Strongness of companion bases for cluster-tilted algebras of finite type
Karin Baur, Alireza Nasr-Isfahani
TL;DR
The paper proves that all companion bases for cluster-tilted algebras of simply-laced Dynkin type are strong, confirming Parsons's conjecture and linking them to indecomposable module dimension vectors.
Contribution
It establishes that every companion basis of such algebras is strong, providing a proof of Parsons's conjecture and clarifying their role in module theory.
Findings
Every companion basis of a cluster-tilted algebra of simply-laced Dynkin type is strong.
Provides a proof of Parsons's conjecture.
Connects companion bases to indecomposable module dimension vectors.
Abstract
For every cluster-tilted algebra of simply-laced Dynkin type we provide a companion basis which is strong, i.e. gives the set of dimension vectors of the finitely generated indecomposable modules for the cluster-tilted algebra. This shows in particular that every companion basis of a cluster-tilted algebra of simply-laced Dynkin type is strong. Thus we give a proof of Parsons's conjecture.
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