Coloured quivers of type A and the cell-growth problem

Coloured quiver mutation for higher cluster categories

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.

$\mathbb{D}_n$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。

Finite mutation classes of coloured quivers

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.

$\boldsymbol{\rm A}_n$型箭图的着色突变类

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。

Coloured quivers for rigid objects and partial triangulations: The unpunctured case

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.

A geometric realization of the $m$-cluster category of type $\tilde{A}$

We give a geometric realization of a subcategory of the $m$-cluster category $\mathcal{C}^m$ of type $\widetilde{A}_{p,q}$, by using $(m+2)$-angulations of an annulus with $p+q$ marked points. We also give a bijection between an equivalence class of $(m+2)$-angulations and the mutation class of coloured quivers of type $\widetilde{A}_{p,q}$.