Dimension vectors of $\tau$-rigid modules and $f$-vectors of cluster monomials from triangulated surfaces

TL;DR

This paper characterizes when different cluster monomials and tau-rigid modules have unique f-vectors or dimension vectors in cluster algebras from triangulated surfaces, using intersection numbers, and confirms the denominator conjecture in specific cases.

Contribution

It provides new criteria for the uniqueness of f-vectors and dimension vectors in cluster algebras and associated Jacobian algebras from triangulated surfaces, utilizing intersection numbers.

Findings

Unique f-vectors for non-initial cluster monomials characterized by surface properties.

Unique dimension vectors for tau-rigid modules linked to surface triangulations.

Verification of the denominator conjecture for certain closed surfaces with one puncture.

Abstract

For the cluster algebra $\mathcal{A}$ associated with a triangulated surface, we give a characterization of the triangulated surface such that different non-initial cluster monomials in $\mathcal{A}$ have different $f$-vectors. Similarly, for the associated Jacobian algebra $J$, we give a characterization of the triangulated surface such that different $\tau$-rigid $J$-modules have different dimension vectors. Moreover, we also show that different basic support $\tau$-tilting $J$-modules have different dimension vectors. Our main ingredient is a notion of intersection numbers defined by Qiu and Zhou. As an application, we show that the denominator conjecture holds for $\mathcal{A}$ if the marked surface is a closed surface with exactly one puncture, or the given tagged triangulation has neither loops nor tagged arcs connecting punctures.