On the combinatorics of gentle algebras

TL;DR

This paper explores the combinatorial structures of gentle algebras, providing bases for Hom and Ext spaces, and describing support τ-tilting modules through combinatorial realizations, extending previous restricted models.

Contribution

It introduces a combinatorial basis for Hom and Ext spaces and characterizes support τ-tilting modules for gentle algebras, extending McConville's constructions.

Findings

Constructed a combinatorial basis for Hom(X, τY).

Described support τ-tilting modules combinatorially.

Extended some McConville's constructions to general gentle algebras.

Abstract

For $A$ a gentle algebra, and $X$ and $Y$ string modules, we construct a combinatorial basis for Hom($X,\tau Y$). We use this to describe support $\tau$-tilting modules for $A$. We give a combinatorial realization of maps in both directions realizing the bijection between support $\tau$-tilting modules and functorially finite torsion classes. We give an explicit basis of Ext$^1(Y,X)$ as short exact sequences. We analyze several constructions given in a more restricted, combinatorial setting by McConville, showing that many but not all of them can be extended to general gentle algebras.