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

## TL;DR

This paper proves that indecomposable tau-rigid modules over tame cluster-tilted algebras have unique dimension vectors and applies this to establish a weak form of the denominator conjecture for cluster algebras of tame type, using triangulated category subfactors.

## Contribution

It demonstrates the uniqueness of dimension vectors for indecomposable tau-rigid modules in tame cases and links this to a weak denominator conjecture for cluster algebras.

## Key findings

- Different indecomposable tau-rigid modules have distinct dimension vectors.
- Different cluster variables have different denominators with respect to a given cluster in tame type.
- Description of subfactors of cluster categories of tame type with respect to indecomposable rigid objects.

## 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.