On support $\tau$-tilting graphs of gentle algebras

TL;DR

This paper studies the combinatorial structure of support τ-tilting graphs for gentle algebras, proving their connectivity and a property related to cluster algebra exchange graphs, thus advancing understanding of their algebraic and combinatorial features.

Contribution

It establishes the connectivity and reachable-in-face property of support τ-tilting graphs for gentle algebras, confirming a conjecture related to cluster algebra exchange graphs.

Findings

Support τ-tilting graph of gentle algebra is connected.

Support τ-tilting graph has the reachable-in-face property.

Confirmed a conjecture by Fomin and Zelevinsky for this class of algebras.

Abstract

Let $A$ be a finite-dimensional gentle algebra over an algebraically closed field. We investigate the combinatorial properties of support $\tau$-tilting graph of $A$. In particular, it is proved that the support $\tau$-tilting graph of $A$ is connected and has the so-called reachable-in-face property. This property was conjectured by Fomin and Zelevinsky for exchange graphs of cluster algebras which was recently confirmed by Cao and Li.