Related papers: On Maximal Green Sequences
Maximal green sequences are important objects in representation theory, cluster algebras, and string theory. It is an open problem to determine what lengths are achieved by the maximal green sequences of a quiver. We combine the…
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…
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 show that, for any cluster-tilted algebra of finite representation type over an algebraically closed field, the following three definitions of a maximal green sequence are equivalent: (1) the usual definition in terms of Fomin-Zelevinsky…
A maximal green sequence introduced by B. Keller is a certain sequence of quiver mutations at green vertices. T. Br\"ustle, G. Dupont and M. P\'erotin showed that for an acyclic quiver, maximal green sequences are realized as maximal paths…
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-C\'ordova-Vafa in the context of supersymmetric gauge…
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 $t$…
In this paper we completely describe maximal green sequences (MGS) of acyclic quivers with multiple edges in terms of maximal green sequences of their multiple edge-free (ME-free) versions. In particular we establish that any MGS of a…
We use semi-invariant pictures to prove two conjectures about maximal green sequences. First: if $Q$ is any acyclic valued quiver with an arrow $j\to i$ of infinite type then any maximal green sequence for $Q$ must mutate at $i$ before…
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…
Maximal green sequences were introduced as combinatorical counterpart for Donaldson-Thomas invariants for 2-acyclic quivers with potential by B. Keller. We take the categorical notion and introduce maximal green sequences for hearts of…
It is known that the existence of a maximal green sequence for a quiver associated to surfaces is equivalent to the equality of the cluster algebra and upper cluster algebra generated by the quiver. This paper makes the first steps in…
We investigate the existence and non-existence of maximal green sequences for quivers arising from weighted projective lines. Let $Q$ be the Gabreil quiver of the endomorphism algebra of a basic cluster-tilting object in the cluster…
In this paper, we study the maximal length of maximal green sequences for quivers of type $\widetilde{\mathbf{D}}$ and $\widetilde{\mathbf{E}}$ by using the theory of tilting mutation. We show that the maximal length does not depend on the…
We introduce $\mathcal{Q}^N$ quivers and construct maximal green sequences for these quivers. We prove that any finite connected full subquiver of the quivers defined by Hernandez and Leclerc, arising in monoidal categorifications of…
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…
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…
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…
We prove that the quantum and classical cluster algebras for all members of the axiomatically defined classes of symmetric quantum and Poisson Cauchon-Goodearl-Letzter extensions possess maximal green sequences in the sense of Keller.…
It is well known that any triangulation of a marked surface produces a quiver. In this paper we will provide a triangulation for orientable surfaces of genus $n$ with an arbitrary number interior marked points (called punctures) whose…