带环箭图的排序元素

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。

具有2环的有向图的突变

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。

Grothendieck classes of quiver cycles as iterated residues

In the case of Dynkin quivers we establish a formula for the Grothendieck class of a quiver cycle as the iterated residue of a certain rational function, for which we provide an explicit combinatorial construction. Moreover, we utilize a new definition of the double stable Grothendieck polynomials due to Rimanyi and Szenes in terms of iterated residues to exhibit how the computation of quiver coefficients can be reduced to computing coefficients in Laurent expansions of certain rational functions.

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.

Quiver coefficients of Dynkin type

We study the Grothendieck classes of quiver cycles, i.e. invariant closed subvarieties of the representation space of a quiver. For quivers without oriented loops we show that the class of a quiver cycle is determined by quiver coefficients, which generalize the earlier studied quiver coefficients for equioriented quivers of type A. We conjecture that quiver coefficients satisfy positivity and finiteness properties. Our main result is a formula for the quiver coefficients for orbit closures of Dynkin type with rational singularities, which confirms the finiteness conjecture. This formula is ba…

有向图表示的迹恒等式

Trace identities for quiver representations

研究通过迹计算有向图表示的扭拉普拉斯算子行列式，提出块矩阵行列式的一般恒等式，并给出枚举和生成函数两种证明方法。

Subquivers of mutation-acyclic quivers are mutation-acyclic

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.

Universal derived equivalences of posets of cluster tilting objects

We show that for two quivers without oriented cycles related by a BGP reflection, the posets of their cluster tilting objects are related by a simple combinatorial construction, which we call a flip-flop. We deduce that the posets of cluster tilting objects of derived equivalent path algebras of quivers without oriented cycles are universally derived equivalent. In particular, all Cambrian lattices corresponding to the various orientations of the same Dynkin diagram are universally derived equivalent.

Quotient closed subcategories of quiver representations

Let Q be a finite quiver without oriented cycles, and let k be an algebraically closed field. The main result in this paper is that there is a natural bijection between the elements in the associated Coxeter group W_Q and the cofinite additive quotient-closed subcategories of the category of finite dimensional right modules over kQ. We prove this correspondence by linking these subcategories to certain ideals in the preprojective algebra associated to Q, which are also indexed by elements of W_Q.

Noncrossing partitions and representations of quivers

We situate the noncrossing partitions associated to a finite Coxeter group within the context of the representation theory of quivers. We describe Reading's bijection between noncrossing partitions and clusters in this context, and show that it extends to the extended Dynkin case. Our setup also yields a new proof that the noncrossing partitions associated to a finite Coxeter group form a lattice. We also prove some new results within the theory of quiver representations. We show that the finitely generated, exact abelian, and extension-closed subcategories of the representations of a quiver $…

等定向循环的若干拉格朗日箭图格拉斯曼簇

Some Lagrangian quiver Grassmannians for the equioriented cycle

本文证明等定向循环箭图格拉斯曼簇的辛子簇 $X(n,2n)^{sp}$ 上轨道闭包序与辛抛接模式序一致，计算轨道维数，并证明其 GKM 性质。

极小表示Cartan平方的箭图基

Quiver Bases of Cartan Squares of Minuscule Representations

本文为极小表示的Cartan平方构造箭图基，统一证明任意极小偏序集上分段线性Coxeter运动与行运动满足循环筛选现象，并指出典范基仅与A型长循环相容。

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…