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

TL;DR

This paper proves the equivalence of three definitions of maximal green sequences for finite type cluster-tilted algebras, advancing the understanding of their structure and lengths.

Contribution

It establishes the equivalence of different characterizations of maximal green sequences in finite type cluster-tilted algebras, completing foundational work for classifying their lengths.

Findings

Equivalence of three definitions of maximal green sequences

Complete characterization of maximal green sequences for finite type cases

Conjectural description of maximum length sequences

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 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". In an Appendix, written jointly with G. Todorov, we give a conjectural description of maximal green sequences of maximum length for any cluster-tilted algebra of finite representation type.