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Shengfei Geng

Publications and source records attributed to Shengfei Geng.

At least 19 recordsLinked to original sources
arXiv2026

Geometric models for endomorphism algebras of tilting modules over gentle algebras

This paper investigates tilting modules over gentle algebras and their endomorphism algebras within the framework of marked surfaces and tilings introduced by Baur and Sim\~{o}es. Faithful dissections of a tiling are shown to correspond to tilting modules. For a faithful dissection, we define an auxiliary algebra and prove that it is isomorphic to the endomorphism algebra of the corresponding tilting module. We also construct a new tiling realizing this endomorphism algebra and introduce a flip preserving tilting modules.

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arXiv2025

Intersection vectors over skew-tilings

We prove that under a mild condition, a multiset of tagged permissible arcs over a skew-tiling is uniquely determined by its intersection vector. As an application, it is proved that -- up to isomorphism -- different $\tau$-rigid modules over a skew-gentle algebra $A$ arising from a skew-triple $(Q,Sp,I)$ have different dimension vectors if and only if $(Q,I)$ has no minimal oriented cycle of even-length with full zero relations. This generalizes a recent work of Fu-Geng for gentle algebras.

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arXiv2025

Functor-induced isomorphisms and $G$-matrices

In this paper, we explore how functor-induced isomorphisms are encoded by $G$-matrices. We first show that the Grothendieck group isomorphism induced by a tilting module can be realized via the $G$-matrix of this tilting module. Building on this, we compare $g$-vectors for a tilted algebra and its associated hereditary algebra, and provide $G$-matrix interpretations of the Coxeter transformation, the Nakayama functor, and the Auslander-Reiten translation for suitable algebras. Furthermore, we demonstrate that every element of any symmetric group and Weyl group can be expressed as the transpose of the $G$-matrix of some tilting module or support $\tau$-tilting module. Finally, we show that the Grothendieck group isomorphism induced by a $2$-term silting complex can also be realized via the $G$-matrix of this $2$-term silting complex.

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arXiv2024

On denominator conjecture for cluster algebras of finite type

We continue our investigation on denominator conjecture of Fomin and Zelevinsky for cluster algebras via geometric models initialed in \cite{FG22}. In this paper, we confirm the denominator conjecture for cluster algebras of finite type. The new contribution is a proof of this conjecture for cluster algebras of type $\mathbb{D}$ and an algorithm for the exceptional types. For the type $\mathbb{D}$ cases, our approach involves geometric model provided by discs with a puncture. By removing the puncture or changing the puncture to an unmarked boundary component, this also yields an alternative proof for the denominator conjecture of cluster algebras of type $\mathbb{A}$ and $\mathbb{C}$ respectively.

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arXiv2024

Denominator conjecture for some surface cluster algebras

The denominator conjecture, proposed by Fomin and Zelevinsky, says that for a cluster algebra, the cluster monomials are uniquely determined by their denominator vectors with respect to an initial cluster. In this paper, for a cluster algebra from a marked surface with at least three boundary marked points, we establish this conjecture with respect to a given strong admissible tagged triangulation.

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arXiv2024

On $\tau$-tilting graphs for quasi-silted algebras

We prove that the $\tau$-tilting graph of any quasi-silted algebra is connected and has the reachable-in-face property. Our approach utilizes $\tau$-reduction and wall and chamber structures. In particular, we observe a sufficient condition on the wall and chamber structure under which the connectivity of $\tau$-tilting graphs is preserved under taking quotients of algebras. As an immediate consequence, the connectivity of $\tau$-tilting graphs is also established for several new classes of algebras.

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arXiv2023

Intersection vectors over tilings with applications to gentle algebras and cluster algebras

It is proved that a multiset of permissible arcs over a tiling is uniquely determined by its intersection vector under a mild condition. This generalizes a classical result over marked surfaces with triangulations. We apply this result to study $τ$-tilting theory of gentle algebras and denominator conjecture in cluster algebras. In the case of gentle algebras, it is proved that different $τ$-rigid $A$-modules over a gentle algebra $A$ have different dimension vectors if and only if $A$ has no even oriented cycle with full relations. For cluster algebras, the denominator conjecture has been established for cluster algebras of type $\mathbb{A}\mathbb{B}\mathbb{C}$.

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arXiv2021

On support $τ$-tilting graphs of gentle algebras

Let $A$ be a finite-dimensional gentle algebra over an algebraically closed field. We investigate the combinatorial properties of support $τ$-tilting graph of $A$. In particular, it is proved that the support $τ$-tilting graph of $A$ is connected and has the so-called reachable-in-face property. This property was conjectured by Fomin and Zelevinsky for exchange graphs of cluster algebras which was recently confirmed by Cao and Li.

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arXiv2021

On maximal green sequence for quivers arising from weighted projective lines

We investigate the existence and non-existence of maximal green sequences for quivers arising from weighted projective lines. Let $Q$ be the Gabreil quiver of the endomorphism algebra of a basic cluster-tilting object in the cluster category $\mathcal{C}_\mathbb{X}$ of a weighted projective line $\mathbb{X}$. It is proved that there exists a quiver $Q'$ in the mutation equivalence class $\operatorname{Mut}(Q)$ such that $Q'$ admits a maximal green sequence. On the other hand, there is a quiver in $\operatorname{Mut}(Q)$ which does not admit a maximal green sequence if and only if $\mathbb{X}$ is of wild type.

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arXiv2020

Cluster algebras arising from cluster tubes II: the Caldero-Chapoton map

We continue our investigation on cluster algebras arising from cluster tubes. Let $\mathcal{C}$ be a cluster tube of rank $n+1$. For an arbitrary basic maximal rigid object $T$ of $\mathcal{C}$, one may associate a skew-symmetrizable integer matrix $B_T$ and hence a cluster algebra $\mathcal{A}(B_T)$ to $T$. We define an analogue Caldero-Chapoton map $\mathbb{X}_M^T$ for each indecomposable rigid object $M\in \mathcal{C}$ and prove that $\mathbb{X}_?^T$ yields a bijection between the indecomposable rigid objects of $\mathcal{C}$ and the cluster variables of the cluster algebra $\mathcal{A}(B_T)$. The construction of the Caldero-Chapoton map involves Grassmanians of locally free submodules over the endomorphism algebra of $T$. We also show that there is a non-trivial $\mathbb{C}^{\times}$-action on the Grassmanians of locally free submodules, which is of independent interest.

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arXiv2020

Cluster algebras arising from cluster tubes I: integer vectors

We study cluster algebras arising from cluster tubes. We obtain categorical interpretations for $g$-vectors, $c$-vectors and denominator vectors for cluster algebras of type $\mathrm{C}$ with respect to arbitrary initial seeds. In particular, a denominator theorem has been proved, which enables us to establish the linearly independence of denominator vectors of cluster variables from the same cluster for cluster algebras of type $\mathrm{A}\mathrm{B}\mathrm{C}$. This strengthens the link between cluster tubes and cluster algebras of type $\mathrm{C}$ initiated by Buan, Marsh and Vatne.

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arXiv2020

On cluster categories of weighted projective lines with at most three weights

Let $\mathbb{X}$ be a weighted projective line and $\mathcal{C}_\mathbb{X}$ the associated cluster category. It is known that $\mathcal{C}_\mathbb{X}$ can be realized as a generalized cluster category of quiver with potential. In this note, under the assumption that $\mathbb{X}$ has at most three weights or is of tubular type, we prove that if the generalized cluster category $\mathcal{C}_{(Q,W)}$ of a Jacobi-finite non-degenerate quiver with potential $(Q,W)$ shares a $2$-CY tilted algebra with $\mathcal{C}_\mathbb{X}$, then $\mathcal{C}_{(Q,W)}$ is triangle equivalent to $\mathcal{C}_\mathbb{X}$. As a byproduct, a $2$-CY tilted algebra of $\mathcal{C}_\mathbb{X}$ is determined by its quiver provided that $\mathbb{X}$ has at most three weights. To this end, for any weighted projective line $\mathbb{X}$ with at most three weights, we also obtain a realization of $\mathcal{C}_\mathbb{X}$ via Buan-Iyama-Reiten-Scott's construction of $2$-CY categories arising from preprojective algebras.

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arXiv2019

Mutation of tilting bundles of tubular type

Let $\mathbb{X}$ be a weighted projective line of tubular type and $\operatorname{coh}\mathbb{X}$ the category of coherent sheaves on $\mathbb{X}$. The main purpose of this note is to show that the subgraph of the tilting graph consisting of all basic tilting bundles in $\operatorname{coh}\mathbb{X}$ is connected. This yields an alternative proof for the connectedness of the tilting graph of $\operatorname{coh}\mathbb{X}$. Our approach leads to the investigation of the change of slopes of a tilting sheaf in $\operatorname{coh}\mathbb{X}$ under (co-)APR mutations, which may be of independent interest.

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arXiv2018

On cluster-tilting graphs for hereditary categories

Let $\mathcal{H}$ be a connected hereditary abelian category with tilting objects. It is proved that the cluster-tilting graph associated with $\mathcal{H}$ is always connected. As a consequence, we establish the connectedness of the tilting graph for the category $\operatorname{coh}\mathbb{X}$ of coherent sheaves over a weighted projective line $\mathbb{X}$ of wild type. The connectedness of tilting graphs for such categories was conjectured by Happel-Unger, which has immediately applications in cluster algebras. For instance, we deduce that there is a bijection between the set of indecomposable rigid objects of the cluster category $\mathcal{C}_{\mathbb{X}}$ of $\operatorname{coh}\mathbb{X}$ and the set of cluster variables of the cluster algebra $\mathcal{A}_{\mathbb{X}}$ associated with $\operatorname{coh}\mathbb{X}$.

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arXiv2018

Tilting modules and support $τ$-tilting modules over preprojective algebras associated with symmetrizable Cartan matrices

For any given symmetrizable Cartan matrix $C$ with a symmetrizer $D$, Geiß~ et al. (2016) introduced a generalized preprojective algebra $Π(C, D)$. We study tilting modules and support $τ$-tilting modules for the generalized preprojective algebra $Π(C, D)$ and show that there is a bijection between the set of all cofinite tilting ideals of $Π(C,D)$ and the corresponding Weyl group $W(C)$ provided that $C$ has no component of Dynkin type. When $C$ is of Dynkin type, we also establish a bijection between the set of all basic support $τ$-tilting $Π(C,D)$-modules and the corresponding Weyl group $W(C)$. These results generalize the classification results of Buan et al. (Compos. Math. 145(4), 1035-1079, 2009) and Mizuno (Math. Zeit. 277(3), 665-690, 2014) over classical preprojective algebras.

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arXiv2018

Relative rigid objects in triangulated categories

Let $\mathcal{T}$ be a Krull-Schmidt, Hom-finite triangulated category with suspension functor $[1]$. Let $R$ be a basic rigid object, $Γ$ the endomorphism algebra of $R$, and $\operatorname{\mathsf{pr}}(R)\subseteq \mathcal{T}$ the subcategory of objects finitely presented by $R$. We investigate the relative rigid objects, \ie $R[1]$-rigid objects of $\mathcal{T}$. Our main results show that the $R[1]$-rigid objects in $\operatorname{\mathsf{pr}}(R)$ are in bijection with $τ$-rigid $Γ$-modules, and the maximal $R[1]$-rigid objects with respect to $\operatorname{\mathsf{pr}}(R)$ are in bijection with support $τ$-tilting $Γ$-modules. We also show that various previously known bijections involving support $τ$-tilting modules are recovered under respective assumptions.

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