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For a given cluster-tilted algebra $A$ of tame type, it is proved thatdifferent indecomposable $\tau$-rigid $A$-modules have different dimensionvectors. This is motivated by Fomin-Zelevinsky's denominator conjecture forcluster algebras. As an application, we establish a weak version of thedenominator conjecture for cluster algebras of tame type. Namely, we show thatdifferent cluster variables have different denominators with respect to a givencluster for a cluster algebra of tame type. Our approach involvesIyama-Yoshino's construction of subfactors of triangulated categories. Inparticular,we obtain a description of the subfactors of cluster categories oftame type with respect to an indecomposable rigid object, which is ofindependent interest.

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