Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On support $τ$-tilting graphs of gentle algebras

Changjian Fu, Shengfei Geng, Pin Liu, Yu Zhou

Abstract

Let $A$ be a finite-dimensional gentle algebra over an algebraically closed field. We investigate the combinatorial properties of support $τ$-tilting graph of $A$. In particular, it is proved that the support $τ$-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.

On support $τ$-tilting graphs of gentle algebras

Abstract

Let be a finite-dimensional gentle algebra over an algebraically closed field. We investigate the combinatorial properties of support -tilting graph of . In particular, it is proved that the support -tilting graph of 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.

Paper Structure

This paper contains 12 sections, 20 theorems, 20 equations, 9 figures.

Table of Contents

  1. Introduction and main results
  2. Background
  3. Main results
  4. Gentle algebras and their geometric models
  5. Gentle algebras
  6. Tilings and permissible curves
  7. Geometric interpretation of $\tau$-tilting theory
  8. $\tau$-reachable property for gentle algebras
  9. Definition of $\tau$-reachability
  10. $\tau$-reachability of gentle algebras
  11. Proofs of the main results
  12. Reduction

Key Result

Theorem 1.3

Let $A$ be a finite-dimensional gentle algebra over an algebraically closed field $k$. The support $\tau$-tilting graph $\mathcal{H}(\operatorname{\mathsf{s\tau-tilt}} A)$ of $A$ has precisely one connected component.

Figures (9)

  • Figure 1: Tiles of type I and II
  • Figure 4: Notations for a permissible arc $\gamma$
  • Figure 5: Cases in Lemma \ref{['l:concatenation']}
  • Figure 6: Case 1 in the proof of Lemma \ref{['l:sincere-reachable']}
  • Figure 7: Case 2 in the proof of Lemma \ref{['l:sincere-reachable']}
  • ...and 4 more figures

Theorems & Definitions (43)

  • Conjecture 1.1
  • Definition 1.2: support $\tau$-tilting graph
  • Theorem 1.3
  • Definition 1.4: reachable-in-face
  • Theorem 1.5
  • Theorem 1.6: Theorem \ref{['t:totally-equivalent-to-connected']}
  • Lemma 2.1
  • Lemma 2.2: S
  • Definition 2.3: BS and HZZ
  • Lemma 2.4
  • ...and 33 more