On indecomposable $\tau$-rigid modules over cluster-tilted algebras of tame type

Abstract

For a given cluster-tilted algebra $A$ of tame type, it is proved that different indecomposable $\tau$-rigid $A$-modules have different dimension vectors. This is motivated by Fomin-Zelevinsky's denominator conjecture for cluster algebras. As an application, we establish a weak version of the denominator conjecture for cluster algebras of tame type. Namely, we show that different cluster variables have different denominators with respect to a given cluster for a cluster algebra of tame type. Our approach involves Iyama-Yoshino's construction of subfactors of triangulated categories. In particular,we obtain a description of the subfactors of cluster categories of tame type with respect to an indecomposable rigid object, which is of independent interest.

24 pages, exposition is improved

References in corpus (3)

• On triangulated orbit categories
• Cluster structures for 2-Calabi-Yau categories and unipotent groups
• Denominators of cluster variables

Cited by in corpus (2)

• On cluster-tilting graphs for hereditary categories
• Mutation of tilting bundles of tubular type