Rigid and Schurian modules over cluster-tilted algebras of tame type

TL;DR

This paper explores the relationship between denominator vectors in cluster algebras and dimension vectors of indecomposable modules over a specific class of cluster-tilted algebras, providing new insights and classifications.

Contribution

It presents an example where the denominator vector does not match any indecomposable module's dimension vector and classifies indecomposable rigid modules for a tame cluster-tilted algebra.

Findings

Existence of denominator vectors not corresponding to any indecomposable module

Any denominator vector can be expressed as a sum of up to three indecomposable rigid modules

Classification of indecomposable rigid modules over the considered algebra

Abstract

We give an example of a cluster-tilted algebra A with quiver Q, such that the associated cluster algebra has a denominator vector which is not the dimension vector of any indecomposable A-module. This answers a question posed by T. Nakanishi. The relevant example is a cluster-tilted algebra associated with a tame hereditary algebra. We show that for such a cluster-tilted algebra A, we can write any denominator vector as a sum of the dimension vectors of at most three indecomposable rigid A-modules. In order to do this it is necessary, and of independent interest, to first classify the indecomposable rigid A-modules in this case.