Related papers: Quivers of finite mutation type and skew-symmetric…
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…
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…
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…
We give a precise definition of folded quivers and folded cluster algebras. We give many examples of including some with finite mutation structure that do not have analogues in the unfolded cases. We relate these examples to the finite…
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…
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…
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…
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 $\pi$-rotations for the cyclic…
This if the final paper in the seriesContinuous Quivers of Type $A$. In this part, we generalize existing geometric models of type $A$ cluster structures to the new $\mathbf E$-clusters introduced in part (III). We also introduce an…
We investigate the cluster-tilted algebras of finite representation type over an algebraically closed field. We give an explicit description of the relations for the quivers for finite representation type. As a consequence we show that a…
Some skew-symmetrizable integer exchange matrices are associated to ideal (tagged) triangulations of marked bordered surfaces. These exchange matrices admits unfoldings to skew-symmetric matrices. We develop an combinatorial algorithm that…
We define several topological spaces whose points are quivers with a given infinite vertex set $X$. In the special case when $X$ is countably infinite, we show that two of the spaces of interest are homeomorphic to the Baire space…
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…
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.
In 2003, Fomin and Zelevinsky obtained Cartan-Killing type classification of all cluster algebras of finite type, i.e. cluster algebras having only finitely many distinct cluster variables. A wider class of cluster algebras is formed by…
We classify mutation-finite cluster algebras with arbitrary coefficients of geometric type.
We give a precise definition of mutation of skew symmetrizable matrices over group rings and relate it to folding and mutation of quivers with symmetries. These matrices can have non-zero diagonal entries and we explain a mutation rule in…
We consider ruled surfaces with finite multiplicity. We study behaviors of the striction curves and the singularities of the ruled surfaces. We also give geometric meanings of invariants related to the ruled surfaces.
We attempt to relate two recent developments: cluster algebras associated to triangulations of surfaces by Fomin-Shapiro-Thurston, and quivers with potentials and their mutations introduced by Derksen-Weyman-Zelevinsky. To each ideal…
In 2011, Barot and Marsh provided an explicit construction of presentation of a finite Weyl group $W$ by any quiver mutation-equivalent to an orientation of a Dynkin diagram with Weyl group $W$. The construction was extended by the authors…