither rigid nor Jacobi-finite, the frozen-vertex version cannot carry the same uniqueness theorem; a separate theory would be needed to decide whether iced quivers also determine their potentials.
A practical test of the uniqueness theorem: take the initial quiver $Q_{\mathrm{ini}}$ for a small pair like $(k,n)=(3,7)$, enumerate non-degenerate potentials on $Q_{\mathrm{ini}}$ by computer, and check right-equivalence to $W_{\mathrm{ini}}$; the criterion predicts that every sufficiently long-cycle perturbation is absorbed by a right-equivalence.
The category-algebra isomorphism suggests that cluster automorphism groups of Grassmannian cluster algebras can be computed by classifying auto-equivalences of the generalized cluster category, a problem where tilting theory supplies tools that are not available on the algebra side.

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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—

The paper does two things. First, it gives an explicit QP associated to a Postnikov diagram and proves mutation of the QP matches geometric exchange. That part is mostly a detailed write-up of a known picture, with the initial diagram spelled out. Second, it proves uniqueness of the non-degenerate QP on the quiver of a Grassmannian cluster algebra up to right-equivalence, and uses that to get AutT(C) ≅ Aut(Aφ(T)). The uniqueness result is genuinely new, as the authors acknowledge rigidity was already known from Buan–Iyama–Reiten–Smith and Kulkarni. The explicit initial QP description is useful and is what carries the new argument.

The main soft spot is real. In the proof of Theorem 2.14, Step I, the authors claim short(ηα) + short(∂αW_ini) > 4m, but their own calculation gives 4m−3 + 3 = 4m, equality. They are quoting Proposition 2.13 with a strict inequality. So condition (2) is not verified as written. This is a load-bearing gap: Theorem 2.14 is the foundation for Corollary 0.3 and Theorem 3.5. It may be fixable—if the original Geiß–Labardini–Schröer theorem actually uses ≥, then it is a misquotation that can be corrected; if the strict inequality is genuinely required, the argument needs a different idea—but as written it does not go through.

There are smaller issues. Definition 2.5 silently assumes the geometric exchange graph of reduced Postnikov diagrams is connected, so the QP of C[Gr(k,n)] is well-defined; that fact is standard but should be stated or cited. Theorem 2.4 proves one local configuration and says the rest "can be proved similarly"; for a claim that is this central, that is not shameful, but it is an omission a referee will want filled. The paper also has unresolved cross-references and a typo or two, indicating it was not fully polished for submission.

The citation pattern is fine: prior work gets credit, the new claims are flagged as new. The proof is a derivation from definitions, no fitted parameters.

Who is this for? People working on cluster categories, Jacobian algebras, and Grassmannian cluster algebras. The application to automorphism groups is a nice bonus. The paper deserves a serious referee—the uniqueness result is important if the gap is repairable—but the referee should focus on the strict inequality issue first.

I would not cite the central theorem until the gap is addressed, but I would send it out.

Referee Report

5 major / 6 minor

Summary. The paper associates to each Postnikov diagram D a quiver with potential (Q(D),W(D)) and an iced version (Q(D),F(D),W(D)), and claims that mutations of these QPs are compatible with geometric exchanges of Postnikov diagrams (Theorem 0.1/2.4). This compatibility is used to define 'a QP of C[Gr(k,n)]' up to mutation-equivalence (Definition 2.5). The authors then prove rigidity and Jacobi-finiteness of these QPs (Theorems 2.7, 2.10) and, via a criterion of Geiß-Labardini-Schröer, uniqueness of the non-degenerate (hence rigid) QP on a Grassmannian cluster algebra quiver up to right-equivalence (Theorem 2.14). From uniqueness they derive Corollary 0.3 and, in Section 3, an isomorphism between the auto-equivalence group of the generalized cluster category and the cluster automorphism group of the associated cluster algebra (Theorem 3.5).

Significance. If the main results hold, this is a substantial contribution: it gives an explicit quiver-with-potential model for Grassmannian cluster algebras, proves rigidity and uniqueness of the potential, and extends the known isomorphism between cluster automorphisms and categorical auto-equivalences to the Grassmannian case. The paper also demonstrates a useful method: reducing a two-dimensional combinatorial problem to checking a list of local configurations and then invoking a general criterion. The authors are careful to distinguish the new results from existing rigidity proofs by Buan-Iyama-Reiten-Smith and Kulkarni. However, the central uniqueness theorem rests on a verification of a stated inequality that is not actually established, and other load-bearing steps are either only sketched or depend on unstated assumptions. The manuscript is therefore not yet ready for publication in its current form.

major comments (5)

[§2.3, proof of Theorem 2.14, Step I] The proof of condition (2) of Proposition 2.13 is not correct as written. The paper states that paths in η_α have length 4m−3 and paths in ∂_α W_ini have length 3, so short(η_α)+short(∂_α W_ini)=4m. But condition (2) of Proposition 2.13 (quoted from [17, Theorem 8.20]) requires this sum to be strictly greater than length(l)=length(ω^m)=4m. The displayed computation gives equality, not strict inequality. The same issue occurs in Step II, where the additional term p q ∂_α W_ini has degree exactly length(l). Thus the verification does not satisfy the stated hypothesis of Proposition 2.13. This is load-bearing because Theorem 2.14, and consequently Corollary 0.3 and Theorem 3.5, depend on it. The authors must either prove the strict inequality, replace the quoted criterion by the correct version from [17] (if the correct version uses a non-strict inequality), or supply an alternative argument.
[§2.3, proof of Theorem 2.14, condition (3)] The verification of condition (3) of Proposition 2.13 is too compressed to be valid as written. The proof says condition (3) 'follows immediately' from Proposition 2.12 and from the fact that every non-fundamental cycle has length greater than 4. These observations alone do not show that an arbitrary non-degenerate potential on Q_ini is right-equivalent to W_ini plus terms of degree greater than long(W_ini). One still needs to show that the degree-4 part of any non-degenerate potential is cyclically equivalent to the signed sum of fundamental cycles that defines W_ini, including control of coefficients and signs. This is a nontrivial step and must be spelled out.
[Definition 2.5] The definition of a QP of C[Gr(k,n)] as a QP mutation-equivalent to (Q(D),W(D)) depends on the class being independent of the choice of Postnikov diagram D. The paper does not state or prove that the graph of reduced Postnikov diagrams connected by geometric exchanges is connected for each (k,n). Since the compatibility result Theorem 2.4 is local, it cannot by itself establish well-definedness without this global connectivity assertion. The authors should either prove the connectivity or give a precise reference and explain how it follows.
[§2.1, proof of Theorem 2.4] The proof of Theorem 2.4 explicitly treats only one of the three local configurations shown in Figure 8 and says the others 'can be proved similarly.' Because Theorem 2.4 is the basis for the definition of the QP and IQP for Grassmannian cluster algebras, the omitted cases are load-bearing. The authors should at least state the structure of the remaining computations or give a clear symmetry/rotation argument that reduces all cases to the displayed one.
[§2.2, proof of Theorem 2.7, Step 2] Step 2 of the proof of Theorem 2.7 asserts that any fundamental cycle ω_2 can be connected to the bottom-left fundamental cycle ω_1 by repeatedly applying Lemma 2.8, but no proof is given that the dual graph of fundamental cycles sharing an arrow is connected. Since Lemma 2.8 only propagates rigidity from one fundamental cycle to another sharing an arrow, the argument requires this connectivity as an explicit premise. The authors should either prove the connectivity or cite a result that implies it.

minor comments (6)

[Abstract] There is a typo in the abstract: 'Postnilov Diagram' should be 'Postnikov Diagram'.
[Remark 1.1 and text after Definition 2.1] Several cross-references appear as 'Remarks ??, 2.9 and 2.11' or 'Remarks ??' with unresolved placeholders; these should be fixed.
[Introduction, paragraph after Theorem 0.1] The name 'Buar-King-Marsh' is a typo for 'Baur-King-Marsh'.
[§2.1, proof of Theorem 2.4] In the displayed potential after the pre-mutation, the term '−ξt' is used; it would help to define the path t explicitly at that point, since t appears both as a path and as a target map in the quiver conventions.
[§2.2, Lemma 2.6] The construction in Step 1 says 'we repeat above construction until a is never a vertex on a cycle l′', but the termination of this process is asserted rather than proved. A short termination argument using the finiteness of the quiver and monotonicity of the relevant coordinate would improve clarity.
[§2.3, Theorem 2.14] The phrase 'unique rigid QP' in Theorem 2.14 is stronger than what is proved unless rigidity coincides with non-degeneracy for the QPs in question; the paper notes rigid implies non-degenerate, but the converse is not established. The statement should be phrased carefully as 'unique rigid among QPs on Q up to right-equivalence' only if the proof covers it, or a reference should be given.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main claims are derived from explicit combinatorial data and an external uniqueness criterion, not from self-referential definitions or fitted parameters.

full rationale

The paper's central derivation is self-contained in the relevant sense. The quiver with potential (Q(D), W(D)) is defined explicitly as a signed sum of fundamental cycles associated to a Postnikov diagram, and the compatibility of mutations with geometric exchanges is verified by direct local computations with reductions and right-equivalences. Rigidity is proved by explicit manipulation of cycles in the initial quiver, using Lemmas 2.6 and 2.8 and the mutation-invariance of rigidity. The uniqueness claim rests on Proposition 2.13, quoted from Geiß-Labardini-Schröer [17], which is an external general criterion; the paper then attempts to verify its hypotheses for the initial QP. No parameter is fitted to a subset of data and then renamed as a prediction, and no equation is shown to be equivalent to its own input by construction. The authors' own earlier works are cited only in background or application contexts and are not load-bearing for the main derivation. Two possible concerns are noted but are not circularity: Definition 2.5 depends on the unstated connectedness of the geometric exchange graph of reduced Postnikov diagrams, and the proof of Theorem 2.14 appears to verify a non-strict inequality where the quoted criterion asks for a strict one. These are correctness or well-definedness issues, not instances of a conclusion being imported through self-citation or definitional equivalence. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numeric free parameters are fitted; the construction uses no invented entities. The proof relies on standard external theorems and one unstated connectedness assumption.

assumptions (5)

standard math Mutation theory of quivers with potentials, including reduction and right-equivalence, as developed in [12] and [27].
Invoked in Section 1.1 and used throughout the paper.
domain assumption Scott's theorem that C[Gr(k,n)] has a cluster algebra structure via Postnikov diagrams [28].
The whole paper is built on this correspondence; cited in Section 1.2.
standard math The uniqueness criterion for non-degenerate QPs, Proposition 2.13 and related results from [17].
Used as the black box in Theorem 2.14 to prove uniqueness.
standard math Labardini-Fragoso's planar graph result [25, Section 10], used in Lemma 2.3 to realize sign changes by right-equivalences.
Load-bearing for Lemma 2.3, which is needed for the mutation compatibility theorem.
domain assumption Reduced Postnikov diagrams for fixed (k,n) are connected by geometric exchanges, so the mutation class in Definition 2.5 is independent of the diagram.
Unstated but necessary for the well-definedness of the QP of a Grassmannian cluster algebra.

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Pith. "Pith review of Quivers with potentials for Grassmannian cluster algebras." pith.science (2026). https://pith.science/paper/24A2SPAW

@misc{pith2026190810103,
  author       = {Pith},
  title        = {Pith review of: Quivers with potentials for Grassmannian cluster algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/24A2SPAW}},
  note         = {Machine review of arXiv:1908.10103}
}

abstract

We consider (iced) quiver with potential $(\bar{Q}(D), F(D), \bar{W}(D))$ associated to a Postnilov Diagram $D$ and prove the mutation of the quiver with potential $(\bare{Q}(D), F(D), \bar{W}(D))$ is compatible with the geometric exchange of the Postnikov diagram $D$. This ensures we may define a quiver with potential for a Grassmannian cluster algebra. We show such quiver with potential is always rigid (thus non-degenerate) and Jacobian-finite. And in fact, it is the unique non-degenerate (thus unique rigid) quiver with potential associated to the Grassmannian cluster algebra up to right-equivalence, by using a general result of Gei\ss-Labardini-Schr\"oer. As an application, we verify that the auto-equivalence group of the generalized cluster category ${\mathcal{C}}_{(Q, W)}$ is isomorphic to the cluster automorphism group of the associated Grassmannian cluster algebra ${{\mathcal{A}}_{(Q, W)}}$ with trivial coefficients.

Figures

Figures reproduced from arXiv: 1908.10103 by the authors.

**Figure 1.** Forbidden crossing Postnikov diagrams are identified up to isotopy. We say that a Postnikov diagram is of reduced type if no untwisting move shown in [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

**Figure 2.** Untwisting move condition ensures that the strands divide the disc into two types of regions: oriented regions, where all the strands on their boundary circle clockwise or counterclockwise, and alternating oriented regions, where the adjacent strands alternate directions. A region is said to be internal if it is not adjacent to the boundary of the disk, and the other regions are referred to as boundary regions. See … view at source ↗

**Figure 3.** A (3, 7)-Postnikov diagram j i s t R D j i s t µeR(D) [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: Pre-geometric exchange Given a Postnikov diagram D and an alternating oriented quadrilateral cell R inside D, a new Postnikov diagram µeR(D) is constructed by the local rearrangement shown in [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: The iced quiver (Q(D), F(D)) of the Postnikov diagram D in [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: Arrow orientations in the quiver Q(D). The figures can also occur in the opposite sense, which means inverting the orientations of the strands and the arrows simultaneously. alternating oriented regions correspond to the frozen vertices, and the frozen arrows are all t…

**Figure 7.** Figure 7: The fundamental cycle, where the horizonal dashed line is the boundary of the diagram, and other dashed lines are the strands. The omitted part is at the interior of the diagram. The regions are anti-clockwise. When the regions are clockwise, the figures occur in the o…

**Figure 8.** Figure 8: Local configuration of a fundamental circle through the vertex a. The reflections and the rotations are also allowed. The p in the first picture is an unfrozen arrow, while the p in the third picture is a frozen arrow. The p in the second picture is a path with length …

**Figure 9.** Figure 9: Mutations and geometric exchanges automorphism φ on Chhµea(Q(D))ii, where φ([γβ]) = [γβ] + t, φ(ξ) = ξ − β ∗ γ ∗ , φ([αβ]) = −[αβ], φ(u) = u for other arrows u in µea((Q(D)). Then φ(µea(W(D))) = ξ[γβ] +[αδ](ζ + δ ∗α ∗ ) + [αβ](η + β ∗α ∗ ) +β ∗γ ∗ t − s[γδ] + δ ∗γ ∗ [γ…

**Figure 10.** Figure 10: Initial Postnikov diagram respectively. We call width(l) = i2 − i1 the width of l, and call height(l) = j4 − j3 the height of l. To prove the rigidity and Jacobi-finiteness property of (Q, F, W), we need the following lemma. Lemma 2.6. Let l be a cycle in Qini with en…

**Figure 11.** Figure 11: Initial quiver Qini (k odd, n odd) Let a = a(i, j) be a highest vertex of l. If j = 2, then l itself already satisfies the conditions of ξ. So we assume j ≥ 2. Then up to the left-right symmetries, we may assume the local configuration of l is as in [PITH_FULL_IMAGE:…

**Figure 12.** Figure 12: Local configuration neighbouring the highest vertex a of a cycle Because the QP-mutations preserve rigidity, it suffices to prove the theorem for the initial QP (Qini, Wini). So we have to show that any cycle in Qini is cyclically equivalent to a cycle in the Jacobian…

**Figure 13.** Figure 13: Jacobi-finiteness of the QP 2.3. The uniqueness. We study in this subsection the uniqueness of the QPs of a Grassmannian cluster algebra. This is based on a general result of Geiß-LabardiniSchr¨oer [17]. They give a criterion which guarantees the uniqueness of a non-…

**Figure 14.** Figure 14: Uniqueness of the QP Note that the quiver we consider is the principal part Q, so (6) and (7) in [PITH_FULL_IMAGE:figures/full_fig_p021_14.png]

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Works this paper leans on

30 extracted references · 29 canonical work pages

[17]

C. Geiß, D. Labardini-Fragoso, J. Schr¨ oer, The representation type of Jacobian algebras, Adv. Math. 290 (2016), 364-452. 1, 4, 19, 23

work page 2016
[1]

Amiot, Cluster categories for algebras of global dimension 2 and quivers with potential

C. Amiot, Cluster categories for algebras of global dimension 2 and quivers with potential. Ann. Inst. Fourier (Grenoble) 59 (2009), no. 6, 2525-2590. 21, 22, 23

work page 2009
[2]

Amiot, On generalized cluster categories, Representations of algebras and related topics, EMS Ser

C. Amiot, On generalized cluster categories, Representations of algebras and related topics, EMS Ser. Congr.Rep., Eur. Math. Soc., Z¨ urich, 2011, pp. 1-53 22

work page 2011
[3]

Amiot, O

C. Amiot, O. Iyama, I. Reiten, Stable categories of Cohen-Macaulay modules and cluster categories, Amer. J. Math. 137(2015), no. 3, 813-857. 22, 23

work page 2015
[4]

Amiot, I

C. Amiot, I. Reiten, G. Todorov, The ubiquity of generalized cluster categories. Adv. Math. 226 (2011), no. 4, 3813-3849. 22 24 WEN CHANG AND JIE ZHANG

work page 2011
[5]

Assem, R

I. Assem, R. Schiﬄer, V. Shramchenko. Cluster automorphisms. Proceedings of the London Math- ematical Society, 2012, 104(6):1271-1302. 3, 22

work page 2012
[6]

K. Baur, A. King, R. Marsh, Dimer models and cluster categories of Grassmannians, Proc. Lond. Math. Soc. (3) 113 (2016), no. 2, 213-260. 2, 10, 11, 21

work page 2016
[7]

A. B. Buan, O. Iyama, I. Reiten, J. Scott, Cluster structures for 2-Calabi-Yau categories and unipotent groups. Compos. Math. 145 (2009), no. 4, 1035-1079. 3, 21, 23

work page 2009

Show all 30 references

[8]

A. B. Buan, O. Iyama, I. Reiten, D. Smith, Mutation of cluster-tilting objects and potentials, Amer. J. Math. 133 (2011), no. 4, 835-887. 2, 3, 4, 22

2011
[9]

Broomhead Dimer models and Calabi-Yau algebras, Mem

N. Broomhead Dimer models and Calabi-Yau algebras, Mem. Amer. Math. Soc. 215 (2012), no. 1011, viii+86 pp. ISBN: 978-0-8218-5308-5. 3

2012
[10]

Chang, B

W. Chang, B. Zhu, Cluster automorphism groups of cluster algebras with coeﬃcients, Sci. China Math. 59 (2016), no. 10, 1919-1936. 22

2016
[11]

Chang, B

W. Chang, B. Zhu, Cluster automorphism groups of cluster algebras of ﬁnite type, J. Algebra 447 (2016), 490-515. 22

2016
[12]

Derksen, J

H. Derksen, J. Weyman, A. Zelevinsky, Quivers with potentials and their representations I: Muta- tions, Selecta Math. 14 (2008), no. 1, 59-119. 2, 4, 6

2008
[13]

Fomin, A

S. Fomin, A. Zelevinsky, Cluster algebras. I. Foundations, Journal of the American Mathematical Society, 2002, 15(2), 497-529. 1

2002
[14]

Fomin, A

S. Fomin, A. Zelevinsky, Cluster algebras. II. Finite type classiﬁcation, Invent. Math. 154(1), 63-121,
[15]

Fomin, A

S. Fomin, A. Zelevinsky, Cluster algebras IV: Coeﬃcients, Compositio Mathematica, 2007, 143:112-

2007
[16]

C. Fu, B. Keller, On cluster algebras with coeﬃcients and 2-Calabi-Yau categories, Trans. Amer. Math. Soc. 362 (2010), no. 2, 859-895. 21

2010
[18]

C. Geiß, B. Leclerc, J. Schr¨ oer, Partial ﬂag varieties and preprojective algebras, Ann. Inst. Fourier 58 (2008) 825-876. 10, 21

2008
[19]

Hanany, Y

A. Hanany, Y. He, V. Jejjala, J. Pasukonis, S. Ramgoolam, D. Rodriguez-Gomez, Invariants of toric Seiberg duality, Internat. J. Modern Phys. A 27 (2012), no. 1, 1250002, 38 pp. 2

2012
[20]

Jensen, A

B. Jensen, A. King, X. Su, A categoriﬁcation of Grassmannian Cluster Algebras, Proc. Lond. Math. Soc. (3) 113 (2016), no. 2, 185-212 10, 21

2016
[21]

Kulkarni, Dimer models on cylinders over Dynkin diagrams and cluster algebras, Proc

M. Kulkarni, Dimer models on cylinders over Dynkin diagrams and cluster algebras, Proc. Amer. Math. Soc. 147 (2019), no. 3, 921-932. 3

2019
[22]

Keller, I

B. Keller, I. Reiten, Acyclic Calabi-Yau categories. With an appendix by Michel Van den Bergh. Compos. Math. 144 (2008), no. 5, 1332-1348. 22

2008
[23]

Keller, D

B. Keller, D. Yang, Derived equivalences from mutations of quivers with potential. Adv. Math. 226 (2011), no. 3, 2118-2168. 23

2011
[24]

Labardini-Fragoso Quivers with potentials associated to triangulated surfaces, Proc

D. Labardini-Fragoso Quivers with potentials associated to triangulated surfaces, Proc. Lond. Math. Soc. (3) 98 (2009), no. 3, 797-839. 3

2009
[25]

Labardini-Fragoso Quivers with potentials associated to triangulated surfaces, part IV: Removing boundary assumptions, Selecta Math

D. Labardini-Fragoso Quivers with potentials associated to triangulated surfaces, part IV: Removing boundary assumptions, Selecta Math. (N.S.) 22 (2016), no. 1, 145-189. 3, 11

2016
[26]

Postnikov, Total positivity, Grassmannians, and networks, arXiv:math/0609764v1

A. Postnikov, Total positivity, Grassmannians, and networks, arXiv:math/0609764v1. 7

arXiv
[27]

Pressland, Mutation of frozen Jacobian algebras, J

M. Pressland, Mutation of frozen Jacobian algebras, J. Algebra 546 (2020), 236-273. 2, 4, 6, 11, 18

2020
[28]

Scott, Grassmannians and cluster algebras, Proc

J. Scott, Grassmannians and cluster algebras, Proc. London Math. Soc. (3) 92 (2006), no. 2, 345-380. 2, 7, 9, 14, 23

2006
[29]

Yang, The interplay between 2-and-3-Calabi–Yau triangulated categories, arXiv:1811.07553

D. Yang, The interplay between 2-and-3-Calabi–Yau triangulated categories, arXiv:1811.07553. 22

arXiv
[30]

Vit´ oria, Mutations vs

J. Vit´ oria, Mutations vs. Seiberg duality, J. Algebra 321 (2009), no. 3, 816-828. 2 School of Mathematics and Statistics, Shaanxi Normal University, Xi’an 710062, China. Email address : changwen161@163.com School of Mathematics and Statistics, Beijing institute of technology...

2009

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [17]

C. Geiß, D. Labardini-Fragoso, J. Schr¨ oer, The representation type of Jacobian algebras, Adv. Math. 290 (2016), 364-452. 1, 4, 19, 23

work page 2016

[2] [1]

Amiot, Cluster categories for algebras of global dimension 2 and quivers with potential

C. Amiot, Cluster categories for algebras of global dimension 2 and quivers with potential. Ann. Inst. Fourier (Grenoble) 59 (2009), no. 6, 2525-2590. 21, 22, 23

work page 2009

[3] [2]

Amiot, On generalized cluster categories, Representations of algebras and related topics, EMS Ser

C. Amiot, On generalized cluster categories, Representations of algebras and related topics, EMS Ser. Congr.Rep., Eur. Math. Soc., Z¨ urich, 2011, pp. 1-53 22

work page 2011

[4] [3]

Amiot, O

C. Amiot, O. Iyama, I. Reiten, Stable categories of Cohen-Macaulay modules and cluster categories, Amer. J. Math. 137(2015), no. 3, 813-857. 22, 23

work page 2015

[5] [4]

Amiot, I

C. Amiot, I. Reiten, G. Todorov, The ubiquity of generalized cluster categories. Adv. Math. 226 (2011), no. 4, 3813-3849. 22 24 WEN CHANG AND JIE ZHANG

work page 2011

[6] [5]

Assem, R

I. Assem, R. Schiﬄer, V. Shramchenko. Cluster automorphisms. Proceedings of the London Math- ematical Society, 2012, 104(6):1271-1302. 3, 22

work page 2012

[7] [6]

K. Baur, A. King, R. Marsh, Dimer models and cluster categories of Grassmannians, Proc. Lond. Math. Soc. (3) 113 (2016), no. 2, 213-260. 2, 10, 11, 21

work page 2016

[8] [7]

A. B. Buan, O. Iyama, I. Reiten, J. Scott, Cluster structures for 2-Calabi-Yau categories and unipotent groups. Compos. Math. 145 (2009), no. 4, 1035-1079. 3, 21, 23

work page 2009