On indecomposable $τ$-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 $τ$-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.

Figures (2)

• Figure 1: The AR quiver of a tube $\mathcal{T}$ of rank $3$. The quasi-simple modules are $(1,1), (2,1)$ and $(3,1)$.
• Figure 2: $\mathcal{T}_{M,\mathcal{C}}\cup {\mathcal{W}}_M$ consists of objects which do not lie in $L_M$ and $R_M$.

Theorems & Definitions (53)

• Theorem 1.1: Theorem \ref{['t:main-theorem-dimension-vector']}
• Theorem 1.2: Theorem \ref{['t:denominator-conjecture-tame-type']}
• Theorem 1.3: Theorem \ref{['t:subfactor-cluster-category']}
• Theorem 1.4: Theorem \ref{['t:subfactor-tame-type']}
• Theorem 2.1
• Proposition 2.2
• Lemma 2.3
• Definition 2.4
• Lemma 2.5
• Lemma 2.6