Maximal Green Sequences Research Articles

Red Sizes of Quivers

In this article, we will expand on the notions of maximal green and reddening sequences for quivers associated to cluster algebras. The existence of these sequences has been studied for a variety of applications related to Fomin and Zelevinsky's cluster algebras. Ahmad and Li considered a numerical measure of how close a quiver is to admitting a maximal green sequence called a red number. In this paper we generalized this notion to what we call unrestricted red numbers which are related to reddening sequences. In addition to establishing this more general framework we completely determine the red numbers and unrestricted red numbers for all finite mutation type quivers. Furthermore, we give conjectures on the possible values of red numbers and unrestricted red numbers in general.

On Maximal Green Sequence for Quivers Arising from Weighted Projective Lines

We investigate the existence and non-existence of maximal green sequences for quivers arising from weighted projective lines. Let Q be the Gabriel quiver of the endomorphism algebra of a basic cluster-tilting object in the cluster category \(\mathcal {C}_{\mathbb {X}}\) of a weighted projective line \(\mathbb {X}\). It is proved that there exists a quiver \(Q^{\prime }\) in the mutation equivalence class Mut(Q) of Q such that \(Q^{\prime }\) admits a maximal green sequence. Furthermore, there is a quiver in Mut(Q) which does not admit a maximal green sequence if and only if \(\mathbb {X}\) is of wild type.

Stability conditions and maximal green sequences in abelian categories

In this paper we study the stability functions on abelian categories introduced by Rudakov in \cite{Ru} and their relation with torsion classes and maximal green sequences. Moreover we introduce a new kind of stability function which is compatible with the wall and chamber of the category.

Maximal green sequences for string algebras

Maximal green sequences are important objects in representation theory, cluster algebras, and string theory. The two fundamental questions about maximal green sequences are whether a given algebra admits such sequences and, if so, does it admit only finitely many. We study maximal green sequences in the case of string algebras and give sufficient conditions on the algebra that ensure an affirmative answer to these questions.

Lengths of maximal green sequences for tame path algebras

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 choice of the orientation and determine it explicitly. Moreover, we give a program which counts all maximal green sequences by length for a given Dynkin/extended Dynkin quiver.

On maximal green sequences in abelian length categories

In this article, we study the relationship among maximal green sequences, complete forward hom-orthogonal sequences and stability functions in abelian length categories. Mainly, we firstly give a one-to-one correspondence between maximal green sequences and complete forward hom-orthogonal sequences via mutual constructions, and then prove that a maximal green sequence can be induced by a central charge if and only if it satisfies crossing inequalities. As applications, we show that crossing inequalities can be computed by c-vectors for finite dimensional algebras; finally, we give the Rotation Lemma for finite dimensional Jacobian algebras.

Picture groups and maximal green sequences

<p style='text-indent:20px;'>We show that picture groups are directly related to maximal green sequences for valued Dynkin quivers of finite type. Namely, there is a bijection between maximal green sequences and positive expressions (words in the generators without inverses) for the Coxeter element of the picture group. We actually prove the theorem for the more general set up of finite "vertically and laterally ordered" sets of positive real Schur roots for any hereditary algebra (not necessarily of finite type).</p><p style='text-indent:20px;'>Furthermore, we show that every picture for such a set of positive roots is a linear combination of "atoms" and we give a precise description of atoms as special semi-invariant pictures.</p>

Building maximal green sequences via component preserving mutations

We introduce a new method for producing both maximal green and reddening sequences of quivers. The method, called component preserving mutations, generalizes the notion of direct sums of quivers and can be used as a tool to both recover known reddening sequences as well as find reddening sequences that were previously unknown. We use the method to produce and recover maximal green sequences for many bipartite recurrent quivers that show up in the study of periodicity of $T$-systems and $Y$-systems. Additionally, we show how our method relates to the dominance phenomenon recently considered by Reading. Given a maximal green sequence produced by our method, this relation to dominance gives a maximal green sequence for infinitely many other quivers. Other applications of this new methodology are explored including computing of quantum dilogarithm identities and determining minimal length maximal green sequences.

Sign-Coherence of C-Vectors and Maximal Green Sequences for Acyclic Sign-Skew-Symmetric Matrices

In this paper we construct an unfolding for c −vectors of acyclic sign-skew symmetric matrices and we also prove that the sign-coherence property holds for acyclic sign-skew-symmetric matrices. Then we prove that every acyclic sign-skew-symmetric matrix admits a maximal green sequence.

Polyhedral parametrizations of canonical bases & cluster duality

We establish the relation of Berenstein–Kazhdan's decoration function and Gross–Hacking–Keel–Kontsevich's potential on the open double Bruhat cell in the base affine space G/N of a simple, simply connected, simply laced algebraic group G. As a byproduct we derive explicit identifications of polyhedral parametrization of canonical bases of the ring of regular functions on G/N arising from the tropicalizations of the potential and decoration function with the classical string and Lusztig parametrizations. In the appendix we construct maximal green sequences for the open double Bruhat cell in G/N which is a crucial assumption for Gross–Hacking–Keel–Kontsevich's construction.

Maximal green sequences for arbitrary triangulations of marked surfaces (Extended Abstract)

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 associated to most marked surfaces. We develop a procedure to find maximal green sequences for cluster quivers associated to an arbitrary triangulation of closed higher genus marked surfaces with at least two punctures. As a corollary, it follows that any triangulation of a marked surface with at least one boundary component has a maximal green sequence.

Linearity of stability conditions

For modules over an artin algebra, a linear stability condition is given by a “central charge” and a nonlinear stability condition is given by the wall-crossing sequence of a “green path.” Finite Harder-Narasimhan stratifications of the module category, maximal forward hom-orthogonal sequences and maximal green sequences, defined using Fomin-Zelevinsky quiver mutation are shown to be equivalent to finite nonlinear stability conditions when the algebra is hereditary. This is the first of a series of three papers whose purpose is to determine all maximal green sequences of maximal length for quivers of affine type A and determine which are linear.

Jacobian matrices of Y-seed mutations

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 Jacobian matrix of the cluster transformation. Furthermore, we show that this relationship reduces to a relationship between the pair of matrices and the C-matrix of the cluster transformation in a certain limit of cluster variables. As an application, we prove that quivers associated with once-punctured surfaces do not have maximal green or reddening sequences.

Stability Conditions for Affine Type A

We construct maximal green sequences of maximal length for any affine quiver of type A. We determine which sets of modules (equivalently c-vectors) can occur in such sequences and, among these, which are given by a linear stability condition (also called a central charge). There is always at least one such maximal set which is linear. The proofs use representation theory and three kinds of diagrams shown in Fig. 1. Background material is reviewed with details presented in two separate papers Igusa (2017a, b).

Wall and chamber structure for finite-dimensional algebras

We use τ-tilting theory to give a description of the wall and chamber structure of a finite-dimensional algebra. We also study D-generic paths in the wall and chamber structure of an algebra A and show that every maximal green sequence in mod A is induced by a D-generic path.

Maximal forward hom-orthogonal sequences for cluster-tilted algebras of finite type

Let $\Lambda$ be a cluster-tilted algebra of finite type over an algebraically closed field and $B$ be one of the associated tilted algebras. We show that the $B$-modules, ordered form right to left in the Auslander-Reiten quiver of $\Lambda$ form a maximal forward hom-orthogonal sequence of $\Lambda$-modules whose dimension vectors form the $c$-vectors of a maximal green sequence for $\Lambda$. Thus we give a proof of Igusa-Todorov's conjecture.

Upper cluster algebras and choice of ground ring

We initiate a study of the dependence on the choice of ground ring on the question of whether a cluster algebra is equal to its upper cluster algebra. A condition for when there is equality of the cluster algebra and upper cluster algebra is given by using a variation of Muller's theory of cluster localization. An explicit example exhibiting dependence on the ground ring is provided. We also present a maximal green sequence for this example.

Uniform column sign-coherence and the existence of maximal green sequences

In this paper, we prove that each matrix in $$M_{m\times n}({\mathbb {Z}}_{\ge 0})$$ is uniformly column sign-coherent (Definition 2.2 (ii)) with respect to any $$n\times n$$ skew-symmetrizable integer matrix (Corollary 3.3 (ii)). Using such matrices, we introduce the definition of irreducible skew-symmetrizable matrix (Definition 4.1). Based on this, the existence of maximal green sequences for skew-symmetrizable matrices is reduced to the existence of maximal green sequences for irreducible skew-symmetrizable matrices.

Maximal green sequences for cluster-tilted algebras of finite representation type

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 mutation of the extended exchange matrix, (2) a complete forward hom-orthogonal sequence of Schurian modules, (3) the sequence of wall crossings of a generic green path. Together with [24], this completes the foundational work needed to support the author’s work with P. J. Apruzzese [1], namely, to determine all lengths of all maximal green sequences for all quivers whose underlying graph is an oriented or unoriented cycle and to determine which are “linear”.

Horizontal and vertical mutation fans

We introduce diagrams for m-cluster categories which we call “horizontal” and “vertical” mutation fans. These are analogous to the mutation fans (also known as “semi-invariant pictures” or “scattering diagrams”) for the standard (m = 1) cluster case, which are dual to the poset of finitely generated torsion classes. The purpose of these diagrams is to visualize mutations and analogues of maximal green sequences in the m-cluster category with special emphasis on the c-vectors (the “brick” labels).