On the Combinatorics of Gentle Algebras

Abstract

For A a gentle algebra, and X and Y string modules, we construct a combinatorial basis for Hom(X, τ Y ). We use this to describe support τ -tilting modules for A. We give a combinatorial realization of maps in both directions realizing the bijection between support τ -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 [McC], showing that many but not all of them can be extended to general gentle algebras.

Oriented Flip Graphs, Noncrossing Tree Partitions, and Representation Theory of Tiling Algebras

“…REMARK 8.13. While this manuscript was under review, several authors have studied the combinatorial aspects of the representation theory of gentle algebras; for instance, see [4,2,31,32]. In [2,31], it is shown that all gentle algebras can be realized as tiling algebras associated with unpunctured surfaces.…”

On Extensions for Gentle Algebras

“…We note that a basis for extensions between string modules over gentle algebras is also given, by different techniques, in [6] building on the work in [19].…”

A Geometric Model for the Module Category of a Gentle Algebra

“…Gentle algebras are Gorenstein [24], closed under tilting and derived equivalence [38,39], and they are ubiquitous, for instance they occur in contexts such as Fukaya categories [27], dimer models [8], enveloping algebras of Lie algebras [29] and cluster theory. As such, there has been widespread interest in this class of algebras; see for example [10,14,15,37] for recent developments in this area.…”

“…10. A tiling algebra A P is of finite representation type if and only if every permissible simple closed curve c is such that |I P (c)| 1.…”