Maximal Green Sequences Research Articles

The No Gap Conjecture for tame hereditary algebras

The “No Gap Conjecture” of Brüstle–Dupont–Pérotin states that the set of lengths of maximal green sequences for hereditary algebras over an algebraically closed field has no gaps. This follows from a stronger conjecture that any two maximal green sequences can be “polygonally deformed” into each other. We prove this stronger conjecture for all tame hereditary algebras over any field, equivalently, for any acyclic tame skew-symmetrizable exchange matrix.

Minimal length 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 combinatorics of surface triangulations and the basics of scattering diagrams to address this problem. Our main result is a formula for the length of minimal length maximal green sequences of quivers defined by triangulations of an annulus or a punctured disk.

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 Br\"ustle 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 graphs with finitely many elements are semidistributive lattices. Furthermore, if the quiver is mutation-equivalent to a type A Dynkin quiver or is an oriented cycle, then the oriented exchange graph is a lattice quotient of a lattice of biclosed subcategories of modules over the cluster-tilted algebra, generalizing Reading's Cambrian lattices in type A. We also apply our results to address a conjecture of Br\"ustle, Dupont, and P\'erotin on the lengths of maximal green sequences.

Properties of minimal mutation-infinite quivers

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 by these quivers are locally acyclic and hence equal to their upper cluster algebra. We also study which quivers in a mutation-class have a maximal green sequence. For any rank 3 quiver there are at most 6 quivers in its mutation class that admit a maximal green sequence. We also show that for every rank 4 minimal mutation-infinite quiver there is a finite connected subgraph of the unlabelled exchange graph consisting of quivers that admit a maximal green sequence.

Maximal green sequences for quivers of finite mutation type

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 quiver has a maximal green sequence unless it arises from a once-punctured closed marked surface, or one of the two quivers in the mutation class of X7. We develop a procedure to explicitly find maximal green sequences for cluster quivers associated to arbitrary triangulations of closed marked surfaces with at least two punctures. As a corollary, it follows that any triangulation of a marked surface with boundary has a maximal green sequence. We also compute explicit maximal green sequences for exceptional quivers of finite mutation type.

Maximal green sequences for cluster algebras associated with the n-torus with arbitrary punctures

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 corresponding quiver has a maximal green sequence.

Cluster Automorphisms and the Marked Exchange Graphs of Skew-Symmetrizable Cluster Algebras

Cluster automorphisms have been shown to have links to the mapping class groups of surfaces, maximal green sequences and to exchange graph automorphisms for skew-symmetric cluster algebras. In this paper we generalise these results to the skew-symmetrizable case by introducing a marking on the exchange graph. Many skew-symmetrizable matrices unfold to skew-symmetric matrices and we consider how cluster automorphisms behave under this unfolding with applications to coverings of orbifolds by surfaces.

Semi-invariant pictures and two conjectures on maximal green sequences

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→i of infinite type then any maximal green sequence for Q must mutate at i before mutating at j. Second: for any quiver Q′ obtained by mutating an acyclic valued quiver Q of tame type, there are only finitely many maximal green sequences for Q′. Both statements follow from the Rotation Lemma for reddening sequences and this in turn follows from the Mutation Formula for the semi-invariant picture for Q.

On maximal green sequences for type $$\mathbb {A}$$ quivers

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 representation theory and physics as they have numerous applications, including the computations of spectrums of BPS states, Donaldson-Thomas invariants, tilting of hearts in the derived category, and quantum dilogarithm identities. In this paper, we study such sequences and construct a maximal green sequence for every quiver mutation-equivalent to an orientation of a type A Dynkin diagram.

Maximal Green Sequences for Cluster Algebras Associated to Orientable Surfaces with Empty Boundary

Given a marked surface (S, M) we can add arcs to the surface to create a triangulation, T, of that surface. For each triangulation, T, we can associate a cluster algebra. In this paper we will consider orientable surfaces of genus n with two interior marked points and no boundary component. We will construct a specific triangulation of this surface which yields a quiver. Then in the sense of work by Keller we will produce a maximal green sequence for this quiver. Since all finite mutation type cluster algebras can be associated to a surface, with some rare exceptions, this work along with previous work by others seeks to establish a base case in answering the question of whether a given finite mutation type cluster algebra exhibits a maximal green sequence.

Maximal Green Sequences for Preprojective Algebras

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 bounded t-structures of triangulated categories that can be tilted indefinitely. We study the case where the heart is the category of modules over the preprojective algebra of a quiver without loops. The combinatorical counterpart of maximal green sequences for Dynkin quivers are maximal chains in the Hasse quiver of basic support τ-tilting modules. We show that a quiver has a maximal green sequence if and only if it is of Dynkin type. More generally, we study module categories for finite-dimensional algebras with finitely many bricks.

Minimal length maximal green sequences and triangulations of polygons

We use combinatorics of quivers and the corresponding surfaces to study maximal green sequences of minimal length for quivers of type $$\mathbb {A}$$A. We prove that such sequences have length $$n+t$$n+t, where n is the number of vertices and t is the number of 3-cycles in the quiver. Moreover, we develop a procedure that yields these minimal length maximal green sequences.

The Existence of a Maximal Green Sequence is not Invariant under Quiver Mutation

This note provides a quiver which does not admit a maximal green sequence, but which is mutation-equivalent to a quiver which does admit a maximal green sequence. The proof uses the `scattering diagrams' of Gross-Hacking-Keel-Kontsevich to show that a maximal green sequence for a quiver determines a maximal green sequence for any induced subquiver.

Twists of Plücker Coordinates as Dimer Partition Functions

The homogeneous coordinate ring of the Grassmannian Gr k,n has a cluster structure defined in terms of planar diagrams known as Postnikov diagrams. The cluster corresponding to such a diagram consists entirely of Plucker coordinates. We introduce a twist map on Gr k,n , related to the Berenstein–Fomin–Zelevinsky-twist, and give an explicit Laurent expansion for the twist of an arbitrary Plucker coordinate in terms of the cluster variables associated with a fixed Postnikov diagram. The expansion arises as a (scaled) dimer partition function of a weighted version of the bipartite graph dual to the Postnikov diagram, modified by a boundary condition determined by the Plucker coordinate. We also relate the twist map to a maximal green sequence.

Maximal Green Sequences of Exceptional Finite Mutation Type Quivers

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 theory. The existence of maximal green sequences for exceptional finite mutation type quivers has been shown by Alim-Cecotti-C\'ordova-Espahbodi-Rastogi-Vafa except for the quiver $X_7$. In this paper we show that the quiver $X_7$ does not have any maximal green sequences. We also generalize the idea of the proof to give sufficient conditions for the non-existence of maximal green sequences for an arbitrary quiver.

Maximal green sequences of skew-symmetrizable [formula omitted] matrices

Maximal green sequences are particular sequences of mutations of skew-symmetrizable matrices which were introduced by Keller in the context of quantum dilogarithm identities and independently by Cecotti–Córdova–Vafa in the context of supersymmetric gauge theory. In this paper we study maximal green sequences of skew-symmetrizable 3×3 matrices. We show that such a matrix with a mutation-cyclic diagram does not have any maximal green sequence. We also obtain some basic properties of maximal green sequences of skew-symmetrizable matrices with mutation-acyclic diagrams.

On Maximal Green Sequences

Maximal green sequences are particular sequences of quiver mutations appearing in the context of quantum dilogarithm identities and supersymmetric gauge theory. Interpreting maximal green sequences as paths in various natural posets arising in representation theory, we prove the finiteness of the number of maximal green sequences for cluster finite quivers, affine quivers, and acyclic quivers with at most three vertices. We also give results concerning the possible numbers and lengths of these maximal green sequences.