Euler characteristics of varieties of composition series

Jan Schröer

This is joint work with Christof Geiss (UNAM) and Bernard Leclerc (Caen).

Given a finite-dimensional module M over an algebra A, one can study the projective variety of composition series of M such that the simple subfactors of the composition series occur in a given order.

So to each composition series type, there is a map which associates to a module the Euler characteristic of the corresponding variety of composition series. These maps generate (as a vector space) the complex Ringel-Hall algebra C(A) associated to A.

Our aim is to obtain a better understanding of such Euler characteristics. We focus on some subcategories of modules over preprojective algebras. Especially for many rigid modules we obtain algorithms which compute the Euler characteristics explicitly. Via a generalization of Berenstein and Zelevinsky's chamber ansatz, this leads to a new geometric description of cluster variables for many cluster algebras (e.g. acyclic cluster algebras).