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

Robert J. Marsh, Idun Reiten

Published 2014-12-19Version 1

We give an example of a cluster-tilted algebra $\Lambda$ with quiver Q, such that the associated cluster algebra A(Q) has a denominator vector which is not the dimension vector of any indecomposable $\Lambda$-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 $\Lambda$, we can write any denominator vector as a sum of the dimension vectors of at most three indecomposable rigid $\Lambda$-modules. In order to do this it is necessary, and of independent interest, to first classify the indecomposable rigid $\Lambda$-modules in this case.