Related papers: Maximal green sequences for $\mathcal{Q}^N$ quiver…

We use combinatorics of quivers and the corresponding surfaces to study maximal green sequences of minimal length for quivers of type $\mathbb{A}$. We prove that such sequences have length $n+t$, where $n$ is the number of vertices and $t$…

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…

Maximal green sequences appear in the study of Fomin-Zelevinsky's cluster algebras. They are useful for computing refined Donaldson-Thomas invariants, constructing twist automorphisms and proving the existence of theta bases and generic…

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…

Let $K$ be a field, $Q$ a quiver, and $\mathcal{A}$ the ideal of the path algebra $KQ$ that is generated by the arrows of $Q$. We present old and new results about the representation theories of the truncations $KQ/\mathcal{A}^L$, $L \in…

We prove quantum dilogarithm identities for $n$-cycle quivers. By the combinatorial approach of Keller, each side of our identity determines a maximal green sequence of quiver mutations. Thus we interpret our identities as factorizations of…

For any acyclic quiver $Q$ without multiple edges, we construct a monoidal category $\mathcal{R}_Q$ whose indecomposable objects are tensor products (over the base field) of finite-dimensional modules over the path algebra of $Q$. We show…

In this note, we introduce monoidal subcategories of the tensor category of finite-dimensional representations of a simply-laced quantum affine algebra, parametrized by arbitrary Dynkin quivers. For linearly oriented quivers of types A and…

Exceptional sequences are certain ordered sequences of quiver representations. We use noncrossing edge-labeled trees in a disk with boundary vertices (expanding on T. Araya's work) to classify exceptional sequences of representations of Q,…

Let $Q$ be a finite quiver without oriented cycles and $k$ an algebraically closed field.In this paper we establish a connection between cluster algebras and the representation theory of the path algebra $kQ$, in terms of the spectral…

In the derived category of mod-KQ for Dynkin quiver Q, we construct a full subcategory in a canonical way, so that its endomorphism algebra is a higher Auslander algebra of global dimension $3k+2$ for any $k\geq 1$. Furthermore, we extend…

We show that a finite, connected quiver Q without oriented cycles is a Dynkin or Euclidean quiver if and only if all orbit semigroups of representations of Q are saturated.

Maximal green sequences are particular sequences of mutations which were introduced by Keller in the context of quantum dilogarithm identities and independently by Cecotti-Cordova-Vafa in the context of supersymmetric gauge theory. In this…

We develop basic cluster theory from an elementary point of view using a variation of binary trees which we call mixed cobinary trees. We show that the number of isomorphism classes of such trees is given by the Catalan number Cn where n is…

Let Q be a Dynkin quiver of type A. The bounded derived category of the path algebra of Q has an autoequivalence given by the composition of the Auslander-Reiten translate and the square of the shift functor. We classify the maximal rigid…

Let $Q$ be an acyclic quiver, it is classical that certain truncations of the translation quiver $\mathbb Z Q$ appear in the Auslander-Reiten quiver of the path algebra $kQ$. The stable $n$-translation quiver $\mathbb Z|_{n-1} Q$ is…

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…

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…

We show that a reddening sequence exists for any quiver which is Banff. Our proof is combinatorial and relies on the triangular extension construction for quivers. The other facts needed are that the existence of a reddening sequence is…

We compute the quiver of any monoid that has a basic algebra over an algebraically closed field of characteristic zero. More generally, we reduce the computation of the quiver over a splitting field of a class of monoids that we term…