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

Kiyoshi Igusa

doi:10.5802/alco.61

Abstract

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”.

Highlights

This paper is the second of three papers on the problem of “linearity” of stability conditions, namely: Is the longest maximal green sequence for an algebra equivalent to one given by a “central charge”? we do not address this question in this paper, we explain the motivation behind the series of papers of which this is a part
The question originates from a conjecture by Reineke [27] which asks if, for a Dynkin quiver, there is a “slope function” making all modules stable
Reineke wanted such a result because, when it holds, his formulas would give an explicit description of a PBW basis for the quantum group Uv(n+) for the Dynkin quiver

Summary

Introduction

The main purpose of the first two papers is to prove, in the three cases considered in [1], that the wall crossing description is equivalent to the usual definition of a maximal green sequence in terms of Fomin–Zelevinsky mutation of a skew-symmetrizable matrix called the “exchange matrix” [22]. This definition is reviewed in the example below. In this paper we restrict to the case of cluster algebras of finite type coming from skew-symmetric matrices To each such algebra there is an associated quiver with potential [20]. The sequence M1, . . . , M5 is a complete forward hom-orthogonal sequence

Forward hom-orthogonal sequences

Semistability sets for algebras of finite representation type