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Wen Chang

Publications and source records attributed to Wen Chang.

At least 19 recordsLinked to original sources
arXiv2026

Obstructions to Jacobi-Finiteness of Quivers with Potentials

We show that Jacobi-finite potentials need not exist on finite $2$-acyclic quivers. Our main tool is a matrix-valued Golod--Shafarevich--Vinberg inequality for quotients of completed path algebras by finitely many, possibly nonhomogeneous, topological relations. Applied to cyclic derivatives, it yields a potential-dependent obstruction to the finite-dimensionality of completed Jacobian algebras. We then construct a purely quiver-level criterion excluding every Jacobi-finite potential on a given quiver, and exhibit a family of quivers for which every potential has an infinite-dimensional Jacobian algebra.

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arXiv2026

Tilting Completion and Full-Rank Self-Orthogonal Modules

We give negative answers to two tilting-completion questions for finite-dimensional algebras. We construct two finite-dimensional basic connected quasi-hereditary $\mathbb C$-algebras. The first admits a faithful basic full-rank pretilting module with no tilting completion; the second admits an almost-tilting module with no tilting completion. The full-rank example also yields counterexamples to two conjectures: Enomoto's Self-orthogonal Wakamatsu-tilting Conjecture and the Self-orthogonal Faithful Conjecture of Chen, Li, Zhang, and Zhao. We further show that the Self-orthogonal Wakamatsu-tilting Conjecture holds for all finite-dimensional algebras if and only if the Self-orthogonal Faithful Conjecture holds for all finite-dimensional algebras. The construction ultimately stems from Krah's non-full exceptional collection of maximal length on a rational surface and Kalck's associated full-rank presilting example.

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arXiv2026

Infinitely Many Components in Auslander--Reiten Quivers of Representation-Infinite Algebras over Perfect Fields

Let $k$ be a perfect field and let $A$ be a representation-infinite finite-dimensional $k$-algebra. We prove that the Auslander--Reiten quiver of $A$ has infinitely many connected components. This establishes, for finite-dimensional algebras over perfect fields, a conjecture of Auslander, Reiten, and Smal\o{} concerning Artin algebras. Over an algebraically closed field, the proof combines a localized polynomial representation embedding with semilinear twists induced by field automorphisms. The passage from a perfect field to its algebraic closure is obtained by separable base change: we prove that if the Auslander--Reiten quiver of $A$ has only finitely many components, then the same holds for the scalar extension to the algebraic closure.

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arXiv2025

On the $\tau$-tilting finiteness and silting-discreteness of graded (skew-) gentle algebras

This paper investigates finiteness conditions for gentle and skew-gentle algebras. First, we prove that a skew-gentle algebra is $\tau$-tilting finite if and only if it is representation-finite, which extends the result for gentle algebras by Plamondon (2019). Second, using surface models, we characterize silting-discreteness for the perfect derived categories of graded gentle and skew-gentle algebras. Specifically, for a graded gentle algebra, silting-discreteness is equivalent to its associated surface being of genus zero with non-zero winding numbers for all simple closed curves. We further extend this geometric characterization to graded skew-gentle algebras via orbifold surface models.

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arXiv2025

Entropy of the Serre functor for partially wrapped Fukaya categories of surfaces with stops

We prove that the entropy of the Serre functor $\mathbb{S}$ in the partially wrapped Fukaya category of a graded surface $\Sigma$ with stops is given by the function sending $t \in \mathbb{R}$ to $ h_t(\mathbb{S}) = (1-\min \Omega)t$, for $t\geq 0$, and to $h_t(\mathbb{S})=(1-\max \Omega)t$, for $t\leq 0$, where $\Omega = \{\frac{\omega_1}{m_1} \ldots, \frac{\omega_b}{m_b},0\}$, and $\omega_i$ is the winding number of the $i$th boundary component $\partial_i\Sigma$ of the surface with $b$ boundary components and $m_i$ stops on $\partial_i \Sigma$. It then follows that the upper and lower Serre dimensions are given by $1-\min \Omega$ and $1-\max \Omega$, respectively. Furthermore, in the case of a finite dimensional gentle algebra $A$, we show that a Gromov-Yomdin-like equality holds by relating the categorical entropy of the Serre functor of the perfect derived category of $A$ to the logarithm of the spectral radius of the Coxeter transformation.

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arXiv2025

Maximal rigid modules over a gentle algebra and applications to higher Auslander-Reiten theory

We construct a bijective correspondence between the set of rigid modules over a gentle algebra and the set of admissible arc systems on the associated coordinated-marked surface. In particular, a maximal rigid module aligns with an equivalence class of admissible $5$-partial triangulations, which is an (admissible) set of simple arcs dissecting the surface into $s$-gons with $3\leqslant s\leqslant 5$. Furthermore, the rank of the maximal rigid module is equal to the rank of the algebra plus the number of internal $4$-gons and $5$-gons in the associated $5$-partial triangulation. Subsequently, these results facilitate an exploration of the higher Auslander-Reiten theory for gentle algebras with global dimension $n$. The $\tau_m$-closures of injective modules are realized as admissible $(m+2)$-partial triangulations, where $\tau_m$ are higher Auslander-Reiten translations with $2\leqslant m \leqslant n$. Finally, we provide a complete classification of gentle algebras that are $\tau_n$-finite or $n$-complete introduced by Iyama [I11].

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arXiv2024

Tilting-completion for gentle algebras

It is demonstrated that any almost-tilting module over a gentle algebra is indeed partial-tilting, meaning it can be completed as a tilting module. Furthermore, such a module has at most $2n$ possible complements, thereby confirming a (modified) conjecture of Happel for the case of gentle algebras. Additionally, for any $n\geq 3$ and $1\leq m \leq n-2$, there always exists a (connected) gentle algebra with rank $n$ and a pre-tilting module of rank $m$ which is not partial-tilting. The tool we use is the surface model associated with the module category of a gentle algebra. The main technique is an induction process involving surface cuts, which is hoped to be beneficial for other applications as well.

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arXiv2023

Geometric models for the algebraic hearts in the derived category of a gentle algebra

We give a geometric model for any algebraic heart in the derived category of a gentle algebra, which is equivalent to the module category of some gentle algebra. To do this, we deform the geometric model for the module category of a gentle algebra given in [BC21], and then embed it into the geometric model of the derived category given in [OPS18], in the sense that each so-called zigzag curve on the surface represents an indecomposable module as well as the minimal projective resolution of this module. A key point of this embedding is to give a geometric explanation of the duality between the simple modules and the projective modules. Such a blend of two geometric models provides us with a handy way to describe the homological properties of a module within the framework of the derived category. In particular, we realize any higher Yoneda-extension as a polygon on the surface, and realize the Yoneda-product as gluing of these polygons. As an application, we realize any algebraic heart in the derived category of a gentle algebra on the marked surface.

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arXiv2023

Quasi-homomorphisms of quantum cluster algebras

In this paper, we study quasi-homomorphisms of quantum cluster algebras, which are quantum analogy of quasi-homomorphisms of cluster algebras introduced by Fraser. For a quantum Grassmannian cluster algebra $\mathbb{C}_q[{\rm Gr}(k,n)]$, we show that there is an associated braid group and each generator $\sigma_i$ of the braid group preserves the quasi-commutative relations of quantum Pl\"{u}cker coordinates and exchange relations of the quantum Grassmannian cluster algebra. We conjecture that $\sigma_i$ also preserves $r$-term ($r \ge 4$) quantum Pl\"{u}cker relations of $\mathbb{C}_q[{\rm Gr}(k,n)]$ and other relations which cannot be derived from quantum quantum Pl\"{u}cker relations (if any). Up to this conjecture, we show that $\sigma_i$ is a quasi-automorphism of $\mathbb{C}_q[{\rm Gr}(k,n)]$ and the braid group acts on $\mathbb{C}_q[{\rm Gr}(k,n)]$.

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arXiv2023

Braid group actions on branched coverings and full exceptional sequences

We relate full exceptional sequences in Fukaya categories of surfaces or equivalently in derived categories of graded gentle algebras to branched coverings over the disk, building on a previous classification result of the first and third author. This allows us to apply tools from the theory of branched coverings such as Birman--Hilden theory and Hurwitz systems to study the natural braid group action on exceptional sequences. As an application, counterexamples are given to a conjecture of Bondal--Polishchuk on the transitivity of the braid group action on full exceptional sequences in a triangulated category.

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arXiv2022

Recollements of partially wrapped Fukaya categories and surface cuts

In this paper we use recollements to investigate partially wrapped Fukaya categories of surfaces with marked points. In particular, we show that cutting surfaces gives rise to recollements of the corresponding partially wrapped Fukaya categories. Our approach is based on the fact that the partially wrapped Fukaya category of a surface with marked points is triangle equivalent to the perfect derived category of a homologically smooth and proper graded gentle algebra with zero differential as shown by Haiden, Katzarkov and Kontsevich. Using this, we study particular generators of partially wrapped Fukaya categories, namely full exceptional sequences, silting objects and simple-minded collections. In particular, we fully characterise the existence of full exceptional sequences and we give an example of a partially wrapped Fukaya category which does not admit a silting object, that is a generator with no positive self-extensions.

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arXiv2022

Exceptional sequences in the derived category of a gentle algebra

In this paper, using the correspondence of gentle algebras and dissections of marked surfaces, we study full exceptional sequences in the perfect derived category $\mathsf{K^b(A)}$ of a gentle algebra $\mathsf{A}$. We show that full exceptional sequences in $\mathsf{K^b(A)}$ exist if and only if the associated marked surface has no punctures and has at least two marked points on the boundary. Furthermore, by using induction on cuts of surfaces, we characterise when an exceptional sequence can be completed to a full exceptional sequence. If the genus of the associated surface is zero then we show that the action of the braid group together with the grading shift on full exceptional sequences in $\mathsf{K^b(A)}$ is transitive. For the case of surfaces of higher genus, we reduce the problem of transitivity to the problem of the existence of certain sequences of pairs of exceptional objects. Finally, we interpret the duality of a full exceptional sequence induced by the longest element in the associated symmetric group using Koszul duality.

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arXiv2021

A geometric realization of silting theory for gentle algebras

A gentle algebra gives rise to a dissection of an oriented marked surface with boundary into polygons and the bounded derived category of the gentle algebra has a geometric interpretation in terms of this surface. In this paper we study silting theory in the bounded derived category of a gentle algebra in terms of its underlying surface. In particular, we show how silting mutation corresponds to the changing of graded arcs and that in some cases silting mutation results in the interpretation of the octahedral axioms in terms of the flipping of diagonals in a quadrilateral as in the work of Dyckerhoff-Kapranov in the context of triangulated surfaces. We also show that silting reduction corresponds to the cutting of the underlying surface as is the case for Calabi-Yau reduction of surface cluster categories as shown by Marsh-Palu and Qiu-Zhou.

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arXiv2021

Quivers with potentials for Grassmannian cluster algebras

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 Geiß-Labardini-Schröer. 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.

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arXiv2020

Cluster automorphism groups and automorphism groups of exchange graphs

For a coefficient free cluster algebra $\mathcal{A}$, we study the cluster automorphism group $Aut(\mathcal{A})$ and the automorphism group $Aut(E_{\mathcal{A}})$ of its exchange graph $E_{\mathcal{A}}$. We show that these two groups are isomorphic with each other, if $\mathcal{A}$ is of finite type excepting types of rank two and type $F_4$, or if $\mathcal{A}$ is of skew-symmetric finite mutation type.

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arXiv2019

Quantum affine algebras and Grassmannians

We study the relation between quantum affine algebras of type A and Grassmannian cluster algebras. Hernandez and Leclerc described an isomorphism from the Grothendieck ring of a certain subcategory $\mathcal{C}_{\ell}$ of $U_q(\hat{\mathfrak{sl}_n})$-modules to a quotient of the Grassmannian cluster algebra in which certain frozen variables are set to 1. We explain how this induces an isomorphism between the monoid of dominant monomials, used to parameterize simple modules, and a quotient of the monoid of rectangular semistandard Young tableaux. Via the isomorphism, we define an element ch(T) in a Grassmannian cluster algebra for every rectangular tableau T. By results of Kashiwara, Kim, Oh, and Park, and also of Qin, every Grassmannian cluster monomial is of the form ch(T) for some T. Using formulas of Arakawa-Suzuki, we give an explicit expression for ch(T), and also give explicit q-character formulas for finite-dimensional $U_q(\hat{\mathfrak{sl}_n})$-modules. We give a tableau-theoretic rule for performing mutations in Grassmannian cluster algebras. We suggest how our formulas might be used to study reality and primeness of modules, and compatibility of cluster variables.

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arXiv2019

A note on cluster automorphism groups

We conjecture a characterization of a cluster automorphism as an algebra homomorphism from the cluster algebra to itself that restricts to a bijection between two clusters. This formulation does not require that the map commutes with mutations as in the original definition of cluster automorphisms. We prove the conjecture in the case where at least one of the two clusters is bipartite.

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