Intersection vectors over tilings with applications to gentle algebras and cluster algebras

TL;DR

This paper proves a uniqueness result for intersection vectors of permissible arcs over tilings, extending classical surface results, and applies it to $ au$-tilting theory of gentle algebras and the denominator conjecture in certain cluster algebras.

Contribution

It generalizes a classical intersection vector uniqueness theorem to tilings and applies this to advance understanding in gentle algebras and cluster algebra theory.

Findings

Unique determination of permissible arcs by intersection vectors under mild conditions.

Different $ au$-rigid modules have distinct dimension vectors unless specific cycles exist.

Denominator conjecture verified for cluster algebras of type A, B, and C.

Abstract

It is proved that a multiset of permissible arcs over a tiling is uniquely determined by its intersection vector under a mild condition. This generalizes a classical result over marked surfaces with triangulations. We apply this result to study $\tau$-tilting theory of gentle algebras and denominator conjecture in cluster algebras. In the case of gentle algebras, it is proved that different $\tau$-rigid $A$-modules over a gentle algebra $A$ have different dimension vectors if and only if $A$ has no even oriented cycle with full relations. For cluster algebras, the denominator conjecture has been established for cluster algebras of type $\mathbb{A}\mathbb{B}\mathbb{C}$.