Tubes Containing String Modules in Symmetric Special Multiserial Algebras

Abstract

We provide a method for determining the existence and ranks of tubes in the stable Auslander-Reiten quiver of symmetric special multiserial algebras using only the information from the underlying Brauer configuration. Essentially, we generalise the notion of a Green walk around a Brauer graph to the notion of a Green `hyperwalk' around a Brauer configuration, and show that these walks determine the number and rank of some of the stable tubes of the corresponding algebra. This description includes both tame and wild symmetric special multiserial algebras. We also provide a description of additional rank two tubes in both tame and wild algebras that do not arise from Green hyperwalks, but which nevertheless contain string modules at the mouth.Comment: 45 page