Locations of modules for Brauer tree algebras

Suppose Λ is a Brauer tree algebra. We determine the location of a Λ-module M in the stable Auslander–Reiten quiver of Λ from the description of M as a multi-pushout of elementary modules. This is done by introducing a new combinatorial object associated to M, which is a certain walk around the Brauer tree of Λ. These walks have applications to determining universal deformation rings and stable homomorphism groups.