The mutation class of $D_n$ quivers

核扩展箭图的唯一突变环

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。

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…

$\widetilde{\mathbb{A}}_n$-有向图的着色突变类的显式描述

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。

$\mathbb{D}_n$型有向图的着色突变类及其在$m$-聚类倾斜代数中的应用

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。

红化序列和无限有向图的突变

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。

Counting the number of elements in the mutation classes of \tilde{A}_n-quivers

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.

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…

箭图突变等价的复杂性

Complexity of quiver mutation equivalence

该研究证明了两类箭图突变等价判定问题的NP-hard及强NP-hard难度，并给出了含两个可变顶点的箭图突变类的刻画，丰富了箭图突变理论的复杂性研究结果。

Mutation classes of \tilde{A}_n-quivers and derived equivalence classification of cluster tilted algebras of type \tilde{A}_n

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.

Quivers of finite mutation type and skew-symmetric matrices

In this paper, we study structural properties of finite mutation type quivers. In particular, we obtain a characterization of finite mutation type quivers that are associated with triangulations of surfaces and give a new numerical invariant for their mutation classes.

Quiver mutation and combinatorial DT-invariants

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…

Every finite acyclic quiver is a full subquiver of a quiver mutation equivalent to a bipartite quiver

We give a very short proof of the claim in the title.

Minimal mutation-infinite quivers

Quivers constructed from hyperbolic Coxeter simplices give examples of minimal mutation-infinite quivers, however they are not the only such quivers. We classify minimal mutation-infinite quivers through a number of moves and link the representatives of the classes with the hyperbolic Coxeter simplices, plus exceptional classes which are not related to simplices.

Maximal Green Sequences of Exceptional Finite Mutation Type Quivers

Maximal green sequences are particular sequences of mutations of quivers which were introduced by Keller in the context of quantum dilogarithm identities and independently by Cecotti-Córdova-Vafa in the context of supersymmetric gauge theory. The existence of maximal green sequences for exceptional finite mutation type quivers has been shown by Alim-Cecotti-Córdova-Espahbodi-Rastogi-Vafa except for the quiver $X_7$. In this paper we show that the quiver $X_7$ does not have any maximal green sequences. We also generalize the idea of the proof to give sufficient conditions for the non-existence …

Properties of minimal mutation-infinite quivers

We study properties of minimal mutation-infinite quivers. In particular we show that every minimal-mutation infinite quiver of at least rank 4 is Louise and has a maximal green sequence. It then follows that the cluster algebras generated by these quivers are locally acyclic and hence equal to their upper cluster algebra. We also study which quivers in a mutation-class have a maximal green sequence. For any rank 3 quiver there are at most 6 quivers in its mutation class that admit a maximal green sequence. We also show that for every rank 4 minimal mutation-infinite quiver there is a finite co…

群环上矩阵的突变

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。

Toric Mutations in the dP$_2$ Quiver and Subgraphs of the dP$_2$ Brane Tiling

Brane tilings are infinite, bipartite, periodic, planar graphs that are dual to quivers. In this paper, we examine the del Pezzo 2 (dP$_2$) quiver and its brane tiling, which arise from the physics literature, in terms of toric mutations on its corresponding cluster. Specifically, we give explicit formulas for all cluster variables generated by toric mutation sequences. Moreover, for each such variable, we associate a subgraph of the dP$_2$ brane tiling to it such that its weight matches the variable.

Remarks on lengths of maximal green sequences for quivers of type $\tilde{A}_{n,1}$

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.

Mutation-finite quivers with real weights

We classify all mutation-finite quivers with real weights. We show that every finite mutation class not originating from an integer skew-symmetrizable matrix has a geometric realization by reflections. We also explore the structure of acyclic representatives in finite mutation classes and their relations to acute-angled simplicial domains in the corresponding reflection groups.

Quiver mutation loops and partition q-series

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.

Exchange graphs for mutation-finite non-integer quivers of rank 3

Skew-symmetric non-integer matrices with real entries can be viewed as quivers with non-integer weights of arrows. One can mutate such quivers according to usual rules of quiver mutation. Felikson and Tumarkin show that rank 3 mutation-finite non-integer quivers admit geometric realisations by partial reflections. This allows to define a notion of seeds (as Y-seeds), and hence, to define the exchange graphs for mutation classes. In this paper we study exchange graphs of mutation-finite quivers in rank 3. The concept of finite and affine type generalises naturally to non-integer quivers. In par…

Building maximal green sequences via component preserving mutations

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 …

The existence of a maximal green sequence is not invariant under quiver mutation

This note provides a quiver which does not admit a maximal green sequence, but which is mutation-equivalent to a quiver which does admit a maximal green sequence. The proof uses the `scattering diagrams' of Gross-Hacking-Keel-Kontsevich to show that a maximal green sequence for a quiver determines a maximal green sequence for any induced subquiver.

秩为3的实淬atum的有界性准则

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。

Mutations of reflections and existence of pseudo-acyclic orderings for type $A_n$

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…

平面曲线奇点的单值性与箭图突变

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。

非整数箭图的范畴化：类型 $I_2(2n)$

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。

Cluster algebras associated with extended Dynkin quivers

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.

在有向图突变类的偏序集上的拓扑

A topology on the poset of quiver mutation classes

本文构建了有向图突变类的拓扑结构，揭示了突变不变和继承性质与偏序集下集的对应关系，并探讨了该空间的拓扑性质及未来研究方向。

箭图的机器学习突变无环性

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。

Coxeter groups, quiver mutations and geometric manifolds

We construct finite volume hyperbolic manifolds with large symmetry groups. The construction makes use of the presentations of finite Coxeter groups provided by Barot and Marsh and involves mutations of quivers and diagrams defined in the theory of cluster algebras. We generalize our construction by assigning to every quiver or diagram of finite or affine type a CW-complex with a proper action of a finite (or affine) Coxeter group. These CW-complexes undergo mutations agreeing with mutations of quivers and diagrams. We also generalize the construction to quivers and diagrams originating from u…

Cluster mutation-periodic quivers and associated Laurent sequences

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…

Quiver mutation sequences and $q$-binomial identities

In this paper, first we introduce a quantity called a partition function for a quiver mutation sequence. The partition function is a generating function whose weight is a $q$-binomial associated with each mutation. Then, we show that the partition function can be expressed as a ratio of products of quantum dilogarithms. This provides a systematic way of constructing various $q$-binomial multisum identities.

由Fork（叉）结构产生的新的遗传与突变不变性质

New Hereditary and Mutation-Invariant Properties Arising from Forks

本研究利用fork（叉）箭图类，提出两种新的箭图遗传与突变不变性质，并用初等方法证明其有效性，拓展了箭图突变理论。

Coloured quivers of type A and the cell-growth problem

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.

Geometry of mutation classes of rank $3$ quivers

We present a geometric realization for all mutation classes of quivers of rank $3$ with real weights. This realization is via linear reflection groups for acyclic mutation classes and via groups generated by $π$-rotations for the cyclic ones. The geometric behavior of the model turns out to be controlled by the Markov constant $p^2+q^2+r^2-pqr$, where $p,q,r$ are the elements of exchange matrix. We also classify skew-symmetric mutation-finite real $3\times 3$ matrices and explore the structure of acyclic representatives in finite and infinite mutation classes.

Reflection groups and quiver mutation: Diagrammatics

We extend Carter's notion of admissible diagrams and attach a "Dynkin-like" diagram to each reduced reflection factorization of an element in a finite Weyl group. We give a complete classification for the diagrams attached to reduced reflection factorizations. Remarkably, such a diagram turns out to be cyclically orientable if and only if it is isomorphic to the underlying graph of a quiver which is mutation-equivalent to a Dynkin quiver. Furthermore we show that each diagram encodes a natural presentation of the Weyl group as reflection group. The latter one extends work of Cameron, Seidel an…

On Maximal Green Sequences For Type A Quivers

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…

变异无限箭图$C$-向量与$C$-矩阵的几何

Geometry of $C$-vectors and $C$-Matrices for Mutation-Infinite Quivers

本文研究了变异无限箭图的$c$-向量几何，证明了保叉变异序列产生的$c$-向量满足特定二次方程，并将该结论推广至秩3变异循环箭图。

对无限箭头图的拓扑化及其突变

Topologizing infinite quivers and their mutations

研究无限箭头图的拓扑空间及其突变性质，证明两种空间与Baire空间同胚，并探讨无限突变序列的收敛性及特殊箭头图的对比行为。

由变红序列产生的变异循环

Mutation Cycles from Reddening Sequences

本文研究如何从两个具有变红序列的箭图构造大量变异循环，推广了已有长变异循环构造，并对某些变异无环箭图与叉形箭图的变红序列给出分类结果。

机器与数学突变：使用GNN刻画箭图突变类

Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes

本文利用图神经网络和AI可解释性技术，独立发现类型为$\ ilde{D}$的箭图突变等价准则，并展示模型能隐式学习类型$D$的已知准则，证明机器学习可提取数学中的抽象规则。

簇 DT 变换的熵与无环箭图的有限–驯–野三分法

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。

关于 $4\ imes 4$ 斜对称矩阵在箭图突变下的不变量

On the invariants of $4\times 4$ skew-symmetric matrix under cluster mutations

本文研究 4-quiver 在箭图突变下的不变量，证明其分段多项式且符号固定时为多项式的不变量仅由行列式决定。

Aztec Castles and the dP3 Quiver

Bipartite, periodic, planar graphs known as brane tilings can be associated to a large class of quivers. This paper will explore new algebraic properties of the well-studied del Pezzo 3 quiver and geometric properties of its corresponding brane tiling. In particular, a factorization formula for the cluster variables arising from a large class of mutation sequences (called $τ-$mutation sequences) is proven; this factorization also gives a recursion on the cluster variables produced by such sequences. We can realize these sequences as walks in a triangular lattice using a correspondence between …

Jacobian matrices of Y-seed mutations

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…

Quivers from non-orientable surfaces

We associate a quiver to a quasi-triangulation of a non-orientable marked surface and define a notion of quiver mutation that is compatible with quasi-cluster algebra mutation defined by Dupont and Palesi. Moreover, we use our quiver to show the unistructurality of the quasi-cluster algebra arising from the Mobius strip.

Universal quivers

We show that for any positive integer $n$, there exists a quiver $Q$ with $O(n^2)$ vertices and $O(n^2)$ edges such that any quiver on $n$ vertices is a full subquiver of a quiver mutation equivalent to $Q$. We generalize this statement to skew-symmetrizable matrices and obtain other related results. In particular, we show that any quiver is a full subquiver of a quiver mutation equivalent to a quiver of a plabic graph.

长突变循环

Long mutation cycles

本文研究了图中的突变循环，针对任意固定的 $n\ge 4$，构造了涉及 $n$ 顶点箭图的任意长突变循环，并证明这些循环无法被短突变循环铺砌。

Lattice Properties of Oriented Exchange Graphs and Torsion Classes

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…

Quiver mutations and Boolean reflection monoids

In 2010, Everitt and Fountain introduced the concept of reflection monoids. The Boolean reflection monoids form a family of reflection monoids (symmetric inverse semigroups are Boolean reflection monoids of type $A$). In this paper, we give a family of presentations of Boolean reflection monoids and show how these presentations are compatible with quiver mutations of orientations of Dynkin diagrams with frozen vertices. Our results recover the presentations of Boolean reflection monoids given by Everitt and Fountain and the presentations of symmetric inverse semigroups given by Popova respecti…

On Exchange Spectra of Valued Cluster Quivers and Cluster 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…

Polygonal Quivers

We show that Fano lattice polygons define a class of balanced quivers with interesting properties. The combinatorics of these quivers is related to singularities of the underlying toric Fano surface. This allows us to show that every Fano polygon defines a point on a certain family of algebraic hypersurfaces. Our quivers admit a generalized mutation which preserves balancing and coincides with combinatorial mutation of Fano polygons whenever both operations are defined. We characterize balanced quivers arising from Fano polygons and discuss generalizations to higher dimensions.

穿孔曲面、箭图突变与 Coxeter 群的商

Punctured surfaces, quiver mutations, and quotients of Coxeter groups

本文将 Barot–Marsh 基于箭图突变的 Coxeter 群商构造推广至多数穿孔曲面与轨形，得到几乎所有带标记曲面的新三角剖分不变量。