Intersection vectors over skew-tilings

TL;DR

This paper establishes that intersection vectors uniquely determine tagged permissible arcs over skew-tilings under certain conditions, and applies this to distinguish τ-rigid modules over skew-gentle algebras based on their dimension vectors.

Contribution

It generalizes previous results by proving a uniqueness property for intersection vectors over skew-tilings and applies this to classify τ-rigid modules over skew-gentle algebras.

Findings

Intersection vectors uniquely determine tagged permissible arcs under mild conditions.

Different τ-rigid modules have distinct dimension vectors if the associated quiver has no even-length oriented cycles with full zero relations.

Generalizes Fu-Geng's result from gentle to skew-gentle algebras.

Abstract

We prove that under a mild condition, a multiset of tagged permissible arcs over a skew-tiling is uniquely determined by its intersection vector. As an application, it is proved that -- up to isomorphism -- different $\tau$-rigid modules over a skew-gentle algebra $A$ arising from a skew-triple $(Q,Sp,I)$ have different dimension vectors if and only if $(Q,I)$ has no minimal oriented cycle of even-length with full zero relations. This generalizes a recent work of Fu-Geng for gentle algebras.