A module-theoretic interpretation of quantum expansion formula

Yutong Yu

arxiv: 2509.03909 · v2 · submitted 2025-09-04 · 🧮 math.RT

A module-theoretic interpretation of quantum expansion formula

Yutong Yu This is my paper

Pith reviewed 2026-05-18 19:54 UTC · model grok-4.3

classification 🧮 math.RT

keywords quantum cluster algebrasunpunctured surfacesstring modulesexpansion formulasskein relationsperfect matchingsmodule extensions

0 comments

The pith

String modules interpret the perfect-matching map for expanding quantum cluster variables from unpunctured surfaces.

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

The paper supplies a module-theoretic reading of Huang's combinatorial expansion formula, in which a map on perfect matchings produces the quantum cluster-variable expansions arising from unpunctured surfaces. It does so by identifying the matchings with the structure of string modules whose one-dimensional extension spaces allow a skein-relation multiplication rule. The authors also prove that this module picture recovers the alternative Kronecker-type formula of Canakci and Lampe. A sympathetic reader sees the work as a bridge that lets algebraic module calculations replace direct combinatorial enumeration in these cluster algebras.

Core claim

The expansion formula given by Huang, which sends perfect matchings to monomials in the quantum cluster variables, admits an interpretation in which the matchings index extensions of string modules; when the extension space is one-dimensional the skein relations supply a multiplication formula for the corresponding cluster elements, and the two formulas agree on the Kronecker quiver.

What carries the argument

The map on perfect matchings, realized through extensions of string modules whose one-dimensional Ext spaces permit a skein-relation multiplication rule.

If this is right

Quantum expansions can be read off from the extension spaces of the corresponding string modules.
The skein relations yield a multiplication rule whenever two string modules have one-dimensional extension space.
The Huang formula and the Canakci-Lampe formula coincide for the Kronecker quiver.
The interpretation supplies a representation-theoretic route to the same monomials previously obtained combinatorially.

Where Pith is reading between the lines

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

The same string-module picture may furnish expansions for cluster algebras attached to surfaces with punctures once the appropriate modules are identified.
The multiplication formula could be used to compute higher products or to verify positivity directly from module data.
Similar module-theoretic translations might apply to other combinatorial formulas in cluster algebras that are currently stated only in terms of matchings or triangulations.

Load-bearing premise

The cluster algebras come from unpunctured surfaces and the modules in question are string modules whose extension spaces are one-dimensional.

What would settle it

An explicit quantum cluster variable on an unpunctured surface whose coefficient expansion obtained from perfect matchings differs from the basis elements counted by string-module extensions.

Figures

Figures reproduced from arXiv: 2509.03909 by Yutong Yu.

**Figure 1.** Figure 1: Neighborhood of τk n −(τk, j, P) = 0 if j − 1 ∈ P 1 otherwise Proof. Without loss of generality, assume xj−1 = τk−1, xj+1 = τk+2. The snake graph of G(j − 1), G(j), G(j + 1) is showed in [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗

**Figure 2.** Figure 2: aj−1 and aj both are inverse We want the notion n +(resp.n −) to be the number of edges labeled τk belonging to P on the tiles with diagonal xj+1(resp. xj−1), and we only focus on the edges incident to the diagonal. We illustrate this definition when j is odd(right) and the other case is similar. In this case, the blue edge always belong to P− and the red edge can not belong to P−. Then n + just depends on… view at source ↗

**Figure 3.** Figure 3: aj−1 and aj both are direct Lemma 5.4. If there exists some j with 2 ≤ j ≤ s−1 satisfying xj = τk and aj−1 is inverse and aj is direct, define two quantities to count the number of edges labeled τk in E(G(j + 1)) ∩ P and E(G(j − 1)) ∩ P. n +(τk, j, P) = 0 if j + 1 ∈ P 1 otherwise n −(τk, j, P) = 0 if j − 1 ∈ P 1 otherwise Proof. Without loss of generality, assume xj−1 = τk−1, xj+1 = τk+1. The snake gra… view at source ↗

**Figure 4.** Figure 4: xj−1 ← xj → xj+1 Lemma 5.5. If there exists some j with 2 ≤ j ≤ s−1 satisfying xj = τk and aj−1 is direct and aj is inverse, define two quantities to count the number of edges labeled τk in E(G(j + 1)) ∩ P and E(G(j − 1)) ∩ P. n +(τk, j, P) = 1 if j + 1 ∈ P 0 otherwise n −(τk, j, P) = 1 if j − 1 ∈ P 0 otherwise Proof. Without loss of generality, assume xj−1 = τk−2, xj+1 = τk+2. The snake graph of G(j −… view at source ↗

**Figure 5.** Figure 5: xj−1 → xj ← xj+1 14 [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗

**Figure 6.** Figure 6: xj = τk−1, xj+1 = τk−2 In the following, we need to consider the first or last tile and count the number of edges label τk in these tiles. The strategy is totally the same as above. We show some of example in [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗

**Figure 7.** Figure 7: s is odd the second tile has diagonal τk−2, then the blue edge belongs to P− and the red edge can’t belong to P−. So whether the red edge belongs to P is equivalent to whether it 2 belongs to P. Similarly for the other case. We provide a summary of all results in this case here. Lemma 5.9. If x1 = τk and x2 ∈ {τk−2, τk+2}, then n +(τk, 1, P) = 1 if 2 ∈ P and n +(τk, 1, P) = 0 if 2 ∈/ P. If x2 ∈ {τk−1, τk+1… view at source ↗

**Figure 8.** Figure 8: s is odd Remark 5.11. (1) For the cases where n(τk, j, P) does not present, let n(τk, j, P) = n +(τk, j, P) + n −(τk, j, P). (2) In [11], the equivalence class of edges label τk must be one of the above case. So n(τk, j, P) is the number of edges belonging to P in the equivalence class near xj . Definition 5.12. Here we define the number identical to Definition 4.9 M+(τk, j, w)=#{xi |xi = τk, i > j} M−(τk,… view at source ↗

**Figure 9.** Figure 9: Arrow extension Theorem 7.5. For any two strings v and w have only one arrow extension. The multiplication formula XM(v)XM(w) = q Λ(v,w)−Λ(u1,u2) 2 +1XM(u1)XM(u2)+q Λ(v,w)−Λ(u3,u4) 2 −1XM(u3)XM(u4) 7.2. Overlap extension. Let v and w have an overlap extension. Recall in this case, v = vLbma−1 vR and w = wLd −1mcwR [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗

**Figure 10.** Figure 10: Overlap extension 8. The Kronecker Case For the case when the quiver is 1 ⇒ 2, [20] and [11] both gave the expansion formula of cluster variables in this quantum cluster algebra. In this section, we show they are actually the same. 8.1. Results in [20]. First, observe that the snake graph corresponding to arcs of this type has alternating face weights(the diagonal edge) 1 and 2. Definition 8.1. The snake… view at source ↗

**Figure 11.** Figure 11: They are the snake graph of Gs, Hs, Gs−1 in order, the edges of the minimal perfect matching are showed in blue and the values of α are on the edges. Lemma 8.5. Let P be a perfect matching of Gs with dimension vector (u, w) such that Twist(P) does not contain the last tile. Then P ′ is a perfect matching of Hs corresponding to P. In addition, α(P) = α(P ′ ) − u and v(P) = v(P ′ ) − u. 22 [PITH_FULL_IMAGE… view at source ↗

**Figure 11.** Figure 11: Snake graph and values of α Proof. The correspondence is obvious by definition. We know P and P ′ have the same edge except the last tile label 1. So the dimension vector of P ′ is (u, w). The twist tiles of P and P ′ are the same. Also there are u tiles labeled 1 and w tiles label 2. The values of tiles labeled 1 differ by −1 and the values of tiles labeled 2 are the same. Therefore α(P) = α(P ′ ) − u. F… view at source ↗

**Figure 12.** Figure 12: Snake graph and values of α Lemma 8.7. Let P be a perfect matching of Hs with dimension vector (u, w) such that Twist(P) contains the last tile. Then P ′ is a perfect matching of Gs−1 corresponding to P. In addition, α(P) = α(P ′ )−s+w and v(P) = v(P ′ ) + s − w. Proof. The correspondence is obvious by definition. We know P and P ′ have the same edge except the last tile label 1. So the dimension vector o… view at source ↗

read the original abstract

We provide a module-theoretic interpretation of the expansion formula given by Huang (2022), which defines a map on perfect matchings to compute the expansion of quantum cluster variables in quantum cluster algebras arising from unpunctured surfaces. In addition, we present a multiplication formula for string modules with one-dimensional extension space, derived using the skein relations. For the Kronecker type, an alternative expansion formula was given in Canakci and Lampe (2020), and we show that the two expansion formulas coincide.

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

Yu reinterprets Huang's perfect-matching expansion for quantum cluster variables via string modules and adds a multiplication rule, but the one-dimensional Ext condition looks insufficiently justified beyond the Kronecker case.

read the letter

The core of this paper is a module-theoretic reading of Huang's 2022 expansion formula for quantum cluster variables on unpunctured surfaces. It defines a map sending perfect matchings to string modules over the Jacobian algebra and supplies a multiplication formula for pairs of such modules whose Ext^1 space is one-dimensional, obtained from skein relations. The author also checks that the resulting expansion agrees with the Canakci-Lampe formula in the Kronecker quiver case. That equivalence check is a concrete, verifiable step that was not already in the cited works. The module map itself gives a direct dictionary between the combinatorial side and the representation category, which is a modest but genuine addition for people who already work with both languages. The Kronecker verification is useful because it shows the two formulas produce the same Laurent polynomials in a setting where explicit computations are feasible. The main soft spot is the scope of the multiplication formula. It is stated only for string modules with dim Ext^1 = 1, and the paper does not supply a general argument that every pair appearing in the perfect-matching expansion on an arbitrary unpunctured surface satisfies this dimension condition. The Kronecker case works, but without either explicit module constructions for other surfaces or a proof that the Jacobian algebra forces the required Ext dimension, the identification between the two expansions rests on an unverified assumption for the general case. The abstract states the claims cleanly, yet the absence of derivations or sample calculations in the visible text makes it hard to confirm that the map and the formula line up without extra hypotheses. This is a narrow, technical note aimed at researchers already inside the cluster-algebra-from-surfaces literature who care about representation-theoretic models. A reader who knows Huang's formula and string modules would find the reinterpretation and the Kronecker check worth a look, but the work is not self-contained enough for a wider audience. I would send it to peer review. The central idea is coherent and the special-case verification is solid, so referees can ask for the missing general argument on the Ext dimension without starting from zero.

Referee Report

1 major / 1 minor

Summary. The paper provides a module-theoretic interpretation of the expansion formula given by Huang (2022), which uses a map on perfect matchings to expand quantum cluster variables in quantum cluster algebras arising from unpunctured surfaces. It introduces a multiplication formula for string modules whose Ext^1-space is one-dimensional, derived from skein relations, and shows that the resulting interpretation coincides with the alternative expansion formula of Canakci and Lampe (2020) in the Kronecker case.

Significance. If the identification is valid, the work links the combinatorial perfect-matching expansion to products in the module category of the Jacobian algebra, which could facilitate representation-theoretic computations of quantum cluster variables. The explicit equivalence check for the Kronecker quiver supplies a concrete, falsifiable consistency test against prior literature.

major comments (1)

[Abstract (multiplication formula paragraph)] Abstract (paragraph on multiplication formula): the multiplication formula is stated only for pairs of string modules with dim Ext^1 = 1. The manuscript supplies no general argument that this dimension condition holds for every pair of modules that appears when the Huang map is applied to perfect matchings on an arbitrary unpunctured surface; the condition is verified only for the Kronecker quiver. This assumption is load-bearing for the claimed module-theoretic interpretation.

minor comments (1)

[Abstract] The abstract could briefly indicate the precise class of triangulations or quivers for which the string-module construction is carried out, to help readers assess the scope of the one-dimensional Ext^1 claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the major comment below and will revise the manuscript to incorporate the necessary clarification and supporting argument.

read point-by-point responses

Referee: [Abstract (multiplication formula paragraph)] Abstract (paragraph on multiplication formula): the multiplication formula is stated only for pairs of string modules with dim Ext^1 = 1. The manuscript supplies no general argument that this dimension condition holds for every pair of modules that appears when the Huang map is applied to perfect matchings on an arbitrary unpunctured surface; the condition is verified only for the Kronecker quiver. This assumption is load-bearing for the claimed module-theoretic interpretation.

Authors: We appreciate the referee highlighting this point. The multiplication formula is indeed presented specifically for string modules with dim Ext^1 = 1, as obtained from the skein relations in the surface setting. We agree that the current manuscript does not supply an explicit general argument verifying this dimension condition for all pairs arising from the Huang map on arbitrary unpunctured surfaces, beyond the explicit verification in the Kronecker case. In the revised version we will add a lemma establishing that the string modules corresponding to the perfect matchings under the Huang map satisfy dim Ext^1 = 1. This follows from the combinatorial structure: the perfect matchings encode non-crossing arc configurations on the surface, so the associated string modules in the Jacobian algebra have at most a single overlap contributing to extensions, yielding dimension exactly 1 whenever the modules interact non-trivially in the expansion. We will also revise the abstract to state this condition and its justification more clearly. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the module-theoretic interpretation

full rationale

The paper provides a module-theoretic interpretation of the external expansion formula from Huang (2022) and derives a multiplication formula for string modules from skein relations. It then verifies coincidence with the Canakci-Lampe formula in the Kronecker case. No parameters are fitted inside the paper and then relabeled as predictions, no self-definitional loops appear in the equations, and the central claims rest on cited external results rather than reducing to inputs by construction. The one-dimensional Ext condition is an assumption stated for the multiplication formula, not a fitted quantity whose output is forced to match the input. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard background from cluster algebra theory on surfaces and representation theory of string modules; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)

domain assumption Quantum cluster algebras arise from unpunctured surfaces as defined in Huang (2022)
Stated in the abstract as the setting for the expansion formula.
domain assumption Skein relations hold for the relevant string modules
Invoked to derive the multiplication formula.

pith-pipeline@v0.9.0 · 5593 in / 1335 out tokens · 40381 ms · 2026-05-18T19:54:23.782254+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.2. The quantum cluster variable associated with M(w) can be expressed by XM(w) = sum_{N in CS(M(w))} q^{v_gamma(N)/2} X_Gamma(N)
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

multiplication formula for string modules with one-dimensional extension space, derived using the skein relations

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

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Quantum cluster algebras.Advances in Mathematics , 195(2):405–455, 2005

Arkady Berenstein and Andrei Zelevinsky. Quantum cluster algebras.Advances in Mathematics , 195(2):405–455, 2005

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On the cluster category of a marked surface without punctures

Thomas Brustle and Jie Zhang. On the cluster category of a marked surface without punctures. Algebra & Number Theory , 5:529–566, 2010

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A module-theoretic interpretation of schiffler’s expansion formula

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M. C. R. Butler and Claus Michael Ringel. Auslander-reiten sequences with few middle terms and applications to string algebras. Communications in Algebra , 15:145–179, 1987

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Quantum cluster variables via serre polynomials.Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal) , 2012(668):149–190, 2012

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Cluster algebras and triangulated surfaces

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An expansion formula for quantum cluster algebras from unpunc- tured triangulated surfaces

Min Huang. An expansion formula for quantum cluster algebras from unpunc- tured triangulated surfaces. Selecta Mathematica, 28, 2021

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Bracelets bases are theta bases

Travis Mandel and Fan Qin. Bracelets bases are theta bases. 2023

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Skein and cluster algebras of marked surfaces

Greg Muller. Skein and cluster algebras of marked surfaces. Quantum Topology, 7(3):435–503, 2016

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[14]

Cluster expansion formulas and perfect matchings

Gregg Musiker and Ralf Schiffler. Cluster expansion formulas and perfect matchings. Journal of Algebraic Combinatorics , 32:187–209, 2008

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[15]

Positivity for cluster al- gebras from surfaces

Gregg Musiker, Ralf Schiffler, and Lauren Williams. Positivity for cluster al- gebras from surfaces. Advances in Mathematics , 227(6):2241–2308, 2011

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Cluster characters for 2-calabi–yau triangulated categories.Annales de l’institut Fourier , 58(6):2221–2248, 2008

Yann Palu. Cluster characters for 2-calabi–yau triangulated categories.Annales de l’institut Fourier , 58(6):2221–2248, 2008

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On cluster algebras arising from unpunctured surfaces ii

Ralf Schiffler. On cluster algebras arising from unpunctured surfaces ii. Ad- vances in Mathematics , 223(6):1885–1923, 2010

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On cluster algebras arising from unpunctured surfaces

Ralf Schiffler and Hugh Thomas. On cluster algebras arising from unpunctured surfaces. International Mathematics Research Notices , 2009:3160–3189, 2007

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On extensions for gentle algebras

˙Ilke C ¸ anak¸ cı, David Pauksztello, and Sibylle Schroll. On extensions for gentle algebras. Canadian Journal of Mathematics , 73(1):249–292, 2021

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An expansion formula for type a and kro- necker quantum cluster algebras

˙Ilke C ¸ anak¸ cı and Philipp Lampe. An expansion formula for type a and kro- necker quantum cluster algebras. Journal of Combinatorial Theory, Series A , 171:105132, 2020

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Extensions in jacobian algebras and cluster categories of marked surfaces

˙Ilke C ¸ anak¸ cı and Sibylle Schroll. Extensions in jacobian algebras and cluster categories of marked surfaces. Advances in Mathematics , 313:1–49, 2017

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[22]

Lattice bijections for string modules, snake graphs and the weak bruhat order

˙Ilke C ¸ anak¸ cı and Sibylle Schroll. Lattice bijections for string modules, snake graphs and the weak bruhat order. Advances in Applied Mathematics , 126:102094, 2021. Special Issue: Dedicated to Joseph Kung. 26 Department of Mathematical Sciences, Tsinghua University, Bei- jing 100084, P. R. China Email address : yyt20@mails.tsinghua.edu.cn(Y. Yu) 27

work page 2021

[1] [1]

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

Claire Amiot. Cluster categories for algebras of global dimension 2 and quivers with potential. Annales de l’Institut Fourier , 59:2525–2590, 2008

work page 2008

[2] [2]

Gentle algebras arising from surface triangulations

Ibrahim Assem, Thomas Br¨ ustle, Gabrielle Charbonneau-Jodoin, and Pierre- Guy Plamondon. Gentle algebras arising from surface triangulations. Algebra & Number Theory , 4(2):201 – 229, 2010

work page 2010

[3] [3]

Quantum cluster algebras.Advances in Mathematics , 195(2):405–455, 2005

Arkady Berenstein and Andrei Zelevinsky. Quantum cluster algebras.Advances in Mathematics , 195(2):405–455, 2005

work page 2005

[4] [4]

On the cluster category of a marked surface without punctures

Thomas Brustle and Jie Zhang. On the cluster category of a marked surface without punctures. Algebra & Number Theory , 5:529–566, 2010

work page 2010

[5] [5]

A module-theoretic interpretation of schiffler’s expansion formula

Thomas Br¨ ustle and Jie Zhang. A module-theoretic interpretation of schiffler’s expansion formula. Communications in Algebra , 41:260 – 283, 2011

work page 2011

[6] [6]

M. C. R. Butler and Claus Michael Ringel. Auslander-reiten sequences with few middle terms and applications to string algebras. Communications in Algebra , 15:145–179, 1987

work page 1987

[7] [7]

Quantum cluster variables via serre polynomials.Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal) , 2012(668):149–190, 2012

Qin Fan. Quantum cluster variables via serre polynomials.Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal) , 2012(668):149–190, 2012. With an appendix by Bernhard Keller

work page 2012

[8] [8]

Cluster algebras and triangulated surfaces

Sergey Fomin, Michael Shapiro, and Dylan Thurston. Cluster algebras and triangulated surfaces. Part I: Cluster complexes. Acta Mathematica, 201(1):83 – 146, 2008

work page 2008

[9] [9]

Cluster algebras i: Foundations

Sergey Fomin and Andrei Zelevinsky. Cluster algebras i: Foundations. Journal of the American Mathematical Society , 15:497–529, 2001

work page 2001

[10] [10]

Cluster algebras iv: Coefficients

Sergey Fomin and Andrei Zelevinsky. Cluster algebras iv: Coefficients. Com- positio Mathematica, 143:112 – 164, 2006

work page 2006

[11] [11]

An expansion formula for quantum cluster algebras from unpunc- tured triangulated surfaces

Min Huang. An expansion formula for quantum cluster algebras from unpunc- tured triangulated surfaces. Selecta Mathematica, 28, 2021

work page 2021

[12] [12]

Bracelets bases are theta bases

Travis Mandel and Fan Qin. Bracelets bases are theta bases. 2023

work page 2023

[13] [13]

Skein and cluster algebras of marked surfaces

Greg Muller. Skein and cluster algebras of marked surfaces. Quantum Topology, 7(3):435–503, 2016

work page 2016

[14] [14]

Cluster expansion formulas and perfect matchings

Gregg Musiker and Ralf Schiffler. Cluster expansion formulas and perfect matchings. Journal of Algebraic Combinatorics , 32:187–209, 2008

work page 2008

[15] [15]

Positivity for cluster al- gebras from surfaces

Gregg Musiker, Ralf Schiffler, and Lauren Williams. Positivity for cluster al- gebras from surfaces. Advances in Mathematics , 227(6):2241–2308, 2011

work page 2011

[16] [16]

Cluster characters for 2-calabi–yau triangulated categories.Annales de l’institut Fourier , 58(6):2221–2248, 2008

Yann Palu. Cluster characters for 2-calabi–yau triangulated categories.Annales de l’institut Fourier , 58(6):2221–2248, 2008

work page 2008

[17] [17]

On cluster algebras arising from unpunctured surfaces ii

Ralf Schiffler. On cluster algebras arising from unpunctured surfaces ii. Ad- vances in Mathematics , 223(6):1885–1923, 2010

work page 1923

[18] [18]

On cluster algebras arising from unpunctured surfaces

Ralf Schiffler and Hugh Thomas. On cluster algebras arising from unpunctured surfaces. International Mathematics Research Notices , 2009:3160–3189, 2007

work page 2009

[19] [19]

On extensions for gentle algebras

˙Ilke C ¸ anak¸ cı, David Pauksztello, and Sibylle Schroll. On extensions for gentle algebras. Canadian Journal of Mathematics , 73(1):249–292, 2021

work page 2021

[20] [20]

An expansion formula for type a and kro- necker quantum cluster algebras

˙Ilke C ¸ anak¸ cı and Philipp Lampe. An expansion formula for type a and kro- necker quantum cluster algebras. Journal of Combinatorial Theory, Series A , 171:105132, 2020

work page 2020

[21] [21]

Extensions in jacobian algebras and cluster categories of marked surfaces

˙Ilke C ¸ anak¸ cı and Sibylle Schroll. Extensions in jacobian algebras and cluster categories of marked surfaces. Advances in Mathematics , 313:1–49, 2017

work page 2017

[22] [22]

Lattice bijections for string modules, snake graphs and the weak bruhat order

˙Ilke C ¸ anak¸ cı and Sibylle Schroll. Lattice bijections for string modules, snake graphs and the weak bruhat order. Advances in Applied Mathematics , 126:102094, 2021. Special Issue: Dedicated to Joseph Kung. 26 Department of Mathematical Sciences, Tsinghua University, Bei- jing 100084, P. R. China Email address : yyt20@mails.tsinghua.edu.cn(Y. Yu) 27

work page 2021