A type A path algebra is an algebra whose basis is the set of all paths in an orientation of a type A Dynkin diagram. We introduce a new class of modules over a type A path algebra and call them maximal almost rigid (MAR). They are counted by the Catalan numbers and are naturally modeled by triangulations of a polygon. The endomorphism algebras of the MAR modules are classical tilted algebras of type A. Furthermore, their oriented flip graph is the oriented exchange graph of a smaller type A cluster algebra which is known to define a Tamari or Cambrian poset.

The type A path algebras are special cases of gentle algebras, a family of finite-dimensional algebras whose indecomposable modules are classified by certain walks called strings and bands. We generalize the notion of MAR to this setting. First, we use the surface models studied by Opper, Plamondon, and Schroll and by Baur and Coelho Simões to show that the MAR modules correspond bijectively to triangulations of a marked surface. We then show that the endomorphism algebra of a MAR module is the endomorphism algebra of a tilting module over a bigger gentle algebra. Finally, we define an oriented flip graph of the MAR modules and conjecture that it is acyclic.

This talk is based on joint projects with Emily Barnard, Raquel Coelho Simões, Emily Meehan, and Ralf Schiffler.