Stability Conditions for Affine Type A
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Stability Conditions for Affine Type A P. J. Apruzzese1 · Kiyoshi Igusa2 Received: 28 December 2018 / Accepted: 20 September 2019 / © The Author(s) 2019
Abstract 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). Keywords Maximal green sequences · Cluster mutation · Quivers · c-vectors · Central charge · Wire diagram · Wall crossing Mathematics Subject Classification (2010) 16G20
1 Introduction Maximal green sequences are one incarnation of a variety of related concepts of current interest. The term was defined by Keller [13] who showed that maximal green sequences can be used to compute the Donalson-Thomas invariant of a quiver. Much of the literature is devoted to the existence of such sequence. [9, 21, 27]. For example, [22] shows that the existence of a maximal green sequence, or even a weaker version known as a “reddening sequence” gives a canonical basis for the upper cluster algebra. However, this paper deals with the case of an acyclic quiver where a maximal green sequence always exists. One important class of maximal green sequences come from stability conditions, a concept introduced much earlier by King [20]. This was extended to the study of vector bundles over projective space by Rudakov [26] who showed that a stability condition gives a Harder-
Presented by: Henning Krause. Kiyoshi Igusa
[email protected] P. J. Apruzzese [email protected] 1
Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA
2
Department of Mathematics, Brandeis University, Waltham, MA 02454, USA
P.J. Apruzzese, K. Igusa
Narasimhan (HN)-filtration for a vector bundle. Bridgeland [3, 4] extended these definitions and results to more general triangulated categories. The connection between these concepts comes from the fact that the dimension vectors of stable modules, in order of their “slopes” forms a maximal green sequence. It is shown in [16] that an HN-stratification of the module category of any finite dimensional algebra is equivalence to a notion called a “forward hom-orthogonal sequence of bricks”. In [13] this is shown to be equivalent to a maximal chain in the poset of functorially finite torsion classes. Although maximal green sequences were defined for cluster algebras and cluster-tilted algebras as certain sequences of mutations of clusters in the cluster category [7, 8], the definition can be extended to arbitrary finite dimensional algebras, namely, they can be defined as maximal (finite) chains in the poset of functorially finite torsion classes or as finite wall-crossing sequences in the semi-invariant picture (or scattering
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