Extensions between modules defined by lattice paths in the preprojective algebra

 

In 2001, Fomin and Zelevinsky introduced cluster algebras which appear as coordinate rings of many varieties. We study cluster algebras coming from Richardson varieties. Leclerc gives a cluster structure on Richardson varieties using the representation theory of preprojective algebras. While this construction is very algebraic, we take a more combinatorial approach.

 

The main goal is to find a combinatorial description for when certain cluster variables are compatible, or equivalently when modules defined by lattice paths in the preprojective algebra have trivial extensions. We extend the known results from Geiss, Leclerc, and Schröer that answer this question in the case of the Grassmannian by using various methods. We introduce the notion of extending a module, describe how add or remove operators applied to pairs of modules affects the extension space between them and provide homological and combinatorial conditions that determine when two arbitrary modules have trivial extension.