On the Combinatorics of Gentle Algebras

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 unite torsion classes. We give an explicit basis of Ext(1)(Y, X) as short exact sequences. We analyze several constructions given in amore restricted, combinatorial setting by McConville, showing that many but not all of them can be extended to general gentle algebras.