We introduce a generalization of representations of quivers that contains also representations of posets, vectorspace problems and other matrix problems. Many examples, some of which are given in the paper, show that the language of marked quivers is rather convenient and that the notion of their representations can probably be considered as basic for a general theory of matrix problems. Some historical comments from my personal point of view are added in the Appendix.

2001-12-15math.CTmath.RAmath.RT

带超势的箭图与带标记曲面的$d$-角剖分

The quiver with superpotentials of a $d$-angulation of a marked surface

本文为带标记曲面的$d$-角剖分构造带超势的箭图，证明翻转与Oppermann突变相容，并推广了$(d-2)$-丛范畴中补对象的计数结果。

2025-10-23math.RAmath.RT

Mutation classes of certain quivers with potentials as derived equivalence classes

We characterize the marked bordered unpunctured oriented surfaces with the property that all the Jacobian algebras of the quivers with potentials arising from their triangulations are derived equivalent. These are either surfaces of genus g with b boundary components and one marked point on each component, or the disc with 4 or 5 points on its boundary. We show that for each such marked surface, all the quivers in the mutation class have the same number of arrows, and the corresponding Jacobian algebras constitute a complete derived equivalence class of finite-dimensional algebras whose member…

2011-02-20math.COmath.RT

Maximal green sequences for quivers of finite mutation type

In general, the existence of a maximal green sequence is not mutation invariant. In this paper we show that it is in fact mutation invariant for cluster quivers of finite mutation type. In particular, we show that a mutation finite cluster quiver has a maximal green sequence unless it arises from a once-punctured closed marked surface, or one of the two quivers in the mutation class of X7. We develop a procedure to explicitly find maximal green sequences for cluster quivers associated to arbitrary triangulations of closed marked surfaces with at least two punctures. As a corollary, it follows …

2016-06-13math.COmath.RT

Singular locally-scalar representations of quivers in Hilbert spaces and separating functions

A numeric function $ρ$: $ρ(k)=1+\frac{k-1}{k+1}, k \in N$ was considered in [1]. In its terms criterions of finite representability and tameness of marked quivers, posets with equivalence and dyadic posets can be obtained; Dynkin schemes and extended schemes also can be characterized. In this paper authors consider the connection of function $ρ$ with locally-scalar representations [2] of extended Dynkin graphs. Then a family of functions $ρ_n$ is defined -- a generalization of function $ρ$, which plays an analogous part for more wide class of graphs. Also some properties of functions $ρ$ and $…

2003-09-07math.COmath.RT

Which mutation classes of quivers have constant number of arrows?

We classify the connected quivers with the property that all the quivers in their mutation class have the same number of arrows. These are the ones having at most two vertices, or the ones arising from triangulations of marked bordered oriented surfaces of two kinds: either surfaces with non-empty boundary having exactly one marked point on each boundary component and no punctures, or surfaces without boundary having exactly one puncture. This combinatorial property has also a representation-theoretic counterpart: to each such quiver there is a naturally associated potential such that the Jaco…

2011-04-03math.COmath.RT

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Decorated marked surfaces III: The derived category of a decorated marked surface

We study the Ginzburg dg algebra $Γ_\mathbf{T}$ associated to the quiver with potential arising from a triangulation $\mathbf{T}$ of a decorated marked surface $\mathbf{S}_\bigtriangleup$, in the sense of Qiu. We show that there is a canonical way to identify all finite dimensional derived categories $\mathcal{D}_{fd}(Γ_\mathbf{T})$, denoted by $\mathcal{D}_{fd}(\mathbf{S}_\bigtriangleup)$. As an application, we show that the spherical twist group $\operatorname{ST}(\mathbf{S}_\bigtriangleup)$ associated to $\mathcal{D}_{fd}(\mathbf{S}_\bigtriangleup)$ acts faithfully on its space of stability…

2018-03-31math.RAmath.RT

Maximal Green Sequences for Cluster Algebras Associated to the Orientable Surfaces of Genus n with Arbitrary Punctures

It is well known that any triangulation of a marked surface produces a quiver. In this paper we will provide a triangulation for orientable surfaces of genus $n$ with an arbitrary number interior marked points (called punctures) whose corresponding quiver has a maximal green sequence.

2015-09-28math.COmath.QAmath.RT

Quivers with potentials associated to triangulated surfaces, Part II: Arc representations

This paper is a representation-theoretic extension of Part I. It has been inspired by three recent developments: surface cluster algebras studied by Fomin-Shapiro-Thurston, the mutation theory of quivers with potentials initiated by Derksen-Weyman-Zelevinsky, and string modules associated to arcs on unpunctured surfaces by Assem-Brustle-Charbonneau-Plamondon. Modifying the latter construction, to each arc and each ideal triangulation of a bordered marked surface we associate in an explicit way a representation of the quiver with potential constructed in Part I, so that whenever two ideal trian…

2009-11-19math.RT

On Jacobian algebras from closed surfaces

We show that the quivers with potentials associated to ideal triangulations of marked surfaces with empty boundary are not rigid, and their completed Jacobian algebras are finite-dimensional and symmetric.

2012-07-16math.COmath.RAmath.RT

带势箭图导出的 Calabi-Yau 范畴的 Verdier 商

Verdier quotients of Calabi-Yau categories from quivers with potential

Università di Verona（维罗纳大学）；Tsinghua University（清华大学）；Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）

研究由带势箭图（来自装饰标记曲面）组合描述的 3-Calabi-Yau 范畴的 Verdier 商三角范畴，分析其有界 t-结构，并证明心与倾斜对象的交换图之间的 Koszul 同构在 Verdier 商下保持。

2026-07-01math.RT

Arc representations

This paper was inspired by four articles: surface cluster algebras studied by Fomin-Shapiro-Thurston \cite{fst}, the mutation theory of quivers with potentials initiated by Derksen-Weyman-Zelevinsky \cite{dwz}, string modules associated to arcs on unpunctured surfaces by Assem-Br$\ddot{u}$stle-Charbonneau-Plamondon \cite{acbp} and Quivers with potentials associated to triangulated surfaces, part II: Arc representations by Labardini-Fragoso. \cite{lf2}. For a surface with marked points ($Σ,M$) Labardini-Fragoso associated a quiver with potential $(Q(τ),S(τ))$ then for an ideal triangulation of …

2017-09-25math.RT

矩阵不变量的分离簇

The separating variety for matrix invariants

本文研究一般线性群通过同时共轭作用于矩阵元组时的分离簇，通过构造组合偏序集刻画其不可约分支，确定其维数与子维数，并据此证明不变量代数不含多项式或超曲面分离集，从而在两个特殊箭图情形下证明作者此前提出的猜想。

2025-08-19math.ACmath.COmath.RT

PAGR：证明承载的代数几何检索——面向接地LLM检索的箭图、溯源与层论框架

PAGR: Proof-Carrying Algebraic-Geometric Retrieval: A Quiver-, Provenance-, and Sheaf-Theoretic Framework for Grounded LLM Retrieval

Louisiana State University（路易斯安那州立大学）

PAGR通过箭图、层论和半环溯源分离知识认证、表示学习与多跳组合，确保学习组件不提升为ground truth，提供不变性、完备性等数学保证。

2026-09-09cs.AIcs.IRmath.RT

From groups to clusters

We construct a new class of symmetric algebras of tame representation type that are also the endomorphism algebras of cluster tilting objects in 2-Calabi-Yau triangulated categories, hence all their non-projective indecomposable modules are $Ω$-periodic of period dividing 4. Our construction is based on the combinatorial notion of triangulation quivers, which arise naturally from triangulations of oriented surfaces with marked points. This class of algebras contains the algebras of quaternion type introduced and studied by Erdmann with relation to certain blocks of group algebras. On the other…

2016-08-01math.RAmath.RT

Cluster exchange groupoids and framed quadratic differentials

We introduce the cluster exchange groupoid associated to a non-degenerate quiver with potential, as an enhancement of the cluster exchange graph. In the case that arises from an (unpunctured) marked surface, where the exchange graph is modelled on the graph of triangulations of the marked surface, we show that the universal cover of this groupoid can be constructed using the covering graph of triangulations of the surface with extra decorations. This covering graph is a skeleton for a space of suitably framed quadratic differentials on the surface, which in turn models the space of Bridgeland …

2019-10-16math.AGmath.DSmath.GT

Generic Caldero-Chapoton functions with coefficients and applications to surface cluster algebras

We realize Derksen-Weyman-Zelevinsky's mutations of representations as densely-defined regular maps on representation spaces, and study the generic values of Caldero-Chapoton functions with coefficients, giving, for instance, a sufficient combinatorial condition for their linear independence. For a quiver with potential $(Q,S)$, we show that if $k$ is a vertex not incident to any oriented 2-cycle, then every generically $τ$-reduced irreducible component $Z$ of any affine variety of (decorated) representations has a dense open subset $U$ on which Derksen-Weyman-Zelevinsky's mutation of represen…

2020-07-10math.RAmath.RT

Coxeter-Dynkin代数的标准类型

Coxeter-Dynkin algebras of canonical type

Paderborn University（帕德博恩大学）

本文提出Coxeter-Dynkin代数的标准类型定义，并通过构造倾斜对象证明其终止代数与该代数一致，同时展示了其与Saito分类的联系以及在domestic情况下有限表示类型的结论。

2026-08-28math.RT