Aslak Bakke Buan, Hugh Thomas
We define mutation on coloured quivers associated to tilting objects in higher cluster categories. We show that this operation is compatible with the mutation operation on the tilting objects. This gives a combinatorial approach to tilting in higher cluster categories and especially an algorithm to determine the Gabriel quivers of tilting objects in such categories.
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We use a geometric description of $m$-cluster categories of Dynkin type $A$ to count the the number of coloured quivers in the $m$-mutation class of quivers of Dynkin type $A$. This is related to angulations of polygons and the cell-growth problem.
The coloured mutation class of $\mathbb{D}_n$- quivers and their application to $m$-cluster tilted algebras
本文通过组合方法明确描述了$\mathbb{D}_n$型有向图的$m$-着色突变类中的$m$-着色有向图，进而推导出$\mathbb{D}_n$型$m$-聚类倾斜代数的Gabriel有向图，并扩展了Vatne的结果。
We consider the general notion of coloured quiver mutation and show that the mutation class of a coloured quiver $Q$, arising from an $m$-cluster tilting object associated with $H$, is finite if and only if $H$ is of finite or tame representation type, or it has at most 2 simples. This generalizes a result known for 1-cluster categories.
Quiver mutation plays a crucial role in the definition of cluster algebras by Fomin and Zelevinsky. It induces an equivalence relation on the set of all quivers without loops and two-cycles. A quiver is called mutation-acyclic if it is mutation-equivalent to an acyclic quiver. This note gives a proof that full subquivers of mutation-acyclic quivers are mutation-acyclic.
We give an explicit description of the mutation class of quivers of type D.
Mutation-acyclic quivers are totally proper
本文证明了每个突变无环箭图都是全真的，从而为突变无环性提供了新的必要条件，并推广了马尔可夫不变量至所有突变无环箭图。
The coloured mutation class of $\mathbb{A}_n$ quivers
本文针对$\boldsymbol{\rm A}_n$型箭图的$m$着色突变类中的$m$着色箭图给出明确纯组合刻画，推广了Fomin-Zelevinsky箭图突变及Buan-Vatne的$m=1$情形结果。
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 …
We show that the number of quivers in the mutation class of a quiver of Dynkin type $D_n$ is given by $\sum_{d|n} ϕ(n/d)\binom{2d}{d}/(2n)$ for $n \geq 5$. To obtain this formula, we give a correspondence between the quivers in the mutation class and certain rooted trees.
Mutations of quivers with 2-cycles
本文提出了一种针对具有2环的有向图的突变理论，通过同调结构扩展了Fomin-Zelevinsky模型，并证明了翻转与突变之间的对应关系。
Unique mutation cycles of nucleus-extension quivers
本文证明了Fomin和Neville关于核扩展箭图长突变环的唯一性猜想，通过分析三种突变状态和子箭图，证明了每个核扩展箭图在带标签突变图中具有唯一简单环。
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…
An explicit description of the colored mutation class of $\widetilde{\mathbb{A}}_n$-quivers
本文通过显式组合方法描述了$\widetilde{\mathbb{A}}_n$型有向图的$m$-着色突变类结构，并推广了Bastian [1]的成果。
Reddening sequences and mutation of infinite quivers
本文提出了一种纯组合框架，用于无限有向图的突变，扩展了红化序列的概念，并统一了拓扑和组合视角。
In this article we prove explicit formulae for the number of non-isomorphic cluster-tilted algebras of type \tilde{A}_n in the derived equivalence classes. In particular, we obtain the number of elements in the mutation classes of quivers of type \tilde{A}_n. As a by-product, this provides an alternative proof for the number of quivers of Dynkin type D_n which was first determined by Buan and Torkildsen.
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…
We associate a coloured quiver to a rigid object in a Hom-finite 2-Calabi--Yau triangulated category and to a partial triangulation on a marked (unpunctured) Riemann surface. We show that, in the case where the category is the generalised cluster category associated to a surface, the coloured quivers coincide. We also show that compatible notions of mutation can be defined and give an explicit description in the case of a disk. A partial description is given in the general 2-Calabi-Yau case. We show further that Iyama-Yoshino reduction can be interpreted as cutting along an arc in the surface.
We give an explicit description of the mutation classes of quivers of type \tilde{A}_n. Furthermore, we provide a complete classification of cluster tilted algebras of type \tilde{A}_n up to derived equivalence. We show that the bounded derived category of such an algebra depends on four combinatorial parameters of the corresponding quiver.
A quiver is an oriented graph. Quiver mutation is an elementary operation on quivers. It appeared in physics in Seiberg duality in the nineties and in mathematics in the definition of cluster algebras by Fomin-Zelevinsky in 2002. We show, for large classes of quivers Q, using quiver mutation and quantum dilogarithms, one can construct the combinatorial DT-invariant, a formal power series intrinsically associated with Q. When defined, it coincides with the "total" Donaldson-Thomas invariant of Q (with a generic potential) provided by algebraic geometry (work of Joyce, Kontsevich-Soibelman, Szen…
We give a combinatorial mutation rule for Aihara's and Iyama's silting mutation. As an application, we reprove Keller-Yang's mutation rule for Ginzburg algebras, and obtain an analog of that rule for arbitrary dimension.
On The Mutations Loops of Valued Quivers
本文研究带值箭图的突变环，特别是仅由突变组成的全局突变环，基于突变类的权重和箭图形状，确定了哪些箭图具有此类环并给出构造。
Important objects of study in $τ$-tilting theory include the $τ$-tilting pairs over an algebra on the form $kQ/I$, with $kQ$ being a path algebra and $I$ an admissible ideal. In this paper, we study aspects of the combinatorics of mutation quivers of support $τ$-tilting pairs, simply called mutation quivers. In particular, we are interested in identifying connected components of the underlying graphs of such quivers. We give a class of algebras with two simple modules such that every algebra in the class has at most two connected components in its mutation quiver, generalizing a result by Demo…
For a quiver with non-degenerate potential, we study the associated stability scattering diagram and how it changes under mutations. We show that under mutations the stability scattering diagram behaves like the cluster scattering diagram associated to the same quiver, which gives them identical cluster chamber structures. As an application, we prove if the quiver has a green-to-red sequence, these two scattering diagrams are identical and if the quiver comes from a once-punctured torus, they differ by a central wall-crossing. This verifies in those cases a conjecture of Kontsevich-Soibelman t…
On the Mutations of Gentle Quivers and Admissible Ideals
本文将Fomin-Zelevinsky箭图突变框架推广至带关系的箭图，通过在路径代数可许理想上引入对合突变操作，构建生成可许温和有界箭图代数的系统框架，并提出相关未来研究方向。
Mutation Of Matrices Over Group Rings
本文研究了群环上斜对称可化矩阵的突变定义及其与具有对称性的有向无环图的突变关系，并提出了新的突变规则。
A maximal green sequence introduced by B. Keller is a certain sequence of quiver mutations at green vertices. T. Brüstle, G. Dupont and M. Pérotin showed that for an acyclic quiver, maximal green sequences are realized as maximal paths in the Hasse quiver of the poset of support tilting modules. They also considered possible lengths of maximal green sequences. In this paper, we calculate possible lengths of maximal green sequences for a quiver of type $A$ or of type $\tilde{A}_{n,1}$ by using theory of tilting mutation.
We study silting mutations (Okuyama-Rickard complexes) for selfinjective algebras given by quivers with potential (QPs). We show that silting mutation is compatible with QP mutation. As an application, we get a family of derived equivalences of Jacobian algebras.
We introduce a new method for producing both maximal green and reddening sequences of quivers. The method, called component preserving mutations, generalizes the notion of direct sums of quivers and can be used as a tool to both recover known reddening sequences as well as find reddening sequences that were previously unknown. We use the method to produce and recover maximal green sequences for many bipartite recurrent quivers that show up in the study of periodicity of $T$-systems and $Y$-systems. Additionally, we show how our method relates to the dominance phenomenon recently considered by …
A quiver mutation loop is a sequence of mutations and vertex relabelings, along which a quiver transforms back to the original form. For a given mutation loop, we introduce a quantity called a partition q-series. The partition q-series are invariant under pentagon moves. If the quivers are of Dynkin type or square products thereof, they reproduce so-called parafermionic or quasi-particle character formulas of certain modules associated with affine Lie algebras. They enjoy nice modular properties as expected from the conformal field theory point of view.
Boundedness criteria for real quivers of rank 3
本文研究秩为3的实淬atum的有界性准则，通过特征化实淬atum的有界突变类来确定其有界性条件。
In a recent paper by K.-H. Lee, K. Lee and M. Mills, a mutation of reflections in the universal Coxeter group is defined in association with a mutation of a quiver. A matrix representation of these reflections is determined by a linear ordering on the set of vertices of the quiver. It was conjectured that there exists an ordering (called a pseudo-acyclic ordering in this paper) such that whenever two mutation sequences of a quiver lead to the same labeled seed, the representations of the associated reflections also coincide. In this paper, we prove this conjecture for every quiver mutation-equ…
We study the quiver with relations of the endomorphism algebra of an APR tilting module. We give an explicit description of the quiver with relations by graded quivers with potential (QPs) and mutations. The result also implies that mutations of QPs with algebraic cuts induce tilting mutations between the associated truncated Jacobian algebras. Consequently, mutations of QPs provide a rich source of derived equivalence classes of algebras. As an application, we give a sufficient condition of QPs such that Derksen-Weyman-Zelevinsky's question has a positive answer.
We show that the mutation class of a finite quiver without oriented cycles is finite if and only is the quiver is either Dynkin, extended Dynkin or has at most two vertices.
Monodromy of plane curve singularities and quiver mutation
该研究证明可形变实Morsification的箭图突变类唯一确定平面曲线奇点的整单值性模，建立了Fomin等人猜想在可分不可约情形下的代数到拓扑蕴含关系。
Categorification of ice quiver mutation
本文基于Pressland的带势冰箭图突变规则，证明了相关相对Ginzburg代数间存在导出等价，并给出了冻结点突变的范畴化。
Categorifications of Non-Integer Quivers: Type $I_2(2n)$
本文通过加权展开方法构造类型 $I_2(2n)$ 箭图突变的范畴化，并由此将箭图突变范畴化推广到所有有限类型。
Matrix mutation appears in the definition of cluster algebras of Fomin and Zelevinsky. We give a representation theoretic interpretation of matrix mutation, using tilting theory in cluster categories of hereditary algebras. Using this, we obtain a representation theoretical interpretation of cluster mutation in case of acyclic cluster algebras of finite type.
Mutation of signed valued quivers and presentations of simple complex Lie algebras
本文提出带符号估值有向图的突变规则，并通过构造李代数的方式呈现了简单复李代数的结构。
Quiver quantum mechanics is invariant under Seiberg duality. A mathematical consequence is that the cohomology of the Higgs branch moduli space is invariant under mutations of the quiver. The Coulomb branch formula, on the other hand, conjecturally expresses the Poincaré / Dolbeault polynomial of the Higgs branch moduli space in terms of certain quantities known as single-centered indices. In this work we determine the transformations of these single-centered indices under mutations. Moreover, we generalize these mutations to quivers whose nodes carry single-centered indices different from uni…
Machine Learning Mutation-Acyclicity of Quivers
本文用机器学习方法判定4顶点箭图的突变无环性，通过计算机辅助证明和神经网络、支持向量机实现高效分类，为组合问题提供新途径。
We describe presentations of braid groups of type ADE and show how these presentations are compatible with mutation of quivers, building on work of Barot and Marsh for Coxeter groups. In types A and D these presentations can be understood geometrically using triangulated surfaces. We then give a categorical interpretation of the presentations, with the new generators acting as spherical twists at simple modules on derived categories of Ginzburg dg-algebras of quivers with potential.
We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the vertices, and give families of quivers which have higher periodicity. The periodicity means that sequences given by recurrence relations arise in a natural way from the associated cluster algebras. We present a number of interesting new families of non-linear recurrences, necessarily with the Laurent property, of both the real line and the plane, containing integrable maps as special cases. In particu…
We show that Derksen-Weyman-Zelevinsky's mutations of quivers with potential yield equivalences of suitable 3-Calabi-Yau triangulated categories. Our approach is related to that of Iyama-Reiten and Koszul dual to that of Kontsevich-Soibelman. It improves on previous work by Vitoria. In the appendix, the first-named author studies pseudo-compact derived categories of certain pseudo-compact dg algebras.
In this survey article we give a brief account of constructions and results concerning the quivers with potentials associated to triangulations of surfaces with marked points. Besides the fact that the mutations of these quivers with potentials are compatible with the flips of triangulations, we mention some recent results on the representation type of Jacobian algebras and the uniqueness of non-degenerate potentials. We also mention how the the quivers with potentials associated to triangulations give rise to CY2 and CY3 triangulated categories that have turned out to be useful in the subject…
We study quivers with relations given by non-commutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representation-theoretic interpretation of quiver mutations at arbitrary vertices. This gives a far-reaching generalization of Bernstein-Gelfand-Ponomarev reflection functors. The motivations for this work come from several sources: superpotentials in physics, Calabi-Yau algebras, cluster algebras.
Given a framed quiver, i.e. one with a frozen vertex associated to each mutable vertex, there is a concept of green mutation, as introduced by Keller. Maximal sequences of such mutations, known as maximal green sequences, are important in representation theory and physics as they have numerous applications, including the computations of spectrums of BPS states, Donaldson-Thomas invariants, tilting of hearts in the derived category, and quantum dilogarithm identities. In this paper, we study such sequences and construct a maximal green sequence for every quiver mutation-equivalent to an orienta…
Seiberg Duality conjecture for star-shaped quivers and finiteness of Gromov-Witten thoery for D-type quivers
本文证明星形箭图的Seiberg对偶猜想成立，并进一步证明D_3型箭图簇代数簇的Gromov-Witten理论及Kähler变量在有限次突变后恢复原状。
We present a graded mutation rule for quivers of cluster-tilted algebras. Furthermore, we give a technique to recover a cluster-tilting object from its graded quiver in the cluster category of coh $\mathbb{X}$.
Mutation of $τ$-exceptional sequences for acyclic quivers over local algebras
本文证明局部代数上箭图的 $\tau$-例外序列突变等价于经典例外序列突变，并由此得出辫群在完备 $\tau$-例外序列上传递作用。
In this paper, we study the distribution of the genuses of cluster quivers of finite mutation type. First, we prove that in the $11$ exceptional cases, the distribution of genuses is $0$ or $1$. Next, we consider the relationship between the genus of an oriented surface and that of cluster quivers from this surface. It is verified that the genus of an oriented surface is an upper bound for the genuses of cluster quivers from this surface. Furthermore, for any non-negative integer $n$ and a closed oriented surface of genus $n$, we show that there always exist a set of punctures and a triangulat…
This is an extended abstract of my talk at the Oberwolfach Workshop "Algebraic Groups" (April 22 - 28, 2007). It is based on a joint work with H.Derksen and J.Weyman (arXiv:0704.0649v2 [math.RA]).
Entropy of cluster DT transformations and the finite-tame-wild trichotomy of acyclic quivers
本文以簇 DT 变换 τ 的符号稳定性和簇拉伸因子刻画无环箭图的有限–驯–野三分法，并计算相关变异环熵，证明其代数与范畴熵由 Coxeter 矩阵谱半径对数统一给出。
Continuous Quivers of Type A (IV) Continuous Mutation and Geometric Models of $\mathbf E$-clusters
本文作为A型连续箭图系列终篇，推广了A型簇结构的几何模型至$\boldsymbol{\rm E}$-簇，引入簇理论同构、弱等价及连续突变概念，构建其与几何模型的关联，并提出推广交换图的突变空间，开启连续A型簇结构的分类研究。
Symmetric mutations algebras in the context of sub-cluster algebras
本文研究赋值箭图上有根簇代数的对称突变异子代数，确定了该子代数与原簇代数相等的簇代数类别（含几乎所有有限突变型箭图），还基于权重分类了箭图突变类并给出子代数性质。
For any quiver mutation sequence, we define a pair of matrices that describe a fixed point equation of a cluster transformation determined from the mutation sequence. We give an explicit relationship between this pair of matrices and the Jacobian matrix of the cluster transformation. Furthermore, we show that this relationship reduces to a relationship between the pair of matrices and the $C$-matrix of the cluster transformation in a certain limit of cluster variables. As an application, we prove that quivers associated with once-punctured surfaces do not have maximal green or reddening sequen…
The theory of Caldero-Chapoton algebras of Cerulli-Irelli, Labardini-Fragoso and Schroer leads to a refinement of the notions of cluster variables and clusters, via so called component clusters. In this paper we compare component clusters to classical clusters for the cluster algebra of an acyclic quiver. We propose a definition of mutation between component clusters and determine the mutation relations of component clusters for affine quivers. In the case of a wild quiver, we provide bounds for the size of component clusters.
The exchange graph of a 2-acyclic quiver is the graph of mutation-equivalent quivers whose edges correspond to mutations. When the quiver admits a nondegenerate Jacobi-finite potential, the exchange graph admits a natural acyclic orientation called the oriented exchange graph, as shown by Brüstle and Yang. The oriented exchange graph is isomorphic to the Hasse diagram of the poset of functorially finite torsion classes of a certain finite dimensional algebra. We prove that lattices of torsion classes are semidistributive lattices, and we use this result to conclude that oriented exchange graph…
Tilting mutation is a way of producing new tilting complexes from old ones replacing only one indecomposable summand. In this paper, we give a purely combinatorial procedure for performing tilting mutation of suitable algebras. As an application, we recreate a result due to Ladkani, which states that the path algebra of a quiver shaped like a line (with certain relations) is derived equivalent to the path algebra of a quiver shaped like a rectangle. We will do this by producing an explicit series of tilting mutations going between the two algebras.
Inspirited by the importance of the spectral theory of graphs, we introduce the spectral theory of valued cluster quiver of a cluster algebra. Our aim is to characterize a cluster algebra via its spectrum so as to use the spectral theory as a tool. First, we give the relations between exchange spectrum of a valued cluster quiver and adjacency spectrum of its underlying valued graph, and between exchange spectra of a valued cluster quiver and its full valued subquivers. The key point is to find some invariants from the spectrum theory under mutations of cluster algebras, which is the second par…
Let $\tilde{Q}$ (resp. $Q$) be an extended exchange (resp. exchange) cluster quiver of finite mutation type. We introduce the distribution set of the number of arrows for $Mut[\tilde{Q}]$ (resp. $Mut[Q]$), give the maximum and minimum numbers of the distribution set and establish the existence of an extended complete walk (resp. a complete walk). As a consequence, we prove that the distribution set for $Mut[\tilde{Q}]$ (resp. $Mut[Q]$) is continuous except the exceptional cases. In case of cluster quivers $Q_{inf}$ of infinite mutation type, the number of arrows does not present a continuous d…
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